ar
X
iv
:m
at
h/
92
07
21
2v
1
[m
ath
.A
P]
1
Ju
l 1
99
2
APPEARED IN BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 27, Number 1, July 1992, Pages 1-67
USER’S GUIDE TO VISCOSITY SOLUTIONS
OF SECOND ORDER
PARTIAL DIFFERENTIAL EQUATIONS
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions
Abstract. The notion of viscosity solutions of scalar fully nonlinear partial differ-
ential equations of second order provides a framework in which startling comparison
and uniqueness theorems, existence theorems, and theorems about continuous de-
pendence may now be proved by very efficient and striking arguments. The range of
important applications of these results is enormous. This article is a self-contained
exposition of the basic theory of viscosity solutions.
Introduction
The theory of viscosity solutions applies to certain partial differential equations
of the form F (x, u,Du,D2u) = 0 where F : RN ×R×RN ×S(N) → R and S(N) is
the set of symmetric N ×N matrices. The primary virtues of this theory are that
it allows merely continuous functions to be solutions of fully nonlinear equations
of second order, that it provides very general existence and uniqueness theorems
and that it yields precise formulations of general boundary conditions. Moreover,
these features go hand-in-hand with a great flexibility in passing to limits in various
settings and relatively simple proofs. In the expression F (x, u,Du,D2u) u will be
a real-valued function defined on some subset O of RN , Du corresponds to the
gradient of u and D2u corresponds to the matrix of second derivatives of u. How-
ever, as explained below, Du and D2u will not have classical meanings and, in fact,
the examples exhibited in §1 will convince the reader that the theory encompasses
classes of equations that have no solutions that are differentiable in the classical
sense.
1991 Mathematics Subject Classification. Primary 35D05, 35B50, 35J60, 35K55; Secondary
35B05, 35B25, 35F20, 35J25, 35J70, 35K20, 35K15, 35K65.
Key words and phrases. Viscosity solutions, partial differential equations, fully nonlinear equa-
tions, elliptic equations, parabolic equations, Hamilton-Jacobi equations, dynamic programming,
nonlinear boundary value problems, generalized solutions, maximum principles, comparison the-
orems, Perron’s method.
First author supported in part by the Army Research Office DAAL03-87-K-0043 and 03-90-
G-0102, National Science Foundation DMS-8505531 and 90-02331, and Office of Naval Research
N00014-88-K-0134
Received by the editors November 16, 1990
This paper was given as a Progress in Mathematics Lecture at the August 8–11, 1990 meeting
of the American Mathematical Society in Columbus, Ohio
c©1992 American Mathematical Society
0273-0979/92 $1.00 + $.25 per page
1

2 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
In order that the theory apply to a given equation F = 0, we will require F to
satisfy a fundamental monotonicity condition
(0.1) F (x, r, p,X) ≤ F (x, s, p, Y ) whenever r ≤ s and Y ≤ X ;
where r, s ∈ R, x, p ∈ RN , X,Y ∈ S(N) and S(N) is equipped with its usual order.
Regarding (0.1) as made up of the two conditions
(0.2) F (x, r, p,X) ≤ F (x, s, p,X) whenever r ≤ s,
and
(0.3) F (x, r, p,X) ≤ F (x, r, p, Y ) whenever Y ≤ X,
we will give the name “degenerate ellipticity” to the second. That is, F is said
to be degenerate elliptic if (0.3) holds. When (0.2) also holds (equivalently, (0.1)
holds), we will say that F is proper.
The examples given in §1 will illustrate the fact that the antimonotonicity in
X is indeed an “ellipticity” condition. The possibility of “degeneracies” is clearly
exhibited by considering the case in which F (x, r, p,X) does not depend on X—
it is then degenerate elliptic. The monotonicity in r, while easier to understand,
is a slightly subtle selection criterion that, in particular, excludes the use of the
viscosity theory for first order equations of the form b(u)ux = f(x) in R when b is
not a constant function, since then F (x, r, p) = b(r)p − f(x) is not nondecreasing
in r for all choices of p (scalar conservation laws are outside of the scope of this
theory).
The presentation begins with §1, which, as already mentioned, provides a list of
examples. This rather long list is offered to meet several objectives. First, we seek
to bring the reader to our conviction that the scope of the theory is quite broad while
providing a spectrum of meaningful applications and, at the same time, generating
some insight as regards the fundamental structural assumption (0.1). Finally, in
the presentation of examples involving famous second order equations, the very
act of writing the equations in a form compatible with the theory will induce an
interesting modification of the classical viewpoint concerning them.
In §2 we begin an introductory presentation of the basic facts of the theory. The
style is initially leisurely and expository and technicalities are minimized, although
complete discussions of various key points are given and some simple arguments in-
conveniently scattered in the literature are presented. Results are illustrated with
simple examples making clear their general nature. Section 2 presents the basic no-
tions of solution used in the theory, the analytical heart of which lies in comparison
results. Accordingly, §3 is devoted to explaining comparison results in the simple
setting of the Dirichlet problem; roughly speaking, they are proved by methods
involving extensions of the maximum principle to semicontinuous functions. Once
these comparison results are established, existence assertions can be established by
Perron’s method, a rather striking tale that is told in §4. With this background
in hand, the reader will have an almost complete (sub)story and with some effort
(but not too much!) should be able to absorb in an efficient way some of the more
technical features of the theory that are outlined in the rest of the paper.
Other important ideas are to be found in §6, which is concerned with the is-
sue of taking limits of viscosity solutions and applications of this and in §7, which

USER’S GUIDE TO VISCOSITY SOLUTIONS 3
describes the adaptation of the theory to accommodate problems with other bound-
ary conditions and problems in which the boundary condition cannot be strictly
satisfied. In the later case, the entire problem has a generalized interpretation for
which there is often existence and uniqueness. While the description of these re-
sults is deferred to §7, they are fundamental and dramatic. For example, if G(p,X)
is uniformly continuous, degenerate elliptic, and independent of x and Ω ⊂ RN is
open and bounded, n(x) is the unit exterior normal at the point x in its smooth
boundary ∂Ω and f ∈ C(Ω), then the Neumann problem
u+G(Du,D2u)− f(x) = 0 in Ω, un = 0 on ∂Ω
has a unique properly interpreted solution (which may not satisfy un = 0 on ∂Ω).
Sections 5, 8, 9 discuss variations of the basic material and need not be read in
sequence. Section 10 is devoted to a commentary about applications (which are not
treated in the main text), references, and possible lines of future development of the
subject. We conclude with an appendix where the reader will find a self-contained
presentation of the proof of the analytical heart on the presentation we have chosen.
References are not given in the main text (with the exception of §10), but are
to be found at the end of each section. In particular, the reader should look to the
end of a section for further comments, references that contain details ommitted in
the main text, and technical generality, historical comments, etc. The references,
while numerous, are not intended to be complete, except that we have sought
to represent all the major directions of research and areas of application. There
are original aspects of the current presentation and the reader will note differences
between the flavor and clarity of our presentation and that of many of the citations.
However, equipped with the view presented here, we hope and expect that perusing
the amazing literature that has so quickly matured will be a much more rewarding
and efficient endeavor.
We are grateful to R. Dorroh, M. Kocan and A. Swiech for their kind help in
reducing the number of errors herein.
Contents
1. Examples
2. The notion of viscosity solutions
3. The maximum principle for semicontinuous functions and comparison
for the Dirichlet problem
4. Perron’s method and existence
5. Comparison: Variations on the theme
5.A. Comparison with more regularity
5.B. Estimates from comparison
5.C. Comparison with strict inequalities and without coercivity in u
5.D. Comparison and existence of unbounded solutions on unbounded
domains
6. Limits of viscosity solutions and an application
7. General and generalized boundary conditions
7.A. Boundary conditions in the viscosity sense
7.B. Existence and uniqueness for the Neumann problem
7.C. The generalized Dirichlet problem
7.C′. The state constraints problem

4 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
7.D. A remark on (BC) in the classical sense
7.E. Fully nonlinear boundary conditions
8. Parabolic problems
9. Singular equations: An example from geometry
10. Applications and perspectives
APPENDIX The proof of Theorem 3.2
1. Examples
We will record here many examples of degenerate elliptic equations mentioning,
when appropriate, areas in which they arise. The reader is invited to scan the list
and pause where interested—it is possible to proceed to §2 at any stage. Below we
will say either that a function F (x, r, p,X) is degenerate elliptic or that the equation
F (x, u,Du,D2u) = 0 is degenerate elliptic or that the “operator” or expression
F (x, u,Du,D2u) is degenerate elliptic and always mean the same thing, i.e., (0.3)
holds; the term “proper” is used in a similar fashion.
Example 1.1. Laplace’s equation. We revisit an old friend, the equation
(1.1) −∆u+ c(x)u = f(x)
—note the sign in front of the Laplacian. The corresponding F is naturally given
by F (x, r, p,X) = − trace(X) + c(x)r − f(x), which is proper if c ≥ 0.
Example 1.2. Degenerate elliptic linear equations. Example 1.1 immediately ex-
tends to the more general linear equation
(1.2) −
N∑
i,j=1
ai,j(x)
∂2u
∂xi∂xj
+
N∑
i=1
bi(x)
∂u
∂xi
+ c(x)u(x) = f(x)
where the matrix A(x) = {ai,j(x)} is symmetric; the corresponding F is
(1.3) F (x, r, p,X) = − trace(A(x)X) +
N∑
i=1
bi(x)pi + c(x)r − f(x).
In this case, F is degenerate elliptic if and only if A(x) ≥ 0 and it is proper if also
c(x) ≥ 0. In the event that there are constants λ,Λ > 0 such that λI ≤ A(x) ≤ ΛI
for all x where I is the identity matrix, F is said to be uniformly elliptic.
Of course, the linear equation in divergence form
−
N∑
i,j=1
∂
∂xi
(
ai,j(x)
∂u
∂xj
)
+
N∑
j=1
bj(x)
∂u
∂xi
+ c(x)u(x) = f(x)
can be written as above with
F (x, r, p,X) = − trace(A(x)X) +
N∑
j=1
(
bj(x) −
N∑
i=1
∂ai,j
∂xi
(x)
)
pj + c(x)r − f(x)
provided that the indicated derivatives of the ai,j exist.
We leave the interesting class of linear equations to turn to the totally degenerate
case of first order equations.

USER’S GUIDE TO VISCOSITY SOLUTIONS 5
Example 1.3. First order equations. The main point is that a first order oper-
ator F (x, u,Du) is always degenerate elliptic and thus it is proper if and only if
F (x, r, p) is nondecreasing in r ∈ R. Proper equations of the form F (x, u,Du) = 0
play a fundamental role in the classical Calculus of Variations and in Optimal Con-
trol Theory of ordinary differential equations; in this context they are often called
Bellman or Hamilton-Jacobi equations and then F (x, r, p) is convex in (r, p). These
equations, in the full generality of nearly arbitrary proper functions F , are also
crucial in Differential Games Theory where they are known as Isaacs’s equations.
Example 1.4. Quasilinear elliptic equations in divergence form. The usual notion
of ellipticity for equations of the form
(1.4) −
N∑
i=1
∂
∂xi
(ai(x,Du)) + b(x, u,Du) = 0
is the monotonicity of the vector field a(x, p) in p as a mapping from RN to RN . If
enough regularity is available to carry out the differentiation, we write (1.4) as
(1.5) −
N∑
i,j=1
∂ai
∂pj
(x,Du) ∂
2u
∂xi∂xj
+ b(x, u,Du)−
N∑
i=1
∂ai
∂xi
(x,Du) = 0
and correspondingly set
(1.6) F (x, r, p,X) = − trace((Dpa(x, p))X) + b(x, r, p)−
N∑
i=1
∂ai
∂xi
(x, p).
The monotonicity of a in p is precisely the condition that guarantees that F is
degenerate elliptic, and then it is proper provided we ask that b be nondecreasing
in r.
Two well-known instances are provided by the equation of minimal surfaces in
nonparametric form and the “m-Laplace’s” equation that are given, respectively,
by a(x, p) = (1 + |p|2)−1/2p, b = b(x, u) and a(x, p) = |p|m−2p, b = b(x, r) where
m ∈ (1,∞). Computations show that the corresponding F ’s are, respectively,
F (x, r, p,X) = −(1 + |p|2)− 12 trace(X) + (1 + |p|2)− 32 trace((p⊗ p)X) + b(x, r)
and
F (x, r, p,X) = −|p|m−2 trace(X)− (m− 2)|p|m−4 trace((p⊗ p)X) + b(x, r).
Example 1.5. Quasilinear elliptic equations in nondivergence form. The equation
(1.7) −
N∑
i,j=1
ai,j(x, p)
∂2u
∂xi∂xj
+ b(x, u,Du) = 0,
where A(x, p) = {ai,j(x, p)} ∈ S(N), contains all of the above as special cases and
corresponds to
F (x, r, p,X) = − trace(A(x, p)X) + b(x, r, p),

6 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
which is proper if A ≥ 0 and b is nondecreasing with respect to r. Two relevant
special cases are
−ν∆u+ f(x, u,Du) = 0
with ν > 0 and f nondecreasing in u, which may be regarded as a first-order
Hamilton-Jacobi equation perturbed by an additional “viscosity” term−ν∆u (equa-
tions of this type arise in optimal stochastic control), and the Le´vi’s equation
−
(∂2u
∂x21
+
∂2u
∂x22
)(
1 +
( ∂u
∂x3
)2)
− ∂
2u
∂x23
(( ∂u
∂x1
)2
+
( ∂u
∂x2
)2)
+ 2
∂2u
∂x1∂x3
( ∂u
∂x3
∂u
∂x1
− ∂u∂x2
)
+ 2
∂2u
∂x2∂x3
( ∂u
∂x3
∂u
∂x2
+
∂u
∂x1
)
= 0,
which is the nonparametric formulation for a hypersurface in C2 with vanishing
Le´vi’s form. Note that in this example F = − trace(A(p)X) where
A(p) =


1 + p23 0 p3p1 − p2
0 1 + p23 p3p2 + p1
p3p1 − p2 p3p2 + p1 p21 + p22


so that A ≥ 0 but det(A(p)) = 0 for all p.
Example 1.6. Hamilton-Jacobi-Bellman and Isaacs equations. Hamilton-Jacobi-
Bellman and Isaacs equations are, respectively, the fundamental partial differential
equations for stochastic control and stochastic differential games. The natural
setting involves a collection of elliptic operators of second-order depending either
on one parameter α (in the Hamilton-Jacobi-Bellman case) or two parameters α, β
(in the case of Isaacs’s equations). These parameters lie in index sets we will not
display in the discussion. Thus we take as ingredients proper expressions of the
form
(1.8) Lαu = −
N∑
i,j=1
aαi,j(x)
∂2u
∂xi∂xj
+
N∑
i=1
bαi (x)
∂u
∂xi
+ cα(x)u(x) − f α(x)
or
(1.9) Lα,βu = −
N∑
i,j=1
aα,βi,j (x)
∂2u
∂xi∂xj
+
N∑
i=1
bα,βi (x)
∂u
∂xi
+ cα,β(x)u(x) − f α,β(x)
where all the coefficients are bounded with respect to α or α, β. Hamilton-Jacobi-
Bellman equations include those of the form
(1.10) sup
α
{Lαu} = 0
while
(1.11) sup
α
inf
β
{
Lα,βu
}
= 0

USER’S GUIDE TO VISCOSITY SOLUTIONS 7
is a typical Isaacs’s equation. The corresponding nonlinearities F have the form
F (x, r, p,X) = sup
α
[− trace(Aα(x)X) + 〈bα(x), p〉+ cα(x)r − f α(x)]
and
F (x, r, p,X) = sup
α
inf
β
[− trace(Aα,β(x)X)
+ 〈bα,β(x), p〉 + cα,β(x)r − f α,β(x)],
each of which is clearly also proper. Notice that in the first case F is convex in
(r, p,X) while in the second case this is not so; indeed, if one allows for “unbounded
envelopes” (i.e., coefficients that are unbounded in the parameters), one can show
that essentially any proper F satisfying minor regularity assumptions can be rep-
resented as a “sup inf” of linear expressions as above.
Indeed, the above process is quite general and does not require linear ingredients.
Suppose Fα,β is proper for each α, β. Then F (x, r, p,X) = supα infβ Fα,β(x, r, p,X)
and F (x, r, p,X) = infα supβ Fα,β(x, r, p,X) are also evidently proper (for the mo-
ment, we set aside considerations of finiteness and continuity).
Example 1.7. Obstacle and gradient constraint problems. A special case of the
last remarks above is met in the consideration of “obstacle problems.” Very general
forms of such problems may be written
(1.12) max{F (x, u,Du,D2u), u− f(x)} = 0
or
(1.13) min{F (x, u,Du,D2u), u− f(x)} = 0
or even
(1.14) max{min{F (x, u,Du,D2u), u− f(x)}, u− g(x)} = 0.
In accordance with remarks made in the previous example, if F is proper then so
are (1.12)–(1.14).
Likewise, “gradient constraints” may be imposed in this way. A typical example
corresponds to max{F (x, u,Du,D2u), |Du| − g(x)} = 0.
Example 1.8. Functions of the eigenvalues. ForX ∈ S(N) we let λ1(X), . . . , λN (X)
be its eigenvalues arranged in increasing order, λi(X) ≤ λi+1(X). If g(x, r, p, s1, . . . , sN )
is defined on R3N+1 and is nondecreasing in r and each si, then F (x, r, p,X) =
g(x, r, p,−λ1(X), . . . ,−λN (X)) is proper. For instance, F (X) = −max{λ1(X), . . . , λN (X)} =
−λN (X), F (X) = −min{λ1(X), . . . , λN (X)} = −λ1(X) and F (X) = −(λ2(X))3
are degenerate elliptic. Another example is
F (x, r, p,X) = −| trace(X)|m−1 trace(X) + |p|q + c(x)r − f(x)
where c ≥ 0 and m, q > 0. The corresponding equation is
−|∆u|m−1∆u+ |Du|q + c(x)u = f(x),
which provides another example of the generality we are dealing with, even if we
have no interpretation of this equation in mind.

8 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
Example 1.9. Sums and increasing functions of proper functions. If Fi is proper
for i = 1, . . . ,M , then so is F1 + · · · + FM . More generally, if g(s1, . . . , sM ) is
nondecreasing in each variable, then g(F1, . . . , FM ) is proper. One may build very
complex examples using the cases discussed above and these remarks.
Example 1.10. Parabolic problems. We just observe that if (x, r, p,X) → F (t, x, r,
p,X) is proper for fixed t ∈ [0, T ], then so is the associated “parabolic” problem
(1.15) ut + F (t, x, u,Du,D2u) = 0
when considered as an equation in the N + 1 independent variables (t, x). We
mention only one example (there are, of course, infinitely many) that has some
geometrical interest since it describes the evolution of a surface (given by a level set
of the initial condition) with a motion along its normal with a speed proportional
to the mean curvature
(1.16) ut − |Du| div
( Du
|Du|
)
= 0.
Carrying out the differentiations yields
(1.17) ut −∆u+
N∑
i,j=1
∂2u
∂xi∂xj
∂u
∂xi
∂u
∂xj
|Du|−2 = 0.
This may be written in the form (1.15) with
F (x, p,X) = − trace
((
I − p⊗ p|p|2
)
X
)
.
Example 1.11. Monge Ampe`re equations. The Monge-Ampe`re equation may be
written as
(1.18) u is convex, det(D2u) = f(x, u,Du)
where f(x, r, p) ≥ 0. We are dealing here with the real Monge-Ampe`re equation,
but everything that will be said adapts to the complex case and to other curvature
equations. Allowing F to be discontinuous (even more, to become infinite), we may
write (1.18) in our form by putting
F (x, r, p,X) =
{ − det(X) + f(x, r, p) if X ≥ 0,
+∞ otherwise;
F is then degenerate elliptic. This follows from the fact that
g(s1, · · · , sN ) =
{ ∏N
i=1 si if si ≥ 0, i = 1, . . . , N ,
−∞ otherwise,
is nondecreasing in each of its arguments and Example 1.8.

USER’S GUIDE TO VISCOSITY SOLUTIONS 9
Example 1.12. Uniformly elliptic functions. This “example” is really a definition.
One says that F (x, r, p,X) is uniformly elliptic if there are constants λ,Λ > 0 for
which
λ trace(P ) ≤ F (x, r, p,X − P )− F (x, r, p,X) ≤ Λ trace(P ) for P ≥ 0
and all x, r, p,X and then calls the constants λ, Λ ellipticity constants. We note
that L in Example 1.2 is uniformly elliptic with constants λ, Λ exactly when λ ≤
λ1(A(x)) and λN (A(x)) ≤ Λ. One notes that sums of uniformly elliptic functions
are again uniformly elliptic and that the sup inf process over a family of uniformly
elliptic functions with common ellipticity constants produces another such function.
Notes on §1. We will disappoint the reader in the following sections by not ap-
plying the theory developed therein to the many examples given above. The goal of
this section was to exhibit clearly the breadth and importance of the class of proper
equations. We simply do not have enough space here to develop applications of the
theory of these equations beyond that which follows immediately from the general
results presented.
Most of the examples listed have been considered via classical approaches. We
give some references containing classical presentations: D. Gilbarg and N. S. Trudinger
[81] is a basic source concerning linear and quasi-linear uniformly elliptic equations;
O. A. Ole˘ınik and E. V. Radkevic [139], J. J. Kohn and L. Nirenberg [106], and A. V.
Ivanov [98] treat degenerate elliptic equations; W. H. Fleming and R. Rishel [75], P.
L. Lions [117], and N. V. Krylov [108, 109] are sources for Hamilton-Jacobi-Bellman
equations; S. Benton [36] and P. L. Lions [116] discuss first-order Hamilton-Jacobi
equations. Most of these references present some of the ways these equations arise.
2. The notion of viscosity solutions
It is always assumed that F satisfies (0.1) (i.e., F is proper) and, unless otherwise
said, is continuous. To motivate the notions, we begin by supposing that u is C2
(i.e., twice continuously differentiable) on RN and
F (x, u(x), Du(x), D2u(x)) ≤ 0
holds for all x (that is, u is a classical subsolution of F = 0 or, equivalently, a
classical solution of F ≤ 0 in RN ). Suppose that ϕ is also C2 and xˆ is a local
maximum of u− ϕ. Then calculus implies Du(xˆ) = Dϕ(xˆ) and D2u(xˆ) ≤ D2ϕ(xˆ)
and so, by degenerate ellipticity,
(2.1) F (xˆ, u(xˆ), Dϕ(xˆ), D2ϕ(xˆ)) ≤ F (xˆ, u(xˆ), Du(xˆ), D2u(xˆ)) ≤ 0.
The extremes of this inequality do not depend on the derivatives of u and so we
may consider defining an arbitrary function u to be (some kind of generalized)
subsolution of F = 0 if
(2.2) F (xˆ, u(xˆ), Dϕ(xˆ), D2ϕ(xˆ)) ≤ 0
whenever ϕ is C2 and xˆ is a local maximum of u − ϕ. Before making any formal
definitions, let us also note that u(x) ≤ u(xˆ) − ϕ(xˆ) + ϕ(x) for x near xˆ, ϕ ∈ C2
and Taylor approximation imply
(2.3) u(x) ≤ u(xˆ) + 〈p, x− xˆ〉+ 12 〈X(x− xˆ), x− xˆ〉+ o(|x − xˆ|2) as x→ xˆ

10 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
where p = Dϕ(xˆ) and X = D2ϕ(xˆ). Moreover, if (2.3) holds for some (p,X) ∈
RN × S(N) and u is twice differentiable at xˆ, then p = Du(xˆ) and D2u(xˆ) ≤ X .
Thus if u is a classical solution of F ≤ 0 it follows that F (xˆ, u(xˆ), p,X) ≤ 0 whenever
(2.3) holds; we may also consider basing a definition of nondifferentiable solutions
u of F ≤ 0 on this fact. Roughly speaking, pursuing (2.2) leads to notions based
upon test functions ϕ but does not immediately lead us, as will pursuing (2.3), to
define “(Du,D2u)” for nondifferentiable functions u, which will turn out to be a
good idea. For this reason, we begin by developing the line suggested by (2.3). Next
we introduce a set O ⊂ RN on which F ≤ 0 is to hold and the appropriate notation
to deal with inequalities like (2.3) “on O.” At the moment, O is arbitrary; later we
require it to be locally compact. Taking off from (2.3), if u : O → R, xˆ ∈ O, and
(2.3) holds as O ∋ x → xˆ, we say (p,X) ∈ J2,+O u(xˆ) (the second-order “superjet”
of u at xˆ). This defines a mapping J2,+O u from O to the subsets of RN × S(N).
Example 2.1. By way of illustration, if u is defined on R by
u(x) =
{
0 for x ≤ 0,
ax+ b2x2 for x ≥ 0,
then J2,+[−1,0]u(0) = ((−∞, 0)× R) ∪ ({0} × [0,∞)), while
J2,+R u(0) =



∅ if a > 0,
{0} × [max{0, b},∞) if a = 0,
((a, 0)× R) ∪ ({0} × [0,∞)) ∪ ({a} × [b,∞)) if a < 0.
Having thought through this example, the reader will see that J2,+O u(x) depends
on O but realize it is the same for all sets O for which x is an interior point;
we let J2,+u(x) denote this common value. If we repeat the above discussion after
switching the inequality sign in (2.3), we arrive at the definitions of the second-order
“subjets” J2,−O u, J2,−u; equivalently, J
2,−
O u(x) = −J2,+O (−u)(x), etc.
We are ready to define the notions of viscosity subsolutions, supersolutions, and
solutions. It will be useful to have the notations
USC(O) = { upper semicontinuous functions u : O → R},
LSC(O) = { lower semicontinuous functions u : O → R}.
Definition 2.2. Let F satisfy (0.1) and O ⊂ RN . A viscosity subsolution of F = 0
(equivalently, a viscosity solution of F ≤ 0) on O is a function u ∈ USC(O) such
that
(2.4) F (x, u(x), p,X) ≤ 0 for all x ∈ O and (p,X) ∈ J2,+O u(x).
Similarly, a viscosity supersolution of F = 0 on O is a function u ∈ LSC(O) such
that
(2.5) F (x, u(x), p,X) ≥ 0 for all x ∈ O and (p,X) ∈ J2,−O u(x).
Finally, u is a viscosity solution of F = 0 in O if it is both a viscosity subsolution
and a viscosity supersolution of F = 0 in O.

USER’S GUIDE TO VISCOSITY SOLUTIONS 11
Remarks 2.3. Since these “viscosity notions” are the primary ones for the current
discussion, we immediately agree (at least, we hope you agree) to drop the term
“viscosity” and hereafter simply refer to subsolutions, supersolutions, and solutions.
This is a happy idea, as the term “viscosity,” which lacks elegance, is an artifact of
the origin of this theory in the study of first-order equations and the name was then
motivated by the consistency of the notion with the method of “vanishing viscosity,”
which is irrelevant for many second-order equations. It follows from the discussion
preceding the definition that, for example, if u is a solution of F ≤ 0 in O, ϕ is C2 in
a neighborhood of O, and u−ϕ has a local maximum (relative to O) at xˆ ∈ O, then
(2.2) holds. Analogous remarks hold for supersolutions. These remarks motivate
the requirement that a subsolution be upper semicontinuous, etc., in the sense that
producing maxima of upper semicontinous functions is straightforward. Solutions,
being both upper semicontinuous and lower semicontinuous, are continuous. One
might ask if the validity of (2.2) for all ϕ ∈ C2 (with the maxima relative to O) for
an upper semicontinuous function u is equivalent to u being a subsolution. This is
so. In fact, if xˆ ∈ O then
J2,+O u(xˆ) = {(Dϕ(xˆ), D2ϕ(xˆ)) : ϕ is C2 and u− ϕ has a local maximum at xˆ};
we leave the proof as an interesting exercise.
We next record the definitions of the closures of the set-valued mappings needed
in the next section.
With the above notation, for x ∈ O, we set
(2.6)
J2,+O u(x) = {(p,X) ∈ RN × S(N) : ∃(xn, pn, Xn) ∈ O × RN × S(N) ∋
(pn, Xn) ∈ J2,+O u(xn) and (xn, u(xn), pn, Xn)→(x, u(x), p,X)}
and
(2.7)
J2,−O u(x) = {(p,X) ∈ RN × S(N) : ∃(xn, pn, Xn) ∈ O × RN × S(N) ∋
(pn, Xn) ∈ J2,−O u(xn) and (xn, u(xn), pn, Xn)→(x, u(x), p,X)};
we are abusing standard practice as regards defining closures of set-valued mappings
a bit in that we put the extra condition u(xn) → u(x) in the definitions while the
graphs of the multifunctions J2,+O u, J
2,−
O u do not themselves record the values of
u. The reader may note the use of expressions like “xn → x” as an abbreviation
for “the sequence xn satisfies limn→∞ xn = x,” etc. If x ∈ interior(O), we define
J2,+u(x), J2,−u(x) in the obvious way.
Remark 2.4. If u is a solution of F ≤ 0 on O, then F (x, u(x), p,X) ≤ 0 for
x ∈ O and (p,X) ∈ J2,+O u(x). This remains true, for reasons of continuity, if
(p,X) ∈ J2,+O u(x) and F is continuous (or even lower semicontinuous). Similar
remarks apply to supersolutions and solutions.
Advice. We advise the reader to either skip the following material or to scan it
lightly at the present time and proceed directly to the next section. The comments
collected below can be referred to as needed.
Remark 2.5. While the definitions above may seem reasonable, they contain sub-
tleties. In particular, they do not define “operators” on domains in a familiar way.

USER’S GUIDE TO VISCOSITY SOLUTIONS 13
imply that (q + Dϕ(xˆ), Y + D2ϕ(xˆ)) ∈ J2,+O u(xˆ) and so J2,+O (u − ϕ)(xˆ) ⊂ {(p −
Dϕ(xˆ), X −D2ϕ(x)) : (p,X) ∈ J2,+O u(x)}. The other inclusion follows from this as
well, since J2,+O u(xˆ) = J
2,+
O ((u − ϕ) + ϕ)(xˆ). It is also clear that one always has
J2,+O (u+ v)(x) ⊃ J2,+O u(x) + J2,+O v(x).
(iii) We consider J2,+O ϕ(xˆ) when ϕ ∈ C2(RN ) and will end up with a general
statement corresponding to Example 2.1. In view of (ii), we may as well assume
that ϕ ≡ 0, and we will write “Zero” for the zero function. We know that if
xˆ ∈ interior(O), then J2,+O Zero(xˆ) = J2,+Zero(xˆ) = {(0, X)} : X ≥ 0}. In general,
(p,X) ∈ J2,+O Zero(xˆ) if
(2.9) 0 ≤ 〈p, x− xˆ〉+ 12 〈X(x− xˆ), x− xˆ〉+ o(|x− xˆ|2) as O ∋ x→ xˆ.
Assuming that xn ∈ O, 0 < |xn − xˆ| → 0, and (xn − xˆ)/|xn − xˆ| → q, we may put
x = xn in (2.9), divide the result by |xn − xˆ| and pass to the limit to conclude that
(2.10) 0 ≤ 〈p, q〉 for q ∈ UTO(xˆ)
where UTO(xˆ) is the set of “generalized unit tangents” to O at xˆ; it is given by
(2.11) UTO(xˆ) =
{
q : ∃xn ∈ O\{xˆ}, xn → xˆ, and
xn − xˆ
|xn − xˆ|
→ q
}
.
If O is a smooth N -submanifold of RN with boundary and xˆ ∈ ∂O, then the
generalized tangent cone
TO(xˆ) = convex hull(UTO(xˆ))
is a halfspace and O has an exterior normal ~n at xˆ. In this event, (2.10) says that
p = −λ~n for some λ ≥ 0. Moreover, if p = 0, (2.9) then implies that 0 ≤ 〈Xq, q〉
for q ∈ UTO(xˆ), and we conclude that
(2.12) (0, X) ∈ J2,+O Zero(xˆ) if and only if 0 ≤ X
provided TO(xˆ) is a halfspace.
Life is more complex if p = −λ~n and λ > 0. In this case, (2.9) reads
(2.13) λ〈~n, x− xˆ〉 − 12 〈X(x− xˆ), x− xˆ〉 ≤ o(|x − xˆ|2) as O ∋ x→ xˆ.
We study (2.13) when xˆ = 0 and we can represent O near 0 in the form
(2.14) {(x˜, xN ) : xN ≤ g(x˜)}
where x˜ = (x1, . . . , xN−1), g(x˜) = 12 〈Zx˜, x˜〉+ o(|x˜|2)}, and Z ∈ S(N − 1). That is,
we assume that the boundary of O is twice differentiable at 0 and rotate so that the
normal is ~n = eN = (0, . . . , 0, 1). With these normalizations, we put (x˜, g(x˜)) into
(2.13) to find λ〈Zx˜, x˜〉 − 〈X(x˜, 0), (x˜, 0)〉 ≤ o(|x˜|2) or λZ˜ ≤ PN−1XPN−1 where
Z˜(x˜, xn) = (Zx˜, 0) and PN−1 is the projection on the first N − 1 coordinates. It is
not hard to see that this is also sufficient.

USER’S GUIDE TO VISCOSITY SOLUTIONS 15
The fact that the theory required only appropriate semicontinuity was realized
earlier on, but the first striking applications of the fact (to existence, stability,
optimal control, . . . ) were given in H. Ishii [86, 90], G. Barles and B. Perthame [25,
26].
Let us finally mention that some structural properties of viscosity solutions were
and will be ommitted from our presentation and, in addition to the above references,
some can be found in P. L. Lions [124], R. Jensen and P. E. Souganidis [105], and
H. Frankowska [78].
3. The maximum principle for semicontinuous functions
and comparison for the Dirichlet problem
Let Ω be a bounded open subset of RN , Ω be its closure, and ∂Ω be its boundary.
Suppose u ∈ USC(Ω) is a solution of F ≤ 0 in Ω, v ∈ LSC(Ω) is a solution of F ≥ 0,
and u ≤ v on ∂Ω. We seek to show that u ≤ v on Ω.
In the event u and v are classical sub and supersolutions, we could employ the
classical “maximum principle.” Let us recall some elementary facts in this regard.
Let w be defined in a neighborhood of xˆ ∈ RN . If there exists (p,X) ∈ RN ×S(N)
such that
(3.1) w(x) = w(xˆ) + 〈p, x− xˆ〉+ 12 〈X(x− xˆ), x− xˆ〉+ o(|x − xˆ|2) as x→ xˆ,
we say that w is twice differentiable at xˆ and Dw(xˆ) = p, D2w(xˆ) = X (it being
obvious that p and X are unique if they exist). It is clear that w is twice differ-
entiable at xˆ if and only if J2w(xˆ) ≡ J2,+w(xˆ) ∩ J2,−w(xˆ) is nonempty (in which
case J2w(xˆ) = {(Dw(xˆ), D2w(xˆ))}). Classical implementations of the maximum
principle are based on the fact, already used above, that if w is twice differentiable
at a local maximum xˆ, then Dw(xˆ) = 0 and D2w(xˆ) ≤ 0. Thus, if u and v are
twice differentiable everywhere and w = u − v has a local maximum xˆ ∈ Ω, we
would have Du(xˆ) = Dv(xˆ) and D2u(xˆ) ≤ D2v(xˆ) and then, in view of (0.1),
F (xˆ, u(xˆ), Du(xˆ), D2u(xˆ)) ≤ 0 ≤ F (xˆ, v(xˆ), Dv(xˆ), D2v(xˆ))
≤ F (xˆ, v(xˆ), Du(xˆ), D2u(xˆ)).
In the event that F (x, r, p,X) is strictly nondecreasing in r (a simple but illustrative
case), it follows that u− v is nonpositive at an interior maximum and so u ≤ v in
Ω since u ≤ v holds on ∂Ω.
We seek to extend this argument to the case u ∈ USC(Ω), v ∈ LSC(Ω). We
are unable to simply plug (Du(xˆ), D2u(xˆ)) and (Dv(xˆ), D2v(xˆ)) into F since these
expressions must be replaced by the set-valued functions J2,+u and J2,−v (and
their values may well be empty at many points, including maximum points of
u(x)−v(x)). To use J2,+u and J2,−v, we employ a device that doubles the number
of variables and then penalizes this doubling. More precisely, we maximize the
function u(x) − v(y) − (α/2)|x − y|2 over Ω × Ω; here α > 0 is a parameter. As
α → ∞, we closely approximate maximizing u(x) − v(x) over Ω. More precisely,
we have
Lemma 3.1. Let O be a subset of RN , u ∈ USC(O), v ∈ LSC(O) and
(3.2) Mα = sup
O×O
(u(x) − v(y)− α2 |x− y|2)

16 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
for α > 0. Let Mα <∞ for large α and (xα, yα) be such that
(3.3) lim
α→∞
(Mα − (u(xα)− v(yα)− α2 |xα − yα|2)) = 0.
Then the following holds:
(3.4)



(i) limα→∞ α|xα − yα|2 = 0 and
(ii) limα→∞Mα = u(xˆ)− v(xˆ) = supO(u(x) − v(x))
whenever xˆ ∈ O is a limit point of xα as α→ ∞.
Deferring the elementary proof of the lemma and returning to our sub and su-
persolutions u ∈ USC(Ω), v ∈ LSC(Ω) satisfying u ≤ v on ∂Ω, we note that
Mα = supΩ×Ω(u(x) − v(y) − (α/2)|x − y|2) is finite since u(x) − v(y) is upper
semicontinuous and Ω is compact. Since we seek to prove u ≤ v, we assume to the
contrary that u(z) > v(z) for some z ∈ Ω; it follows that
(3.5) Mα ≥ u(z)− v(z) = δ > 0 for α > 0.
Choosing (xα, yα) so that Mα = u(xα)− v(yα)− (α/2)|xα − yα|2 (the maximum is
achieved in view of upper semicontinuity and compactness), it follows from (3.4)(i),
(ii) and u ≤ v on ∂Ω that (xα, yα) ∈ Ω×Ω for α large. The next step is to use the
equations to estimate Mα and contradict (3.5) for large α. This requires producing
suitable values of J2,+u and J2,−v, and we turn to this question.
To know what to look for, let us proceed more generally and assume that u, v are
defined in neighborhoods of xˆ, yˆ ∈ RN and twice differentiable at xˆ, yˆ respectively.
Assume, moreover, that ϕ is C2 near (xˆ, yˆ) in RN×RN and (xˆ, yˆ) is a local maximum
of u(x)−v(y)−ϕ(x, y). Applying the classical maximum principle to this situation
(in the 2N variables (x, y)), we learn thatDu(xˆ) = Dxϕ(xˆ, yˆ), Dv(yˆ) = −Dyϕ(xˆ, yˆ),
and
(3.6)
(
X 0
0 −Y
)
≤ D2ϕ(xˆ, yˆ)
where X = D2u(xˆ), Y = D2v(yˆ). Note that with the choice ϕ(x, y) = (α/2)|x−y|2
the above reads
(3.7)
(
X 0
0 −Y
)
≤ α
(
I −I
−I I
)
where I will stand for the identity matrix in any dimension and, since the right-
hand side annihilates vectors of the form ( ξξ ) (also written
t(ξ, ξ) where tZ denotes
the transpose of a matrix Z), (3.7) implies X ≤ Y , making further contact with
the maximum principle.
It is a remarkable fact that perturbations of the above results may be obtained
in the class of semicontinuous functions. The main result we use in this direction
is the following theorem, which is formulated in a useful but distracting generality.
(See (2.6) and (2.7) regarding notation below.)

USER’S GUIDE TO VISCOSITY SOLUTIONS 17
Theorem 3.2. Let Oi be a locally compact subset of RNi for i = 1, . . . , k,
O = O1 × · · · × Ok,
ui ∈ USC(Oi), and ϕ be twice continuously differentiable in a neighborhood of O.
Set
w(x) = u1(x1) + · · ·+ uk(xk) for x = (x1, · · · , xk) ∈ O,
and suppose xˆ = (xˆ1, . . . , xˆk) ∈ O is a local maximum of w−ϕ relative to O. Then
for each ε > 0 there exists Xi ∈ S(Ni) such that
(Dxiϕ(xˆ), Xi) ∈ J
2,+
Oi ui(xˆi) for i = 1, . . . , k,
and the block diagonal matrix with entries Xi satisfies
(3.8) −
(
1
ε + ‖A‖
)
I ≤


X1 . . . 0
...
. . .
...
0 . . . Xk

 ≤ A+ εA2
where A = D2ϕ(xˆ) ∈ S(N), N = N1 + · · ·+Nk.
The norm of the symmetric matrix A used in (3.8) is
‖A‖ = sup{|λ| : λ is an eigenvalue of A} = sup{|〈Aξ, ξ〉| : |ξ| ≤ 1}.
We caution the reader that this result is an efficient summarization of the analytical
heart of the theory suitable for the presentation we have chosen and its proof, which
is outlined in the appendix, is deeper and more difficult than its applications that
we give in the main text. See also the notes to this section.
In order to apply Theorem 3.2 in the above situation, we put k = 2, O1 = O2 =
Ω, u1 = u, u2 = −v, ϕ(x, y) = (α/2)|x − y|2, and recall that J
2,−
Ω v = −J
2,+
Ω (−v).
In this case
Dxϕ(xˆ, yˆ) = −Dyϕ(xˆ, yˆ) = α(xˆ− yˆ), A = α
(
I −I
−I I
)
,
A2 = 2αA, ‖A‖ = 2α,
and we conclude that for every ε > 0 there exists X, Y ∈ S(N) such that
(3.9) (α(xˆ − yˆ), X) ∈ J2,+Ω u(xˆ), (α(xˆ − yˆ), Y ) ∈ J
2,−
Ω v(yˆ)
and
−
(
1
ε + 2α
)(
I 0
0 I
)
≤
(
X 0
0 −Y
)
≤ α(1 + 2εα)
(
I −I
−I I
)
.
Choosing ε = 1/α yields the elegant relations
(3.10) −3α
(
I 0
0 I
)
≤
(
X 0
0 −Y
)
≤ 3α
(
I −I
−I I
)
;

USER’S GUIDE TO VISCOSITY SOLUTIONS 19
The last inequality may be written
(
X 0
0 −(Y + εI)
)
≤
(
1 +
‖Y ‖
ε
)
‖Y ‖
(
I −I
−I I
)
.
Thus if α > (1/3)max{‖X‖, ‖Y ‖, (1 + ‖Y ‖/ε)‖Y ‖} and ε is sufficiently small, the
pair X,Y + εI satisfy (3.10). Next we fix x, r, and p, and put y = x− p/α in (3.14)
so that by (3.14)
F
(
x− pα , r, p, Y + εI
)
− F (x, r, p,X) ≤ ω
(
1
α (|p|
2 + |p|)
)
and then we let α→ ∞ and ε ↓ 0 and conclude that F is degenerate elliptic.
Remark 3.5. The proof of Theorem 3.3 adapts to provide modulus of continuity
estimates on solutions of F = 0.
Example 3.6. We turn to some examples in which (3.14) is satisfied. First
notice that (3.14) evidently holds (with ω the modulus of continuity for f) for
F (x, r, p,X) = G(r, p,X)− f(x) if G is degenerate elliptic and f is continuous; this
is because (3.10) implies X ≤ Y .
Secondly, the linear first order expression 〈b(x), p〉 satisfies (3.14) if
α〈b(y)− b(x), x − y〉 ≤ ω(α|x− y|2 + |x− y|)
for some ω and this holds with ω(r) = cr if there is a constant c > 0 such that
〈b(x)− b(y), x− y〉 ≥ −c|x− y|2, i.e., the vector field b+ cI is “monotone.” In fact,
it is not hard to see that this is a necessary condition as well.
Next, the linear expression
(3.16) G(x,X) = − trace(tΣ(x)Σ(x)X),
where Σ maps Ω into the N ×N real matrices, is degenerate elliptic (it is a special
case of Example 1.2) with A(x) = tΣ(x)Σ(x)). We seek to estimate G(y, Y ) −
G(x,X) when (3.10) holds. Multiplying the rightmost inequality in (3.10) by the
nonnegative symmetric matrix
(
tΣ(x)Σ(x) tΣ(y)Σ(x))
tΣ(x)Σ(y) tΣ(y)Σ(y)
)
and taking traces preserves the inequality and yields
(3.17)
G(y, Y )−G(x,X) = trace(tΣ(x)Σ(x)X − tΣ(y)Σ(y)Y )
≤ 3α trace((tΣ(x)− tΣ(y))(Σ(x) − Σ(y)),
so if Σ is Lipschitz continuous with constant L then
G(y, Y )−G(x,X) ≤ 3L2α|x− y|2
and we may choose ω(r) = 3L2r. We note that if Γ(x) : Ω → S(N) is Lipschitz
continuous and Γ(x) > 0 in Ω, then
− trace(Γ(x)X) = − trace(Γ(x)1/2Γ(x)1/2X)

20 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
is an example of the sort just discussed since Γ(x)1/2 is also Lipschitz continu-
ous. Finally, it is known that it Γ(x) ≥ 0 and Γ ∈ W 2,∞, then Γ1/2 is Lipschitz
continuous.
Now we note that the processes of forming sums or sups or sup infs, etc., above
produce examples obeying (3.14) if the ingredients obey (3.14) with a common ω.
Finally, we may produce examples satisfying (3.13) and (3.14) by putting
F (x, r, p,X) = γr +G(x, r, p,X)
whenever G is degenerate elliptic and satisfies (3.14).
We deferred discussion of the elementary Lemma 3.1 since it is better presented
in a generality not required above. Indeed, Lemma 3.1 may be obtained from the
following proposition via the correspondences 2N → M, O × O → O, (x, y) →
x, u(x)− v(y) → Φ(x), (1/2)|x− y|2 → Ψ(x).
Proposition 3.7. Let O be a subset of RM , Φ ∈ USC(O), Ψ ∈ LSC(O), Ψ ≥ 0,
and
(3.18) Mα = sup
O
(Φ(x) − αΨ(x))
for α > 0. Let −∞ < limα→∞Mα <∞ and xα ∈ O be chosen so that
(3.19) lim
α→∞
(Mα − (Φ(xα)− αΨ(xα)) = 0.
Then the following hold:
(3.20)



(i) limα→∞ αΨ(xα) = 0,
(ii) Ψ(xˆ) = 0 and limα→∞Mα = Φ(xˆ) = sup{Ψ(x)=0} Φ(x)
whenever xˆ ∈ O is a limit point of xα as α→ ∞.
Proof. Put
δα = Mα − (Φ(xα)− αΨ(xα))
so that δα → 0 as α → ∞. Since Ψ ≥ 0, Mα decreases as α increases and
limα→∞Mα exists; it is finite by assumption. Moreover,
Mα/2 ≥ Φ(xα)−
α
2
Ψ(xα) ≥ Φ(xα)− αΨ(xα) +
α
2
Ψ(xα) ≥Mα − δα +
α
2
Ψ(xα),
so 2(Mα/2 − Mα + δα) ≥ αΨ(xα), which shows that αΨ(xα) → 0 as α → ∞.
Suppose now that αn → ∞ and xαn → xˆ ∈ O. Then Ψ(xαn) → 0 and by the lower
semicontinuity Ψ(xˆ) = 0. Since
Φ(xαn)− αnΨ(xαn) ≥Mαn − δαn ≥ sup
{Ψ=0}
Φ(x)− δαn
and Φ is upper semicontinuous, (3.20) holds.
Remark 3.8. We record, for later use, some observations concerning maximum
points (xˆ, yˆ) of u(x) − v(y) − ϕ(x − y) over O × O for other choices of ϕ besides

USER’S GUIDE TO VISCOSITY SOLUTIONS 21
(α/2)|x|2 and the implications of Theorem 3.2. We are assuming that u is upper
semicontinuous, v is lower semicontinuous, and ϕ ∈ C2. In this application, the
matrix A of Theorem 3.2 has the form
(3.21) A =
(
Z −Z
−Z Z
)
, where Z = (D2ϕ)(xˆ − yˆ),
so
(3.22) A2 =
(
2Z2 −2Z2
−2Z2 2Z2
)
and ‖A‖ = 2‖Z‖.
Choosing ε = 1‖A‖ in (3.8), we conclude that there are X, Y ∈ S(N) such that
(3.23) (Dϕ(xˆ − yˆ), X) ∈ J2,+O u(xˆ), (Dϕ(xˆ − yˆ), Y ) ∈ J2,−Ω v(yˆ),
and
(3.24) −2‖A‖
(
I 0
0 I
)
≤
(
X 0
0 −Y
)
≤
( Z + 1‖A‖Z2 −(Z + 1‖A‖Z2)
−(Z + 1‖A‖Z2) Z + 1‖A‖Z2
)
;
in particular,
(3.25) ‖X‖, ‖Y ‖ ≤ 2‖A‖ and X ≤ Y.
Notes on §3. This section is self-contained except for the proof of Theorem 3.2,
which is due to M. G. Crandall and H. Ishii [48]. This result distills and sharpens
the essence of a line of development that runs through several of the references
listed below; the proof is explained in the appendix for the reader’s convenience.
We want to emphasize at the outset that our presentation of the proof of the
comparison theorem, based as it is on Theorem 3.2, is but one among several
possibilities. Moreover, other approaches may be useful in other situations. For
example, one may use elements of the proof of Theorem 3.2 directly and look at
corresponding manipulations on solutions—this approach seems to be the leading
presentation for the infinite-dimensional extensions of the theory and allows for
adaptations on the test functions and the notion itself and avoids the “semijets”
or generalized derivatives. Another presentation selects another building block,
namely, regularization by supconvolutions (which occur in the proof of Theorem
3.2) and its effect on viscosity solutions; this procedure is especially helpful in some
particular problems (integrodifferential equations, regularity issues) and stresses
the regularization procedure. This procedure has some analogues with the use of
mollification in the study of linear partial differential equations. With apologies for
confusing the reader, we are saying that a full grasp of all the elements of the proof
of Theorem 3.2 and exposure to other presentations (e.g., R. Jensen [102] and H.
Ishii and P. L. Lions [96]) may prove valuable.
The first uniqueness proofs for viscosity solutions were given for first-order equa-
tions in M. G. Crandall and P. L. Lions [51] and then M. G. Crandall, L. C. Evans,
and P. L. Lions [47]. The second-order case remained open for quite a while during
which the only evidence that a general theory could be developed was in results for

22 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
Hamilton-Jacobi-Bellman equations obtained by P. L. Lions [119, 118]. The proof
in these works involved ad hoc stochastic control verification arguments.
A breakthrough was achieved in the second-order theory by R. Jensen [101]
with the introduction of several key arguments; some of these were simplified in R.
Jensen, P. L. Lions, and P. E. Souganidis [104], P. L. Lions and P. E. Souganidis
[132], and R. Jensen [102]. In particular, the use of the “supconvolution” regu-
larization (see the appendix), which is a standard tool in convex and nonsmooth
analysis, was somewhat “remise au gouˆt dujour” by J. M. Lasry and P. L. Lions
[110].
Progress in understanding these proofs so as to be able to handle more examples
was made by H. Ishii [89] who introduced matrix inequalities of the general form
(3.7). This work contains an example (following [89, Theorem 3.3]) showing the
optimality of condition (3.14) in the sense that we cannot replace the right-hand side
of (3.14) by, for example, ω(α|x−y|θ+|x−y|) with θ < 2 (but see §5.A). Estimating
the left-hand side of (3.14) differently, for example in the form αg(|x−y|)+|x−y|, it
is possible to prove uniqueness in cases in which g(r)/r2 is unbounded near r = 0 by
refining the arguments of [49]. A more complete understanding together with the
sharpest structure conditions (up to the present) were achieved in [96], which also
contains many examples. We mention that the structure condition (3.14) exhibited
in this section pays no attention to whether the equation is of first order or possess
stronger ellipticity properties. In the case of uniformly ellipitic F , much more can
be done [96]. See also Jensen [102] as regards structure conditions.
A useful sharpening and improved organization of the analytical essence of the
theory were contributed by [46]. The presentation here was based on Theorem 3.2;
as remarked above, it follows from [48] and the generality and proofs presented
there represent the current state of the art (see the appendix).
4. Perron’s method and existence
Let Ω be an arbitrary open subset of RN . By a solution (respectively, subsolution,
etc.) of the Dirichlet problem
(DP)
{
F (x, u,Du,D2u) = 0 in Ω,
u = 0 on ∂Ω
we mean a function u ∈ C(Ω) (respectively, u ∈ USC(Ω), etc.) that is a (viscosity)
solution (respectively, subsolution, etc.) of F = 0 in Ω and satisfies u(x) = 0
(respectively, u(x) ≤ 0, etc.) for x ∈ ∂Ω. (We note that this formulation imposes
the boundary condition in a strict sense that will be relaxed in §7.)
Recall that we always assume that F is proper and, unless otherwise said , con-
tinuous . To discuss Perron’s method, we will use the following notations: if
u : O → [−∞,∞] where O ⊂ RN , then
(4.1)
{ u∗(x) = limr↓0 sup{u(y) : y ∈ O and |y − x| ≤ r},
u∗(x) = limr↓0 inf{u(y) : y ∈ O and |y − x| ≤ r}.
One calls u∗ the upper semicontinuous envelope of u; it is the smallest upper semi-
continuous function (with values in [−∞,∞]) satisfying u ≤ u∗. Similarly, u∗ is
the lower semicontinuous envelope of u.

24 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
From the extreme inequalities we learn y = 0, so xˆn → 0 (without passing to
a subsequence), and then from the first inequality and (4.3)(ii) one sees that
v(0) = limn→∞ un(xˆn). Since we have (p + 4δxˆn + Xxˆn, X + 4δI) ∈ J2,+O un(xˆn)
for large n, we are nearly done. To conclude, we merely note that the set of
(q, Y ) ∈ RN × S(N) such that there exists zn ∈ O, (pn, Xn) ∈ J2,+O un(zn) such
that (zn, un(zn), pn, Xn) → (0, v(0), q, Y ) is closed and contains (p,X + 4δI) for
δ > 0 by the above.
Proof of Lemma 4.2. With the notation of the lemma, suppose that z ∈ O and
(p,X) ∈ J2,+O w∗(z). We seek to show that F (z, w∗(z), p,X) ≤ 0. It is clear that we
may choose a sequence (xn, un) ∈ O × F such that (xn, un(xn)) → (z, w∗(z)) and
that (4.3) then holds with v = w∗. Hence, by the existence of data satisfying (4.4)
and the fact that each un is a subsolution, we may pass to the limit in the relation
F (xˆn, un(xˆn), pn, Xn) ≤ 0 to find F (z, w∗(z), p,X) ≤ 0 as desired.
The second step in the proof of Theorem 4.1 is a simple construction that we
now describe. Suppose that Ω is open, u is a solution of F ≤ 0, and u∗ is not a
solution of F ≥ 0; in particular, assume 0 ∈ Ω and we have
(4.7) F (0, u∗(0), p,X) < 0 for some (p,X) ∈ J2,−Ω u∗(0).
Then, by continuity, uδ,γ(x) = u∗(0) + δ + 〈p, x〉 + 12 〈Xx, x〉 − γ|x|2 is a classical
solution of F ≤ 0 in Br = {x : |x| < r} for all small r, δ, γ > 0. Since
u(x) ≥ u∗(x) ≥ u∗(0) + 〈p, x〉+ 12 〈Xx, x〉+ o(|x|2),
if we choose δ = (r2/8)γ then u(x) > uδ,γ(x) for r/2 ≤ |x| ≤ r if r is sufficiently
small and then, by Lemma 4.2, the function
U(x) =
{
max{u(x), uδ,γ(x)} if|x| < r,
u(x) otherwise,
is a solution of F ≤ 0 in Ω. The last observation is that in every neighborhood of
0 there are points such that U(x) > u(x); indeed, by definition, there is a sequence
(xn, u(xn)) convergent to (0, u∗(0)) and then
lim
n→∞
(U(xn)− u(xn)) = uδ,γ(0)− u∗(0) = u∗(0) + δ − u∗(0) > 0.
We summarize what this “bump” construction provides in the following lemma, the
proof of which consists only of choosing r, γ sufficiently small.
Lemma 4.4. Let Ω be open and u be solution of F ≤ 0 in Ω. If u∗ fails to be
a supersolution at some point xˆ, i.e., there exists (p,X) ∈ J2,−Ω u∗(xˆ) for which
F (xˆ, u∗(xˆ), p,X) < 0, then for any small κ > 0 there is a subsolution Uκ of F ≤ 0
in Ω satisfying
(4.8)
{
Uκ(x) ≥ u(x) and supΩ(Uκ − u) > 0,
Uκ(x) = u(x) for x ∈ Ω, |x− xˆ| ≥ κ.
Proof of Theorem 4.1. With the notation of the theorem observe that u∗ ≤W∗ ≤
W ≤ W ∗ ≤ u∗ and, in particular, W∗ = W = W ∗ = 0 on ∂Ω. By Lemma 4.2

USER’S GUIDE TO VISCOSITY SOLUTIONS 27
d; near ∂Ω (which is all we need) this follows from the obvious fact that Dd(x) =
−n(x) on ∂Ω where n(x) is the exterior normal to Ω at x ∈ ∂Ω.
In other cases, second-order terms dominate. If
G(x, p,X) = − trace(A(x)X) + 〈b(x), p〉 − f(x)
where A, b, f are continuous, and
〈A(x)n(x), n(x)〉 > 0 on ∂Ω,
then the left-hand side of (4.14) has the form
cλ2〈A(x)Dd(x), Dd(x)〉 +O(λ) as λ→ ∞.
Since Dd(x) = −n(x) on ∂Ω, the coefficient of λ2 is positive near ∂Ω and it is easy
to achieve (4.14) by taking λ large. Note that we are thus able to assert the unique
existence of a solution to (4.9) in this case (provided comparison holds) even though
A may be completely degenerate inside Ω.
The reader can invent an unlimited array of examples. It should be noted that
one may often produce both a subsolution and a supersolution by constructions
like the above and then one need not assume u ≡ 0 is a subsolution. One may also
usually take maximums and minimums of operators for which (4.14) can be verified
and stay within this class. Thus, for example, if
G(x, p,X) = max{−|p|2θ−ε − | traceX |θ−1 traceX − f(x), |p|α − g(x)}
and 0 < ε ≤ 2θ, α > 0, f, g ∈ C(Ω), and f, g ≥ 0, then (4.9) has a unique solution.
Notes on §4. The combination of Perron’s method and viscosity solutions was
introduced by H. Ishii [86]. The definition of a viscosity subsolution (respectively,
supersolution) u in [86] was that u∗ is a subsolution (respectively, u∗ is a supersolu-
tion) in the current sense. Solutions are then functions that are both a subsolution
and a supersolution and continuity is not required. (Note that then the character-
istic function of the rationals is a solution of u′ = 0). With this setup, Perron’s
method does not require the comparison assumption and the statements become
more elegant (see [86]).
We mention some other approaches to existence, for even if they are in general
much more complicated and of a more limited scope, they can be useful in some
delicate situations. For example, one can use formulas from control and differential
games to write explicit solutions for approximate equations and then use limiting
arguments; this approach is used in M. G. Crandall and P. L. Lions [55, Parts III
and V] and D. Tataru [157].
Two other approximations that have been used are discretization and elliptic
regularization (for first-order equations; this is the method of “vanishing viscosity”
and its relation to the theory accounts for the term “viscosity solutions”). Having
solved an approximate problem, one then needs to pass to the limit (with some a
priori estimates—but see §6!). Existence schemes of this sort have been used in P.
L. Lions [116], M. G. Crandall and P. L. Lions [53], G. Barles [14, 15], H. Ishii [82,
85], P. E. Souganidis [154], and I. Capuzzo-Dolcetta and P. L. Lions [44].

USER’S GUIDE TO VISCOSITY SOLUTIONS 29
Modifying the arguments in the proof of Theorem 3.3 in an obvious way, we will
still be able to establish comparison if we can show that
(5.6)
(F (yˆ, r, α(xˆ− yˆ) +Dψn(yˆ), Y +D2ψn(yˆ))
− F (xˆ, r, α(xˆ− yˆ) +Dψn(xˆ), X +D2ψn(xˆ)))+ → 0
as n, α → ∞ in some appropriate fashion when r remains bounded and (3.10),
(5.3), (5.5) hold.
It is not very informative to try and analyze this condition in general. Instead,
let us note that in the linear case (see Remark 3.6)
F (x, r, p,X) = − trace(tΣ(x)Σ(x)X) + 〈b(x), p〉 + r − f(x)
with f ∈ C(Ω), the first-order terms are harmless from the point of view of verifying
(5.6) if b is continuous. Via Remark 3.6, the second-order terms contribute at most
3α trace(t(Σ(xˆ)− Σ(yˆ))(Σ(xˆ)− Σ(yˆ)) + trace(−A(yˆ)D2ψn(yˆ) +A(xˆ)D2ψn(xˆ))
when estimating (5.6) above. Here we have put A(x) = tΣ(x)Σ(x). Assuming
Σ ∈ Cλ(Ω) (so A ∈ Cλ(Ω)), we invoke (5.3) and estimate the above expression by
a constant times
α|xˆ− yˆ|2λ + n1−γ |xˆ− yˆ|λ + n2−γ |xˆ− yˆ|.
By (5.5), this expression may, in turn, be estimated above in the form
(5.7)
(
α
(n−γ
α
)2λ
+ n1−γ
(n−γ
α
)λ
+ n2−γ
(n−γ
α
))
as n→ ∞.
Putting α = nβ, this becomes
(
nβ−2λ(γ+β) + n−λ(γ+β)+1−γ + n2(1−γ)−β
)
as n→ ∞.
Regarding λ as fixed so as to see what we require of γ, we note that we may make all
of the exponents above vanish by the choices γ = (1− 2λ)/(1−λ), β = 2λ/(1−λ).
(The case λ > 12 was treated above, so here we are concerned with λ ≤ 12 .) If we
increase γ from this value, so that γ > (1− 2λ)/(1− λ), it follows that comparison
holds.
Comparison is thus assured if Σ ∈ Cλ and Du ∈ Cγ under the relation γ >
(1− 2λ)/(1− λ). The limit cases λ = 12 and λ = 0 need to be discussed separately,
for γ = 0 suffices if λ = 12 and γ = 1 suffices if λ = 0. In the event that γ = 0,
the first relation in (5.3) should be replaced by |u − ψn| ≤ o(1) and (5.5) by
α|xˆ − yˆ| ≤ o(1) and then the proof runs as before for γ = 0 but with the constant
o(1) appearing everywhere. To treat the other limiting case, λ = 0, γ = 1, note
that if ρ is a modulus of continuity of Σ and A, (5.7) should be replaced by
αρ(|xˆ − yˆ|)2 + ρ(|xˆ − yˆ|) + n|xˆ− yˆ|
and (5.5) should read α|xˆ − yˆ| ≤ C/n. We conclude upon letting n, α → ∞ in a
manner so that αρ(C/(nα))2 → 0.

30 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
It is worthwhile to compare the uniqueness results above with those that may
be obtained when solutions are more nearly classical in the sense that they possess
second derivatives in a strong enough sense. To simplify the exposition, we will
always assume that (3.13) holds. Then, of course, if we assume that one of u or v is
C2 (or even everywhere twice differentiable), the comparison result holds without
other structure conditions on F beyond properness, since we can work directly with
maxima of u−v. Unfortunately, this regularity is rarely available, even for uniformly
elliptic fully nonlinear equations. If we require less regularity of u and v—but still
much more than assumed in Theorem 3.3—different structure conditions suffice to
guarantee the comparison result via finer considerations about the pointwise twice
differentiability of functions.
We present below a strategy that establishes comparison without further as-
sumptions on F if both u+ c|x|2 and −v+ c|x|2 are convex for some c > 0. It also
shows that comparison holds if u, v ∈ W 2,Nloc (Ω) and one of u or v lies in W
2,p
loc (Ω)
for some p > N and the structure condition
{ F (x, t, p,X) is uniformly continuous in p
uniformly for x ∈ Ω, X ∈ S(N), and r, p bounded
holds.
In both of these cases, the argument runs as follows: we may assume, without
loss of generality, that all maxima of u − v lie in Ω and then general optimization
results imply that there exists a sequence pn → 0 such that u(x) − v(x) − 〈pn, x〉
has a strict maximum at xˆn and xˆn → xˆ where xˆ is a maximum point of u − v
over Ω. Then one knows (cf. Lemma A.3) that if r > 0 is small enough, the set of
maximum points of u(x) − v(x) − 〈pn, x〉 − 〈q, x〉 in |x − xˆn| < r as q ranges over
the ball Bδ contains a set of positive measure if δ < r−1εn where
εn = inf
|y−xˆn|=r
[(u(xˆn)− v(xˆn))− 〈pn, xˆn − y〉 − (u(y)− v(y))] .
Furthermore, either u or v is twice differentiable a.e. (cf. Theorem A.2) and
therefore we can find maxima z with |xˆn − z| < r for some q ∈ Bδ that is a point
where, in the first case, u and v are twice differentiable and in the second case v or
u is twice differentiable. In the first case we have
F (z, u(z), Dv(z) + pn + q,D2v(z)) ≤ 0 ≤ F (z, v(z), Dv(z), D2v(z)).
In the second case, if v is twice differentiable the same inequality holds and otherwise
we have
F (z, u(z), Du(z), D2u(z)) ≤ 0 ≤ F (z, v(z), Du(z)− pn − q,D2u(z)).
The conclusions are easily reached upon sending δ, r to 0+ and then n→ ∞.
5.B. Estimates from comparison. We make some simple remarks that hold
as soon as one has comparison by any means; there are many variants of these
(including parabolic ones). For example, suppose K > 0 and u − v ≤ K on ∂Ω
instead of u ≤ v on ∂Ω. Then u−K is also a subsolution since J2,+Ω (u−K) = J2,+Ω u
and F (x, u − K, p,X) ≤ F (x, u, p,X) by properness. Thus u − v ≤ K in Ω. In

USER’S GUIDE TO VISCOSITY SOLUTIONS 31
particular, the variant of Theorem 3.3 in which u ≤ v on ∂Ω is dropped and the
conclusion is changed to u− v ≤ sup∂Ω(u− v)+ holds.
In a similar spirit, suppose we have solutions u and v of F (x, u,Du,D2u) ≤ 0,
and Fˆ (x, u,Du,D2u) ≥ 0 in Ω where F is proper, satisfies (3.14) and (3.13), K > 0,
and F (x, r, p,X) +K ≥ Fˆ (x, r, p,X). Then w = v+max(sup∂Ω(u− v)+,K/γ) is a
solution of F (x,w,Dw,D2w) ≥ 0 in Ω. Since we have comparison for F , we then
conclude u − w ≤ sup∂Ω(u − w)+ = 0 or u − v ≤ max(sup∂Ω(u − v)+,K/γ). In
particular, if
u+G(x, u,Du,D2u)− f(x) ≤ 0 and v +G(x, v,Dv,D2v)− g(x) ≥ 0 in Ω,
G is proper and satisfies (3.14), and f, g ∈ C(Ω), then
u− v ≤ max(sup
∂Ω
(u− v)+, sup
Ω
(f − g)+).
5.C. Comparison with strict inequalities and without coercivity in u. The
condition (3.13) was used in the proof of Theorem 3.3 in order to have (3.15).
If we simply assume there is a δ > 0 such that either F (x, u,Du,D2u) ≤ −δ
or F (x, v,Dv,D2v) ≥ δ we are in the same situation and do not need (3.13) to
hold. Moreover, if only F (x, u,Du,D2u) ≤ 0 and F (x, v,Dv,D2v) ≥ 0 but for
ε > 0 we can find ψε ∈ C2, δε > 0 such that |ψε| ≤ ε and F (x, u + ψε, D(u +
ψε), D2(u + ψε)) ≤ −δε, we conclude that (u + ψε) − v ≤ sup∂Ω(u + ψε − v) and
then u−v ≤ sup∂Ω(u−v)+2ε and we recover the result as ε ↓ 0. This construction
can be carried out in some cases.
5.D. Comparison and existence of unbounded solutions on unbounded
domains. We first illustrate a method to establish comparison of unbounded solu-
tions in unbounded domains by showing that if u and v grow at most linearly and
solve
(5.8) u+ F (Du,D2u)− f(x) ≤ 0 and v + F (Dv,D2v)− f(x) ≥ 0 in RN
where f is uniformly continuous on RN (i.e., f ∈ UC(RN )), then u ≤ v. After this,
we will prove that if u is a solution of u + F (Du,D2u) = 0 of linear growth, then
u ∈ UC(RN ) and finally that Perron’s method supplies existence. Thus we will
prove
Theorem 5.1. If f ∈ UC(RN ), then u + F (Du,D2u) − f(x) = 0 has a unique
solution u that grows at most linearly and u ∈ UC(RN ).
Proof of comparison. The proof proceeds in two steps. First we note that f ∈
UC(RN ) implies that there is a constant K such that
(5.9) sup
RN×RN
(f(x)− f(y)−K|x− y|) <∞
and then we show that
(5.10) sup
RN×RN
(u(x)− v(y)− 2K|x− y|) <∞.

34 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
Notes on §5. The discussion of §5.A regarding regularity of Du is new in this
framework, but such results were first obtained by R. Jensen [102] in a more com-
plicated presentation. The assertions of §5.A regarding twice differentiable solutions
(classical or W 2,p, etc.) rely on various versions of the maximum principle as it
evolved through the works of Aleksandrov [2, 3], Y. Bakelman [6], C. Pucci [143],
J. M. Bony [37], and P. L. Lions [120].
Section 5.C recalls a classical strategy and corresponds to remarks used in M. G.
Crandall and P. L. Lions [51], H. Ishii [87], H. Ishii and P. L. Lions [96], . . . . More
sophisticated uses of estimates from comparison arguments occur in the study of
regularity questions—see §10 for references.
Section 5.D is concerned with the growth of solutions at infinity and its influence
on comparison-uniqueness results. The relevance of the class UC(RN ) for general
results of this sort was progressively understood in a series of papers by the authors
and we refer to M. G. Crandall and P. L. Lions [55, Part II] for a few examples
showing how natural this class is and how it interacts with structure conditions
on the nonlinearity. Of course, if one restricts the nonlinearity further, other as-
ymptotic behaviors can be allowed. See H. Ishii [83, 89], M. G. Crandall and P.
L. Lions [54, 56], and M. G. Crandall, R. Newcomb, and Y. Tomita [59] for more
information in this direction; in particular, general functions F (x, r, p,X) can be
used in place of F (p,X)− f(x) (although subtleties concerning existence arise).
6. Limit operations with viscosity solutions
Suppose we have a sequence un, n = 1, 2, . . . , of subsolutions of an equation
F = 0 on the locally compact set O. It turns out that the following “limit”
(6.1) U(z) = lim sup
n→∞
∗un(z) = lim
j→∞
sup{un(x) : n ≥ j, x ∈ O, and |z − x| ≤ 1j }
in which, roughly speaking, the “lim sup” operation and the ∗ operations are per-
formed simultaneously, will also be a solution of F ≤ 0 in O. Indeed, since U(x) < r
only if there are ε, j > 0 such that un(z) < r − ε for n ≥ j and z ∈ O with
|z−x| ≤ 1/j, it is clear that {x ∈ O : U(x) < r} is open in O and thus U ∈ USC(O).
We have
Lemma 6.1. Let un ∈ USC(O) for n = 1, 2, . . . and U be given by (6.1), z ∈ O
and U(z) <∞. If (p,X) ∈ J2,+O U(z), then there exist sequences
(6.2) nj → ∞, xˆj ∈ O, (pj , Xj) ∈ J2,+O unj (xˆj)
such that
(6.3) (xˆj , unj (xˆj), pj , Xj) → (z, U(z), p,X).
In particular, if each un is a solution of F ≤ 0 and U < ∞ on O, then U is a
solution of F ≤ 0 on O.
Proof. By definition, there are sequences
nj → ∞, xj ∈ O such that xj → z and unj (xj) → U(z);

USER’S GUIDE TO VISCOSITY SOLUTIONS 35
it is also clear that if zj → x in O, then lim supunj (zj) ≤ U(x). The result now
follows at once from Proposition 4.3.
Remark 6.2. If the assumptions of the lemma are altered to assume instead that
un ∈ LSC(O) and (p,X) ∈ J2,−O v(z), then the assertions are changed by replacing
U by
(6.4) U(z) = lim inf
n→∞∗
un(z) = lim
j→∞
inf{un(x) : n ≥ j, x ∈ O, and |z − x| ≤ 1j },
putting J2,−O in place of J
2,+
O in (6.2) and supersolutions in place of subsolutions.
Remark 6.3. In fact, the above proof shows more—suppose un is a solution of a
proper equation Fn ≤ 0 that varies with n. Then the conclusion is that U is a
solution of G ≤ 0 where
(6.5) G(x, r, p,X) = lim inf
n→∞∗
Fn(x, r, p,X);
note that Fn need not be continuous and that if Fn = F is independent of n but
discontinuous, then G = F∗ is the lower semicontinuous envelope of F . Analogous
remarks hold for supersolutions.
Remark 6.4. The above results are related to uniform convergence as explained
next. Let un be a sequence of functions on O and
(6.6) U(x) = lim sup
n→∞
∗un(x), U(x) = lim inf
n→∞∗
un(x).
Suppose U(x) = U(x) on O, let U(x) denote this common value and assume that
−∞ < U(x) < ∞ on O. Then U is continuous (since it is both upper and lower
semicontinuous) and limn→∞ un(x) = U(x) uniformly on compact sets. Indeed, if
this were not the case and uniform convergence failed on some compact set K, there
would be an ε > 0 and sequences nj → ∞, xj ∈ K such that unj (xj)−U(xj) > ε or
unj (xj)− U(xj) < −ε. Assuming xj → x and using the continuity of U , we would
conclude that |U(x) − U(x)| ≥ ε, a contradiction. In order to prove that U = U
one notes that U ≤ U by definition and typically uses comparison results to prove
the other inequality. The next result provides a simple example; a typical case in
which the hypotheses of this result are easily verified is mentioned after the short
proof.
Theorem 6.5. Let Ω be a bounded open set in RN , H ∈ C(RN ), and f ∈ C(Ω).
Consider the problem
(DP)ε u+H(Du)− ε∆u = f(x) in Ω, u = 0 on ∂Ω,
and assume that (DP)ε has a subsolution u ∈ C(Ω) and a supersolution u ∈ C(Ω)
independent of ε ∈ [0, 1] that vanish on ∂Ω. Then (DP)ε has a solution uε for
ε ∈ [0, 1] and limε↓0 uε(x) = u0(x) uniformly for x ∈ Ω.
Proof. We know that (DP)ε has a solution uε for ε ∈ [0, 1] by Perron’s method and
the assumptions. Moreover, U(x) = lim supε↓0∗uε(x) and U(x) = lim infε↓0∗uε(x)
are a subsolution and a supersolution of (DP)0 by the above and then U ≤ U by
comparison while U ≥ U by definition. Thus u0 = U = U and the convergence is
uniform.
Harking back to the discussion of Example 4.6, the reader may easily verify the
hypotheses of Theorem 6.5 when ∂Ω is of class C2, H satisfies lim|p|→∞H(p) = ∞
and H(0) − f ≤ 0. To do so, set u ≡ 0 and u(x) = (λd(x)) ∧ C where λ is first
chosen sufficiently large and then C is chosen in a suitable manner.

36 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
Notes on §6. The fact that viscosity solutions pass to the limit under uniform
convergence is an immediate consequence of the definition. This consistency and
stability property can also be seen as the analogue (in the space of continuous
functions) of Minty’s device for monotone operators as developed by L. C. Evans
[64, 65].
The idea of using only “half-relaxed” limits like U¯ in (6.1) arises naturally when
attempting to pass to limits with maxima and minima and has been used extensively
in the calculus of variations and homogenization theory. The use of this concept
in the area of viscosity solutions was introduced by G. Barles and B. Perthame
[25, 26] and H. Ishii [90]. This passage to the limit and the associated a posteriori
uniform convergence through comparison of semicontinuous functions has become
one of the main features of the theory. Viscosity solutions were first used to prove
theorems analogous to Theorem 3.2 (but more complicated) with applications by L.
C. Evans and H. Ishii [68]; they were motivated by questions arising in probability
(see also §10). Proofs in the spirit we have presented are much simpler.
7. General and generalized boundary conditions
7.A. Boundary conditions in the viscosity sense. In this section we consider
more general boundary value problems of the form
(BVP)
{
(E) F (x, u,Du,D2u) = 0 in Ω
(BC) B(x, u,Du,D2u) = 0 on ∂Ω
where (E) denotes the equation and (BC) the boundary condition. Here F is proper,
as always, and B is a given function on ∂Ω×R×RN×S(N) that is also to be proper.
While it is convenient to allow B to depend on D2u at this stage, as the reader
will see, when we turn to existence and uniqueness theorems B = B(x, r, p) will
be of first order. For example, the Dirichlet condition u = f(x) on ∂Ω arises from
the choice B(x, r, p,X) = B(x, r, p) = r − f(x) of B while the Neumann condition
un = f(x) (here and later, n(x) denotes the outward unit normal to x ∈ ∂Ω)
arises if B(x, r, p) = 〈n(x), p〉− f(x). The Neumann condition is generalized by the
“oblique derivative” problem, in which B has the form B(x, r, p) = 〈ν(x), p〉− f(x)
where ν(x) is a vector field on ∂Ω satisfying 〈ν(x), n(x)〉 > 0 on ∂Ω.
We have already given a meaning to statements like (BC) in Definition 2.2, which
is, however, clearly inappropriate here since it would only involve the behavior of
u on ∂Ω. The simplest viscosity definition of (BC) we might try is
Definition 7.1. A function u ∈ USC(Ω) is a subsolution of (BC) in the strong
(viscosity) sense if
(7.1) B(x, u(x), p,X) ≤ 0 for x ∈ ∂Ω, (p,X) ∈ J2,+Ω u(x)
and v ∈ LSC(Ω) is a supersolution of (BC) in the strong (viscosity) sense if
(7.2) B(x, v(x), p,X) ≥ 0 for x ∈ ∂Ω, and (p,X) ∈ J2,−Ω v(x).
Finally, u ∈ C(Ω) satisfies (BC) in the strong (viscosity) sense if it is both a
subsolution and a supersolution in the strong sense.

38 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
that (passing to a subsequence if necessary), there exists (pn, Xn) ∈ J
2,+
Ω un(xn)
such that (xn, un(xn), pn, Xn) → (x, u(x), p,X), but we do not know xn ∈ ∂Ω (and
the example below shows this cannot be achieved in general). If xn ∈ Ω, we have
Fn(xn, un(xn), pn, Xn) ≤ 0 (rather than Bn(xn, un(xn), pn, Xn) ≤ 0). However, in
this way we can conclude that either
B(x, u(x), p,X) ≤ 0 or F (x, u(x), p,X) ≤ 0;
in other words, u is a subsolution for the boundary condition
F (x, u,Du,D2u) ∧B(x, u,Du,D2u) = 0
in the strong sense (where a∧ b = min{a, b} and a∨ b = max{a, b}). The following
example, in the context of a Neumann problem, shows that this is the best that
can be expected.
Example 7.3. The linear problem
(7.5) −εu′′ + u′ + u = x+ 1
on (0, 1) ⊂ R with the homogeneous Neumann conditions
(7.6) u′(0) = u′(1) = 0
has a unique classical solution uε for ε > 0 and it follows from Proposition 7.2 that
uε satisfies (7.6) in the strong sense when we chose the function B that defines the
boundary condition (7.6) to be given by B(1, r, p) = p and B(0, r, p) = −p; note
the monotonicity in the direction of the exterior normals to (0, 1). We will see that
the limit limε↓0 uε = u exists uniformly on [0, 1] and the limit u does not satisfy
(7.6) in the strong sense.
It is elementary to compute
uε(x) = x+
eλ− − 1
λ+ (eλ+ − eλ−)
eλ+x + 1− e
λ+
λ− (eλ+ − eλ−)
eλ−x,
where λ± = (1/2ε)
(
1± (1 + 4ε)1/2
)
. Noting that λ+ → ∞ and λ− → −1 as ε ↓ 0,
one sees that
uε(x) → u(x) ≡ x+ e−x uniformly on [0, 1] as ε ↓ 0.
The function u(x) = x+ e−x satisfies
u′ + u = x+ 1 in (0, 1) and u′(0) = 0
in the classical sense. However, u′(1) = 1 − 1/e > 0 and therefore u is not a
subsolution of B(1, u′(1)) = 0 in the strong sense. Note, however, that (u′ + u −
(x+ 1)) ∧ u′ ≤ 0 does hold at x = 1, in agreement with the above discussion.
These considerations suggest the appropriate definitions regarding solutions of
(BVP).

USER’S GUIDE TO VISCOSITY SOLUTIONS 41
where C = C(ε) will be chosen later on. Using (7.14) and (7.15), for x ∈ V and
(p,X) ∈ J2,+Ω uε(x) we have
F (x, uε(x), p,X) ≤ F (x, u(x), p+ εDϕ(x), X + εD2ϕ(x)) − γC + ω(εM),
where M = maxΩ(|Dϕ(x)| + ‖D2ϕ(x)‖) and, if x ∈ ∂Ω,
B(x, uε(x), p) = B(x, uε(x), p+ εDϕ(x)) − ε〈Dϕ(x), n(x)〉
≤ B(x, u(x), p + εDϕ(x)) − ε.
Using next that J2,+Ω uε(x) + (εDϕ(x), εD2ϕ(x)) = J
2,+
Ω u(x) (see Remarks 2.7), we
conclude that if C = ω(εM)/γ, then uε is a subsolution of (BVP) with B + ε in
place of B. Similarly, vε is a supersolution of (BVP) with B replaced by B − ε.
We are reduced to proving that if u is a subsolution and v is a supersolution of
(BVP) with B replaced by B + ε and B − ε respectively, then u ≤ v. Assume, to
the contrary, that maxΩ(u − v) > 0. We know that max∂Ω(u − v) ≥ maxΩ(u − v)
by §5.B, so there must exist z ∈ ∂Ω such that u(z)− v(z) = maxΩ(u− v) > 0. Put
Φ(x, y) = u(x)− v(y)− α
2
|x− y|2 + f(z, u(z))〈n(z), x− y〉 − |x− z|4 on Ω× Ω,
and let (xα, yα) be a maximum point of Φ. By Proposition 3.7, since u(x)− v(x)−
|x− z|4 has x = z as a unique maximum point,
(7.20) xα → z, α|xα − yα|2 → 0, u(xα) → u(z), v(xα) → v(z)
as α→ ∞. For simplicity, we now write (xˆ, yˆ) for (xα, yα) and put
ψ(x, y) = α
2
|x− y|2 − f(z, u(z))〈n(z), x− y〉+ |x− z|4.
We have not yet invoked the exterior sphere condition (7.13) and do so now. Re-
stating (7.13) as
|y − x− rn(x)|2 > r2 for x ∈ ∂Ω and y ∈ Ω,
we see it is equivalent to
(7.21) 〈n(x), y − x〉 < 1
2r |y − x|
2 for x ∈ ∂Ω and y ∈ Ω.
Using (7.21) we compute that if xˆ ∈ ∂Ω then
B(xˆ, u(xˆ), Dxψ(xˆ, yˆ)) = B(xˆ, u(xˆ), α(xˆ− yˆ)
− f(z, u(z))n(z) + 4|xˆ− z|2(xˆ− z))
≥ − α
2r |xˆ− yˆ|
2 − f(z, u(z))〈n(xˆ), n(z)〉
+O(|xˆ − z|3) + f(xˆ, u(xˆ))
and in view of (7.20) this implies
B(xˆ, u(xˆ), Dxψ(xˆ, yˆ)) ≥ o(1) as α→ ∞ if xˆ ∈ ∂Ω.

USER’S GUIDE TO VISCOSITY SOLUTIONS 43
Using (7.16) we therefore have
F (y, r, p, Y + δI)− F (x, r, p,X − δI) ≤ ω((α + 23δ)|x− y|2 + |x− y|(|p|+ 1)).
Next, (7.15) implies
F (y, r, p, Y + δI)− F (x, r, p,X − δI) ≥ F (y, r, p, Y )− F (x, r, p,X)− 2ω(δ)
and the result follows.
We complete the proof of comparison. From the lemma and the developments
above,
0 ≤ F (yˆ, v(yˆ),−Dyψ(xˆ, yˆ), Y )− F (xˆ, u(xˆ), Dxψ(xˆ, yˆ), X)
≤ F (yˆ, u(xˆ),−Dyψ(xˆ, yˆ), Y )− F (xˆ, u(xˆ),−Dyψ(xˆ, yˆ), X)
− γ(u(xˆ)− v(yˆ)) + ω(4|xˆ− z|3)
≤ ω(α|xˆ− yˆ|2 + |xˆ− yˆ|(|Dyψ(xˆ, yˆ)|+ 1))− γ(u(xˆ)− v(yˆ)) + o(1)
as α→ ∞. From this we obtain a contradiction as in the proof of Theorem 3.3.
Proof of existence. In order to establish existence, we may use Perron’s Method
from Theorem 4.1 as supplemented by Remark 4.5 to deduce that it suffices to
produce a subsolution and a supersolution. Using Proposition 7.2, we see it suffices
to produce a classical subsolution and supersolution. Let ϕ ∈ C2(Ω) be as in
Lemma 7.6. Put
u(x) = −Aϕ(x) −B, u(x) = Aϕ(x) +B
where A, B are constants to be determined. Observing that
B(x, u(x), Du(x)) ≥ A〈n(x), Dϕ(x)〉 + f(x, 0) ≥ A+ f(x, 0)
for x ∈ ∂Ω and
F (x, u(x), Du(x), D2u(x)) ≥ F (x, 0, ADϕ(x), AD2ϕ(x)) + γB
for x ∈ Ω we see that if we first choose A and then B by
A = sup
∂Ω
|f(x, 0)|, B = 1γ supΩ,|p|+‖X‖≤AM
|F (x, 0, p,X |,
where M = supΩ(|Dϕ(x) + ‖D2ϕ(x)‖), then u is a supersolution. Likewise, u will
be a subsolution. It remains to prove Lemma 7.6.
Proof of Lemma 7.6. In view of (7.12), there is a C1 function ψ on RN such that
ψ ≥ 0 on RN\Ω, ψ < 0 on Ω, and |Dψ| > 0 on ∂Ω. Multiplying by a constant
if necessary, we may also assume that |Dψ(x)| > 〈ν(x), n(x)〉−1 for x ∈ ∂Ω. By
standard approximation theorems, we see that there is a sequence of C2 functions
ψk such that ψk → ψ in C1(Ω) as k → ∞. In particular, we have
〈ν(x), Dψk(x)〉 → 〈ν(x), Dψ(x)〉 uniformly on ∂Ω as k → ∞.
Since n(x) = Dψ(x)/|Dψ(x)| and |Dψ(x)|〈ν(x), n(x)〉 > 1 for x ∈ ∂Ω, if k and
then C are sufficiently large, ϕ = ψk + C has the desired properties.

44 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
7.C. The generalized Dirichlet problem. We turn to the Dirichlet problem
(7.24)
{ F (x, u,Du,D2u) = 0 in Ω,
u− f(x) = 0 on ∂Ω
where f ∈ C(∂Ω).
The simplest example of this problem with a unique solution that does not
satisfy boundary conditions in the strong sense is the following: F (x, u,Du,D2u) =
u − h(x) and f(x) ≡ 0. If h is continuous on Ω, then u ≡ h is evidently the only
solution of this problem and it does not satisfy u = 0 on ∂Ω in the strong sense
unless h = 0 on ∂Ω.
We will provide a comparison theorem, Theorem 7.9 below, with this example
as a special case. This comparison theorem, however will be different in character
from those we have obtained so far in that it will assert comparison for a continuous
subsolution and supersolution. As we know, the ability to compare semicontinuous
subsolutions and supersolutions allows us to prove existence. Thus the following
counterexample to existence indicates that the restriction to continuous functions
(or some other assumption) is necessary in Theorem 7.9.
Example 7.8. Let N = 2 and Ω = {(x, y) : −1 < x < 1, 0 < y < 1}. Let
f ∈ C(∂Ω) satisfy 0 ≤ f ≤ 1, f(x, 0) = 0, and f(x, 1) = 1 for |x| < 1. We claim
that the problem
(7.25)
{ u+ xuy = 0 in Ω,
u = f on ∂Ω
does not have a solution. To see this, suppose the contrary and let u(x, y) be a
solution. We claim that then for fixed |x| < 1, v(y) = u(x, y) is a solution of
(7.26) v + xv′ = 0 in (0, 1), v(0) = 0, v(1) = 1.
Assume for the moment that this is the case. Then we remark that Theorem 7.9
below shows that v is uniquely determined by (7.26), v ≡ 0 is a solution of (7.26)
if x > 0 and v(y) = e(1−y)/x is a solution of (7.26) if x < 0 (these last assertions
the reader can check by computations). Thus we see that u(x, y) = 0 if x > 0 and
u(x, y) = e(1−y)/x if x < 0; however, u cannot then be continuous at (0, 1) and we
conclude that (7.25) does not have a solution.
It remains to verify that if u is a solution of (7.25), then v is a solution of
(7.26). One way to check this is as follows: Fix x ∈ (−1, 1) and set v(y) =
u(x, y). Let ϕ ∈ C2([0, 1]) and assume that v − ϕ has a unique maximum at
some point y ∈ [0, 1]. For α > 0 let (xˆ, yˆ) be a maximum point of the function
u(x, y)− ϕ(y)− α|x− x|2. It follows from Proposition 3.7 that then xˆ→ x, yˆ → y
as α → ∞; moreover, we have u(xˆ, yˆ) + xˆϕ′(yˆ) ≤ 0. From this information we
deduce that v(y) + xϕ′(y) ≤ 0 and conclude that v is a subsolution of (7.26) with
x = x. In a similar way, one shows that v is a supersolution. Finally we remark
that u ≡ 1, u ≡ 0 are, respectively, a supersolution and a subsolution of (7.25).
Thus comparison of semicontinuous semisolutions of (7.25) must fail. (If the reader
is concerned about regularity questions and the “corners,” note that the corners
can be smoothed up without changing the essential points of the discussion.)
We turn to the comparison result for continuous solutions.

USER’S GUIDE TO VISCOSITY SOLUTIONS 45
Theorem 7.9. Let (7.12), (7.14), (7.15), and (7.16) hold. If u, v ∈ C(Ω), u is a
subsolution of (7.24) and v is a supersolution of (7.24), then u ≤ v.
Proof. As before, we argue by contradiction and so suppose that maxΩ(u− v) >0.
We may assume that maxΩ(u − v) = u(z)− v(z) > 0 for some z ∈ ∂Ω. We divide
our considerations into two cases.
First, we consider the case when v(z) < f(z). For α > 1 and 0 < ε < 1, we
define the function
Φ(x, y) = u(x)− v(y)− |α(x− y) + εn(z)|2 − ε|y − z|2 on Ω× Ω.
Let (xˆ, yˆ) be a maximum point of Φ. We assume α is so large that z−(ε/α)n(z) ∈ Ω.
The inequality Φ(xˆ, yˆ) ≥ Φ(z − (ε/α)n(z), z) reads
|α(xˆ − yˆ) + εn(z)|2 + ε|yˆ − z|2 ≤ u(xˆ)− v(yˆ)− u
(
z − εαn(z)
)
+ v(z)
from which it follows (using the continuity of u!) that if ε is fixed, xˆ, yˆ → z and
α(xˆ− yˆ) + εn(z) → 0 as α→ ∞. Indeed, it is clear that α(xˆ− yˆ) remains bounded
as α → ∞, so assuming that α(xˆ − yˆ) → w, xˆ, yˆ → z˜ (along a subsequence) as
α→ ∞, one finds
|w + εn(z)|2 + ε|z˜ − z|2 ≤ u(z˜)− v(z˜)− u(z) + v(z) ≤ 0,
where the last inequality follows from the definition of z, whence the claim is im-
mediate. From this we see that xˆ = yˆ− (εn(z)+ o(1))/α as α→ ∞ and hence that
xˆ ∈ Ω if α is large enough. Since v(z) < f(z), we have (using the continuity of v)
v(yˆ) < f(yˆ) if α is large enough. Thus, if α is large enough, we have
F (xˆ, u(xˆ), p,X) ≤ 0 for (p,X) ∈ J2,+Ω u(xˆ)
and
F (yˆ, v(yˆ), q, Y ) ≥ 0 for (q, Y ) ∈ J2,−Ω v(yˆ).
Observe that if we set ϕ(x, y) = |α(x − y) + εn(z)|2 + ε|y − z|2, then
Dxϕ(x, y) = 2α(α(x − y) + εn(z)),
−Dyϕ(x, y) = 2α(α(x − y) + εn(z))− 2ε(y − z),
and
D2ϕ(x, y) = 2α2
(
I −I
−I I
)
+ 2ε
(
0 0
0 I
)
.
Now, using Theorem 3.2 together with Lemma 7.7, calculating as usual, sending
α→ ∞ and then ε ↓ 0, we obtain a contradiction.
It remains to treat the case when v(z) ≥ f(z). Since u(z) > v(z) this entails
u(z) > f(z). Replacing the above Φ by
Φ(x, y) = u(x)− v(y)− |α(x− y)− εn(z)|2 − ε|x− z|2,
we argue as above and obtain a contradiction.

46 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
7.C′. The state constraints problem. The problem
(7.27)
{ F (x, u,Du,D2u) ≤ 0 in Ω,
F (x, u,Du,D2u) ≥ 0 in Ω
corresponds to an important problem in optimal control, the so-called state con-
straints problem provided F has the form of the left member of (1.10). This is,
indeed, the extreme form of (7.24) where f(x) ≡ −∞. To make this clear, we give
the definitions precisely. A function u ∈ USC(Ω) (respectively, v ∈ LSC(Ω)) is
called a subsolution (supersolution) of (7.27) if
F (x, u(x), p,X) ≤ 0 for x ∈ Ω and (p,X) ∈ J2,+u(x)
(respectively,
F (x, v(x), p,X) ≥ 0 for x ∈ Ω and (p,X) ∈ J2,−Ω v(x)).
An application of Theorem 7.9 immediately yields
Theorem 7.10. Let (7.12), (7.14), (7.15), and (7.16) hold. If u, v ∈ C(Ω), u is a
subsolution of (7.27) and v is a supersolution of (7.27), then u ≤ v on Ω.
Indeed, v is a supersolution and u is a subsolution of (7.24) with f(x) ≥ u(x) on
∂Ω.
In fact, a subsolution v of (7.27) is a supersolution of (7.24) for any f ∈ C(∂Ω)
and hence a “universal bound” on all solutions. This implies that solutions of (7.27)
do not exist in general. For example, if F is a linear uniformly elliptic operator,
(7.24) is uniquely solvable for arbitrary f ∈ C(∂Ω) and then there can be no uni-
versal bound that is continuous in Ω. On the other hand, if F is suitably degenerate
in the normal direction at ∂Ω, one can sometimes find such a supersolution.
7.D. A remark (BC) in the classical sense. The reader will have noticed by
now that we gave some examples of solutions of (BC) that were smooth but did
not satisfy the boundary condition in the classical sense. In these examples, the
equation (E) is rather degenerate. In fact, “degeneracy” of a sort is necessary for
this to happen, as we now establish.
Proposition 7.11. Let B ∈ C(∂Ω × R × RN ), F be proper, and u ∈ C2(Ω) be a
subsolution (respectively, supersolution) of (BVP). If
(7.28)
{
lim supµ→∞ F (x, r, p,X − µn⊗ n) > 0
for (x, r, p,X) ∈ ∂Ω× R × RN × S(N), n ∈ RN\{0}
(respectively,
(7.29)
{
lim infµ→∞ F (x, r, p,X + µn⊗ n) < 0
for (x, r, p,X) ∈ ∂Ω× R × RN × S(N), n ∈ RN\{0}),
then u is a classical subsolution (respectively, supersolution) of (BC).

USER’S GUIDE TO VISCOSITY SOLUTIONS 47
Proof. It will suffice to treat the subsolution case. We rely again on Remarks
2.7(iii). Let y ∈ RN\Ω and z ∈ ∂Ω be a nearest point to y in Ω. It is clear
that the set of such nearest points (as y varies) is dense in ∂Ω. Then Ω ⊂ {x ∈
RN : |x − y| ≥ |y − z|} = O where the last equality is the definition of O; now
this inclusion evidently implies J2,+O Zero ⊂ J
2,+
Ω Zero(z). On the other hand, we
computed J2,+O Zero(z) in Remark 2.7—with the notations n = (y − z)/|y − z| and
r = |y − z|, this computation shows that
(
−λn, λr I + µn⊗ n
)
∈ J2,+O Zero(z)
for λ > 0 and µ ∈ R. Thus the assumption that u is a supersolution implies that
B(z, u(z), Du(z)− λn) ∧ F
(
z, u(z), Du(z)− λn,D2u(z) + λr I + µn⊗ n
)
≤ 0
for λ > 0 and µ ∈ R. Taking the limit superior as µ→ −∞ and using the fact that
assumption (7.28) implies B(z, u(z), Du(z) − λn) ≤ 0 for λ > 0 and then letting
λ ↓ 0 we find B(z, u(z), Du(z)) ≤ 0. Since z was an arbitrary “nearest point” and
these are dense in ∂Ω, the proof is complete. The reader will notice that it suffices
to choose n in (7.28) and (7.29) from NΩ(x).
7.E. Fully nonlinear boundary conditions. We conclude this section by de-
scribing an extremely general existence and uniqueness result for a fully nonlinear
first-order boundary operator of the form B = B(x, p), B ∈ C(∂Ω× RN ). We will
require that B satisfies
(7.30) B is uniformly continuous in p uniformly in x ∈ ∂Ω
and
(7.31) |B(x, p)−B(y, p)| ≤ ω(|x− y|(1 + |p|))
for some ω : [0,∞) → [0,∞) satisfying ω(0+) = 0 and for some ν > 0
(7.32) B(x, p+ λn(x)) ≥ B(x, p) + νλ for x ∈ ∂Ω, λ ≥ 0, p ∈ RN .
We will also need to strengthen a bit the regularity of Ω by assuming that
(7.33) ∂Ω is of class C1,1.
Observe that (7.33) is stronger than (7.12) and (7.13) since Ω is bounded. The
loss of generality, however, is not really more restrictive for applications, since Ω
typically is either smooth or has “corners,” and corners are not allowed by (7.12)
in any case. (Corners typically require some ad hoc analysis.)
We have
Theorem 7.12. Let (7.30)–(7.33) and (7.14)–(7.16) hold. If u is a subsolution of
(BVP) and v is a supersolution of (BVP), then u ≤ v in Ω. Moreover, (BVP) has
a unique solution.

USER’S GUIDE TO VISCOSITY SOLUTIONS 49
of appropriate super and subsolutions. It was also noted in M. G. Crandall and R.
Newcomb [58] and P. E. Souganidis [156] that some part of the boundary may be
irrelevant (in the case of a first-order equation).
Progress on the understanding of Dirichlet boundary conditions was stimulated
by the “state constraints” problems studied first by M. Soner [149, 150] and later
by I. Capuzzo-Dolcetta and P. L. Lions [44]. This led to the “true” viscosity for-
mulation of Dirichlet conditions as considered in H. Ishii [90] and G. Barles and
B. Perthame [25, 26]. This formulation, in some sense, automatically selects the
relevant part of the boundary for degenerate problems and yields uniqueness for
continuous solutions. However, the existence of a continuous solution fails in gen-
eral (Example 7.8), and the situation is not entirely clear except for first-order
optimal control problems [25, 26, 90].
We also mention at this stage that some uniqueness results for semicontinuous
solutions have begun to emerge (G. Barles and B. Perthame [25, 26] and E. N.
Barron and R. Jensen [33, 34]) for optimal control problems and the study of some
second-order nondegenerate state constraints problems (J. M. Lasry and P. L. Lions
[111]).
Finally, the possibility of solving general equations with general fully nonlinear
oblique derivative type boundary conditions—a rather startling fact—was illus-
trated in G. Barles and P. L. Lions [24] for first-order equations. The full second-
order result of Theorem 7.12 is a generalization by G. Barles [21] of a similar result
of H. Ishii [91].
8. Parabolic problems
In this section we indicate how to extend the results of the preceeding sections
to problems involving the parabolic equation
(PE) ut + F (t, x, u,Du,D2u) = 0
where now u is to be a function of (t, x) andDu,D2umean Dxu(t, x) and D2xu(t, x).
We do this by discussing comparison for the Cauchy-Dirichlet problem on a bounded
domain; it will then be clear how to modify other proofs as well. Let O be a locally
compact subset of RN , T > 0, and OT = (0, T ) × O. We denote by P2,+O , P2,−O
the “parabolic” variants of the semijets J2,+O , J
2,−
O ; for example, if u : OT → R then
P2,+O u is defined by (a, p,X) ∈ R × RN × S(N) lies in P2,+O u(s, z) if (s, z) ∈ OT
and
(8.1)
u(t, x) ≤u(s, z) + a(t− s) + 〈p, x− z〉+ 12 〈X(x− z), x− z〉
+ o(|t− s|+ |x− z|2) as OT ∋ (t, x) → (s, z);
similarly, P2,−O u = −P2,+O (−u). The corresponding definitions of P
2,+
O , P
2,−
O are
then clear.
A subsolution of (PE) on OT is a function u ∈ USC(OT ) such that
(8.2) a+ F (t, x, u(t, x), p,X) ≤ 0 for (t, x) ∈ OT and (a, p,X) ∈ P2,+O u(t, x);
likewise, a supersolution is a function v ∈ LSC(OT ) such that
(8.3) a+ F (t, x, v(t, x), p,X) ≥ 0 for (t, x) ∈ OT and (a, p,X) ∈ P2,−O v(t, x);

USER’S GUIDE TO VISCOSITY SOLUTIONS 51
Observe that the condition (8.5) is guaranteed by having each ui be a subsolution
of a parabolic equation.
Proof of Theorem 8.2. We first observe that for ε > 0, u˜ = u − ε/(T − t) is also a
subsolution of (8.4) and satisfies (PE) with a strict inequality; in fact,
u˜t + F (t, x, u˜,Du˜,D2u˜) ≤ −
ε
(T − t)2 .
Since u ≤ v follows from u˜ ≤ v in the limit ε ↓ 0, it will simply suffice to prove the
comparison under the additional assumptions
(8.7)



(i) ut + F (t, x, u,Du,D2u) ≤ −ε/T 2 < 0 and
(ii) lim
t↑T
u(t, x) = −∞ uniformly on Ω.
We will assume
(8.8) (s, z) ∈ (0, T )× Ω and u(s, z)− v(s, z) = δ > 0
and then contradict this assumption. We may assume that u,−v are bounded
above. Let (tˆ, xˆ, yˆ) be a maximum point of u(t, x) − v(t, y) − (α/2)|x − y|2 over
[0, T )×Ω×Ω where α > 0; such a maximum exists in view of the assumed bound
above on u, −v, the compactness of Ω, and (8.7)(ii). The purpose of the term
(α/2)|x− y|2 is as in the elliptic case. Set
(8.9) Mα = u(tˆ, xˆ)− v(tˆ, yˆ)−
α
2
|xˆ− yˆ|2.
By (8.8), Mα ≥ δ. If tˆ = 0, we have
0 < δ ≤Mα ≤ sup
Ω×Ω
(
ψ(x)− ψ(y)− α
2
|x− y|2
)
;
however, the right-hand side above tends to zero as α→ ∞ by Lemma 3.1, so tˆ > 0
if α is large. Likewise, xˆ, yˆ ∈ Ω if α is large by u ≤ v on [0, T )× ∂Ω. Thus we may
apply Theorem 8.3 at (tˆ, xˆ, yˆ) to learn that there are numbers a, b and X,Y ∈ S(N)
such that
(a, α(xˆ − yˆ), X) ∈ P2,+O u(tˆ, xˆ), (b, α(xˆ− yˆ), Y ) ∈ P
2,−
O v(tˆ, yˆ)
such that
(8.10) a− b = 0 and − 3α
(
I 0
0 I
)
≤
(
X 0
0 −Y
)
≤ 3α
(
I −I
−I I
)
.
The relations
a+ F (tˆ, xˆ, u(tˆ, xˆ), α(xˆ − yˆ), X) ≤ −c,
b+ F (tˆ, yˆ, v(tˆ, yˆ), α(xˆ − yˆ), Y ) ≥ 0,
and (8.10) imply
c ≤ F (tˆ, yˆ, v(tˆ, yˆ), α(xˆ − yˆ), Y )− F (tˆ, xˆ, u(tˆ, xˆ), α(xˆ − yˆ), X)
≤ ω(α|xˆ − yˆ|+ |xˆ− yˆ|)

52 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
which leads to a contradiction as in the proof of Theorem 3.3.
Let us mention a couple of other adaptations of results above to parabolic prob-
lems. Section 5.B may be regarded as establishing continuity of solutions with
respect to boundary data and the equation itself. In the parabolic case, there is the
initial data, the boundary data, and the equation to consider. In the above context,
we may consider a solution of ut+F (t, x,Du,D2u) ≤ 0 in (0, T )×Ω and a solution
of vt + G(t, x,Dv,D2v) ≥ 0 in (0, T ) × Ω with the continuity and boundedness
properties assumed in Theorem 8.2. Suppose g(t) ≥ (G(t, x, p,X)− F (t, x, p,X))+
where g is continuous and (u(t, x) − v(t, x))+ ≤ K1 for (t, x) ∈ (0, T ) × ∂Ω,
(u(0, x) − v(0, x))+ ≤ K2 for x ∈ Ω. Then the function w(x, t) = v(x, t) +
max(K1,K2) +
∫ t
0 g(s) ds is a solution of wt + F (t, x,Dw,D2w) ≥ 0. In this way,
if we have comparison, we conclude that
u(t, x) ≤ v(t, x) + max
(
sup
(0,T )×∂Ω
(u− v)+, sup
{0}×Ω
(u− v)+
)
+
∫ t
0
g(s) ds.
In many cases we may put
g(t) = sup
RN×RN×S(N)
(G(t, ·)− F (t, ·))+,
and a simple example is G(t, x, p,X) = F (t, x, p,X) + f(t, x).
Notes on §8. The presentation follows M. G. Crandall and H. Ishii [48]. Let
us also mention that it is well recognized that most results concerning stationary
equations have straightforward parabolic analogues; indeed, the parabolic situation
is often better, since
ut + F (x, u,Du,D2u) = 0
more or less corresponds to λu+F (x, u,Du,D2u) = 0 with large λ > 0. Moreover,
the special linear dependence on ut in the parabolic case allows one to let F depend
on t in a merely measurable manner; see, e.g., H. Ishii [84], B. Perthame and P. L.
Lions [129], N. Barron and R. Jensen [32], and D. Nunziante [137, 138]. Another
special feature of the parabolic case is that, owing to the special structure, one
can often allow rather singular initial data. For example, infinite values may be
allowed and semicontinuity may suffice, depending on the situation (see, e.g., M.
G. Crandall, P. L. Lions, and P. E. Souganidis [57] and E. N. Barron and R. Jensen
[33, 34]).
9. Singular equations: an example from geometry
Let p⊗ q = {piqj}, the matrix with entries piqj . It can be shown that if ψ is a
smooth function and Dψ does not vanish on the level set Γ = {ψ = c} and u is a
classical solution of the Cauchy problem
(9.1) ut − trace
((
I − Du⊗Du|Du|2
)
D2u
)
= 0, u(0, x) = ψ(x)
on some strip (0, T )×RN , then Γt = {u(t, ·) = c} represents the result of evolving Γ
according to its mean curvature to the time t, whence there is geometrical interest
in (9.1). Indeed, in less regular situations where we have viscosity solutions, Γt has

USER’S GUIDE TO VISCOSITY SOLUTIONS 53
been proposed as a definition of the result of evolving Γ in this way. The nonlinearity
involved, which we hereafter denote F (p,X) = − trace((I − (p ⊗ p)/|p|2)X), is
degenerate elliptic on the set p 6= 0 and undefined at p = 0. Thus the results of
the preceeding sections do not apply immediately. However, this is easy to remedy
using the special form of the equation. The extensions of F to (0, X) given by
(9.2) F (p,X) =
{ F (p,X) if p 6= 0,
−2‖X‖ if p = 0, F (p,X) =
{ F (p,X) if p 6= 0,
2‖X‖ if p = 0
are lower semicontinuous and upper semicontinuous respectively. We define u to
be a subsolution (respectively, supersolution) of ut + F (Du,D2u) = 0 if it is a
subsolution of ut + F (Du,D2u) = 0 (respectively, of ut + F (Du,D2u) = 0) and a
solution if it is both a subsolution and a supersolution. The reason that this will
succeed is roughly that, in the analysis, we will need only to insert X = 0 when we
have to deal with p = 0, so what one does with p = 0 is not important so long as
it is consistent. To illustrate matters in a slightly simpler setting, let us consider
instead the stationary problem
(9.3) u+ F (Du,D2u)− f(x) = 0 in RN ,
where we use the corresponding definitions of subsolutions, etc. Theorem 5.1 re-
mains valid for the current F .
Theorem 9.1. Let f ∈ UC(RN ). Then ((9.3) has a unique solution u ∈ UC(RN ).
We sketch the proof, which proceeds according to the outline given in the proof
of Theorem 5.1 with slight twists.
Proof of comparison. The comparison proof is a slight modification of that of §5.D.
We begin assuming that u, v are a subsolution and a supersolution of (9.3) and
u(x)− v(y) ≤ L(1 + |x|+ |y|) (which is (5.11), and then proceed as before, ending
up with
(9.4)
u(xˆ)− v(yˆ) ≤ f(xˆ)− f(yˆ) + F (pˆ−DβR(yˆ),−Zˆ −D2βR(yˆ))
−F (pˆ+DβR(xˆ), Zˆ +D2βR(xˆ))
in place of (5.14). This still implies a bound on u(x)− v(y)− 2K|x− y| as before.
Then proceeding still further with the proof, we adapt slightly and consider a
maximum point (xˆ, yˆ) of
Φ(x, y) = u(x)− v(y)− (α|x − y|4 + ε(|x|2 + |y|2)),
which will exist by virtue of the bound already obtained. We assume, without loss
of generality for what follows, that Φ(xˆ, yˆ) ≥ 0 so that for some C
(9.5) α|xˆ− yˆ|4 + ε(|xˆ|2 + |yˆ|2) ≤ u(xˆ)− v(yˆ) ≤ 2K|xˆ− yˆ|+ C.
Using Remark 3.8, we then have the existence of X,Y ∈ S(N) such that
(9.6) (pˆ+ 2εxˆ,X + 2εI) ∈ J2,+u(xˆ), (pˆ− 2εyˆ, Y − 2εI) ∈ J2,−v(yˆ)

54 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
where
(9.7)
pˆ = 4α|xˆ− yˆ|2(xˆ− yˆ),
‖X‖, ‖Y ‖ ≤ C1α|xˆ− yˆ|2 and X ≤ Y.
Now (9.5) implies that ε(|xˆ|2 + |yˆ|2) and α|xˆ − yˆ|3, and hence pˆ, are bounded
independently of ε ≤ 1 for fixed α ≥ 1 while α|xˆ− yˆ|4 is also bounded independently
of α. Hence, εxˆ, εyˆ → 0 as ε ↓ 0. We have the following analogue of (5.18)
u(xˆ)− v(yˆ) ≤ (F (pˆ− 2εyˆ, Y − 2εI)− F (pˆ+ 2εxˆ,X + 2εI)) + f(yˆ)− f(xˆ)
that leads, by use of the estimates above, to
lim sup
ε↓0
(u(xˆ)− v(yˆ)) ≤ F (p, Y0)− F (p,X0) + κ(C/(α)1/4)
where C is some constant and (p,X0, Y0) is a limit point of (pˆ, X, Y ) as ε ↓ 0. If
p 6= 0, we are done since
F (p, Y0)− F (p,X0) = F (p, Y0)− F (p,X0) ≤ 0
because X0 ≤ Y0. If p = 0, we use the information 4α|xˆ − yˆ|2(xˆ − yˆ) → p = 0
and (9.7) (recall α is fixed) to conclude that X0 = Y0 = 0, and then F (p, Y0) =
F (p,X0) = 0, and we are still done.
Proof that solutions lie in UC(RN ). This is exactly as in §5.D.
Proof of existence. A supersolution and subsolution are produced exactly as in
§5.D. Perron’s method still applies here, since F is lower semicontinuous and, we
may use Lemma 4.4 as is with F replaced by F and the variant of Lemma 4.4 given
in Remark 4.5 with G− = F and G+ = F .
Notes on §9. As shown in §1, a number of equations arising from geometrical
considerations present singularities at p = 0. The fact that this can easily be
circumvented was shown independently by L. C. Evans and J. Spruck [72] and Y.
Chen, Y. Giga, and S. Goto [45]. We also mention the work of H. M. Soner [152]
on the equation (9.1) and the papers by G. Barles [16] and S. Osher and J. Sethian
[140], which showed how various geometrical questions about “moving fronts” could
be reduced to equations that can be handled by viscosity theory. A general class
of singular equations is treated in Y. Giga, S. Goto, H. Ishii, and M. H. Sato [80]
and M. H. Sato [146]; these works establish existence and uniqueness as well as
convexity properties of solutions.
10. Applications and perspectives
In this section we list some applications of the theory of viscosity solutions and
indicate some of the promising directions for development of the theory in the next
few years. We give some important references but they are not exhaustive.
To begin this rather long list of applications, we recall that perhaps the main
motivation for developing the theory was its relevance for the theories of Optimal
Control and Differential Games. Indeed, as is well known, in the theory of optimal
control of ordinary differential equations or stochastic differential equations (with

USER’S GUIDE TO VISCOSITY SOLUTIONS 55
complete observations) or in the theory of zero sum, two player deterministic or
stochastic differential games, the Dynamic Programming Principle (DPP for short)
states that the associated value functions should be characterized as the solutions
of associated partial differential equations. These equations are called Bellman or
or Hamilton-Jacobi-Bellman (HJB for short) equations in control theory or Isaacs
equations in differential games. The DPP was, however, heuristic and proofs of
it required more regularity of the value functions than they usually enjoy. The
flexibility of the theory of viscosity solutions has completely filled this regularity
gap: roughly speaking, value functions are viscosity solutions and are uniquely
determined by this fact (via the uniqueness of viscosity solutions). See, e.g., [70,
116, 118, 119, 153]. This basic theoretical fact allows a spectacular simplification
of the theory of deterministic differential games [70, 153] and also provided the
possibility of creating sound mathematical foundations for stochastic differential
games [77]. See also [88, 136] in addition to [29, 30].
The generalization of the definition of viscosity solutions to systems in diag-
onal form is rather straightforward and has applications to optimal control and
differential games. For these topics see, e.g., [43, 63, 92, 95, 114, 147]. In the
case of systems, a combination of viscosity solutions and weak solutions based on
distribution theory may define a natural notion of weak solutions [80].
More generally, as usual, a better understanding of existence-uniqueness issues
for classes of equations leads to a better understanding of more specific issues.
Typical examples here are perturbation questions, asymptotic problems, and a more
detailed solution of some specific applications to Engineering or Finance problems
[35, 161, 162]. Also, the part of the theory concerned with boundary conditions
has led to a rather complete theory for problems with state-constraints (at least for
deterministic problems) that are enforced by cost or boundary mechanisms [149,
150, 44, 90, 25, 26, 33, 34]. Let us also mention that an interesting link between
viscosity solutions and the other main argument of Control Theory, the Pontryagin
principle, has been shown [31, 9]. Of course, last but not least, these results have
led to numerical approaches to Control or Differential Games problems via the
resolution of the HJB (or Isaacs) equation.
Indeed, the viscosity solutions theory is intimately connected with numerical
analysis and scientific computing. First of all, it provides efficient tools to per-
form convergence analyses (e.g., [52, 28, 133, 155]). It also indicates how to build
discretization methods or schemes for other general boundary conditions and in
particular for classical boundary conditions when working with rather degenerate
equations [73, 144].
Another consequence of efficient existence, uniqueness, approximation, and con-
vergence results is the possibility of establishing or discovering various qualitative
properties of solutions (formulae, representations, singularities, geometrical prop-
erties, characterizations and properties of semigroups, . . .—[12, 40, 41, 54, 56, 70,
88, 124, 131]). Of course, one of the most important qualitative properties is the
regularity of solutions. Viscosity solutions, because of their flexibility and their
pointwise definition, have led to regularity results that are spectacular either in
their generality (regularizing effects, Lipschitz regularity, or semicontinuity—[122,
116, 57, 20, 18, 17]) or by their originality (C1,α, C2,α or pointwise Lp estimates—
[39, 96, 158, 159, 160, 103]).
The uniqueness and convergence parts of the theory have made possible partial
differential equations approaches to various asymptotic problems like large devia-

56 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
tions [68, 76, 11, 141], geometrical optics [71, 23, 22, 13, 69], or homogenization
problems [66, 123, 128] by arguments that are both powerful and simple.
More specific applications concern the interplay between the behavior of so-
lutions at infinity and structure conditions [83, 54, 56, 134] or the treatment of
integrodifferential operators [112, 148]. Other applications concern some particular
classes of equations arising in Engineering like some models of the propagation of
fronts in Combustion Theory [16, 140], or the so-called shapes from shading models
in Vision Theory [144].
Finally, a large part of the theory (but not yet all of it) has been “raised”
to infinite-dimensional equations both for first-order and second-order equations
[55, 125–127]. In addition to “standard” extensions to infinite-dimensional spaces,
specific applications like the optimal control of partial differential equations or
even stochastic partial differential equations—this last topic being motivated in
particular by the optimal control of the so-called Zakai’s operation, a well-known
formulation of optimal control problems of finite-dimensional diffusion processing
with partial observations—require some new developments of the viscosity solutions
theory in order to accommodate unbounded terms in the equations [55, Parts IV
and V; 126, 8, 7, 151, 42, 157].
The above rather vague and general comments on applications already contain
many hints concerning promising directions the theory of viscosity solutions may
take in the near future. In particular, the infinite-dimensional part of the theory
will most probably explode in view of the unbounded possible avenues of inves-
tigation. Much more progress is also to be expected for degenerate second-order
equations and boundary conditions. In particular, progress is to be made on ex-
istence questions for Dirichlet (and state constraints) boundary conditions—this
might have a considerable impact on various applications like models in Finance.
Similarly, we expect progress on uniqueness and regularity questions for uniformly
elliptic second-order equations.
It is also reasonable to hope that the insight gained by viscosity solutions will
help to devise efficient high-order schemes for numerical approximations and prove
their convergence.
More specific developments should (and will) concern questions of behavior at
infinity, geometrical optics problems, the use and the theory of discontinuous solu-
tions, and the investigation of “second-order” integrodifferential operators . . . .
Of course, the reader should not restrict his imagination to the borders we drew
above.
Appendix. The proof of Theorem 3.2
In this section we sketch the proof of Theorem 3.2, which we reproduce here for
convenience.
Theorem 3.2. Let Oi be a locally compact subset of RNi for i = 1, . . . , k,
O = O1 × · · · × Ok,
ui ∈ USC(Oi), and ϕ be twice continuously differentiable in a neighborhood of O.
Set
w(x) = u1(x1) + · · ·+ uk(xk) for x = (x1, · · · , xk) ∈ O,

60 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
for J(x) ∈ F3. However, if we let ψ(δy) = ϕ(J(x) + δy) and ψ˜(δy) = ϕ(J(x)) +
Dϕ(Jx)δy+ 12 〈(DJ(x)−1 − I)δy, δy〉, we have ψ(0) = ψ˜(0) and for almost all small
δy
Dψ(δy) = Dϕ(J(x) + δy) = Dϕ(J(x)) + (DJ(x))−1δy − δy + o(δy)
= Dψ˜(δy) + o(δy);
thus Ψ(δy) = ψ(δy) − ψ˜(δy) is locally Lipschitz continuous and satisfies Ψ(0) = 0
and DΨ(δy) = o(δy) for almost all small δy. It is clear that then Ψ(δy) = o(|δy|2),
whence the result.
The next result we will need concerning semiconvex functions, which we call
Jensen’s lemma, follows. In the statement, B(x, r) is the closed ball of radius r
centered at x and Br is the ball centered at the origin.
Lemma A.3. Let ϕ : RN → R be semiconvex and xˆ be a strict local maximum
point of ϕ. For p ∈ RN , set ϕp(x) = ϕ(x) + 〈p, x〉. Then for r, δ > 0,
K = {x ∈ B(xˆ, r) : there exists p ∈ Bδ for which ϕp has a local maximum at x}
has positive measure.
Proof. We assume that r is so small that ϕ has xˆ as a unique maximum point in
B(xˆ, r) and assume for the moment that ϕ is C2. It follows from this that if δ is
sufficiently small and p ∈ Bδ, then every maximum of ϕp with respect to B(xˆ, r)
lies in the interior of B(xˆ, r). Since Dϕ + p = 0 holds at maximum points of ϕp,
Dϕ(K) ⊃ Bδ. Let λ ≥ 0 and ϕ(x)+(λ/2)|x|2 be convex; we then have −λI ≤ D2ϕ;
moreover, on K, D2ϕ ≤ 0 and then
−λI ≤ D2ϕ(x) ≤ 0 for x ∈ K.
In particular, | detD2ϕ(x)| ≤ λN for x ∈ K. Thus
meas(Bδ) ≤ meas(Dϕ(K)) ≤
∫
K
| detD2ϕ(x)|dx ≤ meas(K)|λ|N
(see the notes) and we have a lower bound on the measure of K depending only on
λ.
In the general case, in which ϕ need not be smooth, we approximate it via
mollification with smooth functions ϕε that have the same semiconvexity constant
λ and that converge uniformly to ϕ on B(xˆ, r). The corresponding sets Kε obey
the above estimates for small ε and
K ⊃
∞⋂
n=1
∞⋃
m=n
K1/m
is evident. The result now follows.
Step 3. A consequence of Step 2 and magic properties of sup convolution.
Lemma A.4 below applies to wˆ of Step 1 and we shall see it provides us with matrices
Xi ∈ S(Ni) such that (0, Xi) ∈ J
2,+uˆi(0) and (A.3) holds.
We use the notation
(A.8) J2f(z) = J2,+f(z) ∩ J2,−f(z)
from which one defines J2 analogously to J2,+, J2,−. Note that (p,X) ∈ J2f(x)
amounts to
f(y) = f(x) + 〈p, y − x〉+ 12 〈X(y − x), y − x〉+ o(|x− y|2) as y → x;
i.e., f is twice differentiable at x and p = Df(x), X = D2f(x).

62 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
while substituting x = y and ξ = η + α(λ(η − y) + q) yields
(A.11) 0 ≤ α|λ(η − y) + q|2 +O(α2).
The former simply says that (q, Y ) ∈ J2,+v(y) while the latter with small α < 0
implies that λ(η − y) + q = 0 or y = η + q/λ as claimed. The relation vˆ(η) +
(1/2λ)|q|2 = v(η+ q/λ) follows at once. Assume now that (qn, Yn) ∈ J2,+vˆ(ξn) and
(ξn, vˆ(ξ), qn, Yn) → (0, 0, 0, Y ); by the foregoing, (qn, Yn) ∈ J2,+v(ξn + qn/λ) and
v(ξn + qn/λ) = vˆ(ξn) + (1/2λ)|qn|2. By the definitions, v(0) ≤ vˆ(0) and from this,
the upper semicontinuity and the foregoing, we have
v(0) ≥ lim sup
n→∞
v(ξn +
qn
λ ) = lim supn→∞
(vˆ(ξn) +
1
2λ |qn|
2) = vˆ(0) ≥ v(0),
which provides the final piece of information we needed to conclude that (0, Y ) ∈
J2,+v(0).
Remark A.6. We close with a final and more elegant reformulation of what was
proved above: if the ui satisfy the conditions of Theorem 3.2, w(x) = u1(x1) +
· · · + uk(xk) and ((p1, . . . , pk), A) ∈ J
2,+
Ω w(xˆ), then for each ε > 0, there exists
Xi ∈ S(Ni) such that (pi, Xi) ∈ J
2,+
O ui(xˆi) and (A.1) holds.
Notes on the appendix. Above, we provided (for the first time) a self-contained
proof of Theorem 3.2. Except for the two auxiliary results on semiconvex functions,
the main tool is the so-called sup convolution. This approximation procedure (more
often in the guise of inf convolution) is well known in functional analysis and, in
particular, in convex analysis and the theory of maximal monotone operators (see,
for example, the text [38] of H. Brezis). It was noticed in [110] that it may provide
an efficient regularization procedure for (even degenerate) elliptic equations; some
of its properties are given there. See also [104]. Its “magical properties” can be
seen as related to the Lax formula for the solution of
∂w
∂t −
1
2
|∇w|2 = 0 for x ∈ RN , t ≥ 0, w|t=0 = v on RN ,
which is
w(x, t) = sup
y
{
v(y)− 1
2t |x− y|
2
}
.
Indeed, the coincidence of this solution formula and solutions produced by the
method of charteristics leads to the properties used. Of course, this is a heuristic
connection, since characteristic methods require too much regularity to be rigorous
here.
The inf convolution can also be seen as a nonlinear analogue of the standard
mollification when replacing the “linear structure of L2 and its duality” by the
“nonlinear structure of L∞ or C.” One can also interpret this analogy in terms of
the so-called exotic algebra (R, max, +).
Theorem A.2 is a classical result of A. D. Aleksandrov [1]. The proof given here
is in fact a slightly stronger form of this result (Dϕ is differentiable a.e.) and our
proof follows F. Mignot [135] (although we have made his proof more complicated
for pedagogical reasons). In the proof we used the fact that Lipschitz functions are

USER’S GUIDE TO VISCOSITY SOLUTIONS 63
differentiable a.e., which is called Rademacher’s Theorem. A proof may be found in
L. C. Evans and R. Gariepy [67] and F. Mignot [135]. We also used that Lipschitz
functions map null sets to null sets and the solvability of J(x + δy) = J(x) + δx
when DJ(x) exists and is nonsingular. These are proved in, respectively, Lemma
7.25 and the proof of Theorem 7.24 in W. Rudin [145]; see also [135]. We also
used the fact that J({x : DJ(x) exists and is singular}) is null. This follows from
the general formula
∫
RN
#(A ∩ J−1(y)) dy =
∫
A
| detDJ(x)| dx
where # is counting measure. This formula holds for Lipschitz continuous func-
tions and measurable sets A ⊂ RN and is a special case of the Area Formula for
Lipschitzian maps between Euclidean spaces. The Area Formula may be found in
L. C. Evans and R. Gariepy [67] and H. Federer [74].
Jensen’s Lemma (Lemma A.3) is a variation on an aspect of the theme known
as “Aleksandrov’s maximum principle” (see [2, 3, 6, 37, 120, 143]).
References
1. A. D. Aleksandrov, Almost everywhere existence of the second differential of a convex
function and some properties of convex functions, Leningrad. Univ. Ann. (Math. Ser.) 37
(1939), 3–35. (Russian)
2. , Uniqueness conditions and estimates for the solution of the Dirichlet problem,
Amer. Math. Soc. Transl., vol. 68, Amer. Math. Soc., Providence, RI, 1968, pp. 89–119.
3. , Majorization of solutions of second-order linear equations, Amer. Math. Soc.
Transl., vol. 68, Amer. Math. Soc., Providence, RI, 1968, pp. 120–143.
4. R. Alziary de Roquefort, Jeux de poursuite et approximation des fonctions valeur, The`se,
Universite` de Paris-Dauphine, 1990; Mode´l. Math. Anal. Num. 25 (1991), 535–560.
5. S. Aizawa and Y. Tomita, On unbounded viscosity solutions of a semilinear second order
elliptic equation, Funkcial. Ekvac. 31 (1988), 141–160.
6. I. Ja. Bakelman, On the theory of quasilinear elliptic equations, Sibirsk. Mat. Z. 2 (1961),
179–186.
7. V. Barbu, Hamilton-Jacobi equations and nonlinear control problems, preprint.
8. , The dynamic programming equation for the time optimal control problem in infi-
nite dimensions, SIAM J. Control Optim. 29 (1991), 445–456.
9. V. Barbu, E. N. Barron, and R. Jensen, Necessary conditions for optimal control in Hilbert
spaces, J. Math. Anal. Appl. 133 (1988), 151–162.
10. V. Barbu and G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces, Research Notes
in Math., vol. 86, Pitman, Boston, MA, 1983.
11. M. Bardi, An asymptotic formula for the Green’s function of an elliptic operator, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1987), 569–612.
12. M. Bardi and L. C. Evans, On Hopf’s formulas for solutions of Hamilton-Jacobi equations,
Nonlinear Anal. Theory Methods Appl. 8 (1984), 1373–1381.
13. M. Bardi and B. Perthame, Exponential decay to stable states in phase transitions via a
double log-transformation, Comm. Partial Differential Equations 15 (1990), 1649–1669.
14. G. Barles, Existence results for first-order Hamilton-Jacobi equations, Ann. Inst. H. Poincare´
Anal. Non Line´aire 1 (1984), 325–340.
15. , Remarques sue des re´sultats d’existence pour les e´quations de Hamilton-Jacobi
du premier ordre, Ann. I. H. Poincare´ Anal. Non Line´aire 2 (1985), 21–33.
16. , Remark on a flame propagation model, Rapport INRIA, no. 464, 1985.
17. , Interior gradient bounds for the mean curvature equation by viscosity solution
methods, Differential Integral Equations 4 (1991), 263–275.
18. , A weak Bernstein method for fully nonlinear elliptic equations, Differential Inte-
gral Equations 4 (1991), 241–262.

64 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
19. , Uniqueness and regularity results for first-order Hamilton-Jacobi equations, Indi-
ana Univ. Math. J. 39 (1990), 443–466.
20. , Regularity results for first-order Hamilton-Jacobi equations, Differential Integral
Equations 3 (1990), 103–125.
21. , Fully nonlinear Neumann type boundary conditions for second-order elliptic and
parabolic equations, J. Differential Equations (to appear).
22. G. Barles, L. Bronsard, and P. E. Souganidis, Front propagation for reaction-diffusion
equations of bistable type, Analyse Non Line´aire (to appear).
23. G. Barles, L. C. Evans, and P. E. Souganidis, Wave front propagation for reaction-diffusion
systems of PDE’s, Duke Univ. Math. J. 61 (1990), 835–858.
24. G. Barles and P. L. Lions, Fully nonlinear Neumann type boundary conditions for first-
order Hamilton-Jacobi equations, Nonlinear Anal. Theory Methods Appl. 16 (1991), 143–
153.
25. G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping-
time problems, Mode`l. Math. Anal. Num. 21 (1987), 557–579.
26. , Exist time problems in optimal control and the vanishing viscosity method, SIAM
J. Control Optim. 26 (1988), 1133–1148.
27. , Comparison principle for Dirichlet-type H. J. equations and singular perturbations
of degenerate elliptic equations, Appl. Math. Optim. 21 (1990), 21–44.
28. G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear
second order equations, Asymp. Anal. 4 (1991), 271–283.
29. E. N. Barron, Differential games with maximum cost, Nonlinear Anal. Theory Methods
Appl. 14 (1990), 971–989.
30. E. N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost,
Nonlinear Anal. Theory Methods Appl. 13 (1989), 1067–1090.
31. E. N. Barron and R. Jensen, The Pontryagin maximum principle from dynamic pro-
gramming and viscosity solutions to first-order partial differential equations, Trans. Amer.
Math. Soc. 298 (1986), 635–641.
32. , Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable
Hamiltonians, J. Differential Equations 68 (1987), 10–21.
33. , Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex
Hamiltonians, Comm. Partial Differential Equations 15 (1990), 1713–1742.
34. , Optimal control and semicontinuous viscosity solutions, Proc. Amer. Math. Soc.
113 (1991), 397–402.
35. , A stochastic control approach to the pricing of options, Math. Operations Research
15 (1990), 49–79.
36. S. Benton, The Hamilton-Jacobi equation: A global approach, Academic Press, New York,
1977.
37. A. D. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Ser.
A 265 (1967), 333–336.
38. H. Brezis, Operateurs Maximaux Monotones et semigroupes de contractions dans les es-
paces de Hilbert, North-Holland, Amsterdam, 1973.
39. L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Ann.
of Math. (2) 130 (1989), 180–213.
40. P. Cannarsa and M. Soner, On the singularities of the viscosity solutions to Hamilton-
Jacobi-Bellman equations, Indiana Univ. Math. J. 46 (1987), 501–524.
41. , Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and
applications, Nonlinear Anal. Theory Methods Appl. 13 (1989), 305–323.
42. P. Cannarsa, F. Gozzi, and M. Soner, A boundary value problem for Hamilton-Jacobi
equations in Hilbert spaces, Appl. Math. Opt. 24 (1991), 197–220.
43. I. Cappuzzo-Dolcetta and L. C. Evans, Optimal switching for ordinary differential equa-
tions, SIAM J. Optim. Control 22 (1988), 1133–1148.
44. I. Cappuzzo-Dolcetta and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations
and state constraints, Trans. Amer. Math. Soc. 318 (1990), 643–683.
45. Y. G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of
generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749–786.
46. M. G. Crandall, Quadratic forms, semidifferential and viscosity solutions of fully nonlinear
elliptic equations, Ann. I. H. Poincare´ Anal. Non Line´aire 6 (1989), 419–435.

USER’S GUIDE TO VISCOSITY SOLUTIONS 65
47. M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of
Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502.
48. M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Dif-
ferential Integral Equations 3 (1990), 1001–1014.
49. M. G. Crandall, H. Ishii, and P. L. Lions, Uniqueness of viscosity solutions of Hamilton-
Jacobi equations revisited, J. Math. Soc. Japan 39 (1987), 581–596.
50. M. G. Crandall and P. L. Lions, Condition d’unicite´ pour les solutions generalise´es des
e´quations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Se´r. I Math. 292
(1981), 183–186.
51. , Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277
(1983), 1–42.
52. , Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43
(1984), 1–19.
53. , On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlin.
Anal. Theory Methods Appl. 10 (1986), 353–370.
54. , Unbounded viscosity solutions of Hamilton-Jacobi equations, Illinois J. Math. 31
(1987), 665–688.
55. , Hamilton-Jacobi equations in infinite dimensions, Part I. Uniqueness of viscosity
solutions, J. Func. Anal. 62 (1985), 379–396; Part II. Existence of viscosity solutions 65
(1986), 368–405; Part III 68 (1968), 214–247; Part IV. Unbounded linear terms 90 (1990),
237–283; Part V. B-continuous solutions 97 (1991), 417–465.
56. , Quadratic growth of solutions of fully nonlinear second order equations on Rn,
Differential Integral Equations 3 (1990), 601–616.
57. M. G. Crandall, P. L. Lions, and P. E. Souganidis, Maximal solutions and universal bounds
for some quasilinear evolution equations of parabolic type, Arch. Rat. Mech. Anal. 105
(1989), 163–190.
58. M. G. Crandall and R. Newcomb, Viscosity solutions of Hamilton-Jacobi equations at the
boundary, Proc. Amer. Math. Soc. 94 (1985), 283–290.
59. M. G. Crandall, R. Newcomb, and Y. Tomita, Existence and uniqueness of viscosity solu-
tions of degenerate quasilinear elliptic equations in Rn, Appl. Anal. 34 (1989), 1–23.
60. P. Dupuis and H. Ishii, On oblique derivative problems for fully nonlinear second order el-
liptic equations on nonsmooth domains, Nonlinear Anal. Theory Methods Appl. 12 (1991),
1123–1138.
61. , On oblique derivative problems for fully nonlinear second order elliptic PDE’s on
domains with corners, Hokkaido Math. J. 20 (1991), 135–164.
62. P. Dupuis, H. Ishii, and H. M. Soner, A viscosity solution approach to the asymptotic
analysis of queuing systems, Ann. Probab. 18 (1990), 226–255.
63. H. Engler and S. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-
Jacobi equations, Proc. London Math. Soc. 63 (1991), 212–240.
64. L. C. Evans, A convergence theorem for solutions of nonlinear second order elliptic equa-
tions, Indiana Univ. Math. J. 27 (1978), 875–887.
65. , On solving certain nonlinear differential equations by accretive operator methods,
Israel J. Math. 36 (1980), 225–247.
66. , The perturbed test function technique for viscosity solutions of partial differential
equations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 359–375.
67. L. C. Evans and R. Gariepy, Measure theory and fine properties of functions, Studies in
Advanced Math., CRC Press, Ann Arbor, 1992.
68. L. C. Evans and H. Ishii, A pde approach to some asymptotic problems concerning random
differential equations with small noise intensities, Ann. I. H. Poincare´ Anal. Non Line´aire
2 (1985), 1–20.
69. L. C. Evans, M. Soner, and P. E. Souganidis, The Allen-Cahn equation and generalized
motion by mean curvature, preprint.
70. L. C. Evans and P. E. Souganidis, Differential games and representation formulas for
solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J. 33 (1984), 773–
797.
71. , A PDE approach to geometric optics for semilinear parabolic equations, Indiana
Univ. Math. J. 38 (1989), 141–172.

66 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
72. L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom.
33 (1991), 635–681.
73. M. Falcone, A numerical approach to the infinite horizon problem of deterministic control
theory, Appl. Math. Optim. 15 (1987), 1–13.
74. H. Federer, Geometric measure theory, Springer, New York, 1969.
75. W. H. Fleming and R. Rishel, Deterministic and stochastic optimal control, Springer-
Verlag, New York, 1975.
76. W. H. Fleming and P. E. Souganidis, PDE-viscosity solution approach to some problems
of large deviations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 171–192.
77. , On the existence of value functions of two player, zero-sum stochastic differential
games, Indiana Univ. Math. J. 38 (1989), 293–314.
78. H. Frankowska, On the single-valuedness of Hamilton-Jacobi operators, Nonlin. Anal. The-
ory Methods Appl. 10 (1986), 1477–1483.
79. Y. Giga, S. Goto, H. Ishii, and M.-H. Sato, Comparison principle and convexity preserving
properties for singular degenerate parabolic equations on unbounded domains, Indiana U.
Math. J. 40 (1991), 443–470.
80. Y. Giga, S. Goto, and H. Ishii, Global existence of weak solutions for interface equations
coupled with diffusion equations, SIAM J. Math. Anal. (to appear).
81. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd
Ed., Springer-Verlag, New York, 1983.
82. H. Ishii, Remarks on existence and uniqueness of viscosity solutions of Hamilton-Jacobi
equations, Bull. Fac. Sci. Eng. Chuo Univ. 26 (1983), 5–24.
83. , Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, Indi-
ana Univ. Math. J. 33 (1984), 721–748.
84. , Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open
sets, Bull. Fac. Sci. Eng. Chuo Univ. 28 (1985), 33–77.
85. , Existence and uniqueness of solutions of Hamilton-Jacobi equations, Funkcial.
Ekvac. 29 (1986), 167–188.
86. , Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), 369–
384.
87. , A simple direct proof for uniqueness for solutions of the Hamilton-Jacobi equa-
tions of eikonal type, Proc. Amer. Math. Soc. 100 (1987), 247–251.
88. , Representation of solutions of Hamilton-Jacobi equations, Nonlin. Anal. Theory
Methods Appl. 12 (1988), 121–146.
89. , On uniqueness and existence of viscosity solutions of fully nonlinear second-order
elliptic PDE’s, Comm. Pure Appl. Math. 42 (1989), 14–45.
90. , A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 105–135.
91. , Fully nonlinear oblique derivative problems for nonlinear second-order elliptic
PDE’s, Duke Math. J. 62 (1991), 661–663.
92. , Perron’s method for monotone systems of second-order elliptic PDE’s, Diff. Inte-
gral Equations 5 (1992), 1–24.
93. H. Ishii and S. Koike, Remarks on elliptic singular perturbation problems, Appl. Math.
Optim. 23 (1991), 1–15.
94. , Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising
in switching games, Funkcial. Ekvac. 34 (1991), 143–155.
95. , Viscosity solutions for monotone systems of second-order elliptic PDE’s, Comm.
Partial Differential Equations 16 (1991), 1095–1128.
96. H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial
differential equations, J. Differential Equations 83 (1990), 26–78.
97. H. Ishii and N. Yamada, A remark on a system of inequalities with bilateral obstacles,
Nonlinear Anal. Theory Methods Appl. 13 (1989), 1295–1301.
98. A. V. Ivanov, Quasilinear degenerate and nonuniformly elliptic equations of second order,
Proc. Steklov Inst. Math. 1 (1984), 1–287.
99. S. Koike, On the rate of convergence of solutions in singular perturbation problems, J.
Math. Anal. Appl. 157 (1991), 277–292.
100. , An asymptotic formula for solutions of Hamilton-Jacobi-Bellman equations, Non-
linear Anal. Theory Methods Appl. 11 (1987), 429–436.

USER’S GUIDE TO VISCOSITY SOLUTIONS 67
101. R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order
partial differential equations, Arch. Rat. Mech. Anal. 101 (1988), 1–27.
102. , Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial dif-
ferential equations, Indiana Univ. Math. J. 38 (1989), 629–667.
103. , Local quadratic growth estimates for elliptic obstacle problems, preprint.
104. R. Jensen, P. L. Lions, and P. E. Souganidis, A uniqueness result for viscosity solutions
of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc. 102
(1988), 975–978.
105. R. Jensen and P. E. Souganidis, Regularity result for viscosity solutions of Hamilton-Jacobi
equations in one space dimension, Trans. Amer. Math. Soc. 301 (1987), 137–147.
106. J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order,
Comm. Pure Appl. Math. 20 (1967), 797–872.
107. S. N. Kruzkov, First order quasilinear equations in several independent variables, Math.
USSR-Sb. 10 (1970), 217–243.
108. N. V. Krylov, Controlled diffusion processes, Springer-Verlag, Berlin, 1980.
109. , Nonlinear elliptic and parabolic equations of second order, D. Reidel Publishing
Co., Boston, MA, 1987.
110. J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math.
55 (1986), 257–266.
111. , Nonlinear elliptic equations with singular boundary conditions and stochastic con-
trol with state constraints, I, The model problem, Math. Ann. 283 (1989), 583–630.
112. S. Lenhart, Integro-differential operators associated with diffusion processes with jumps,
Appl. Math. Optim. 9 (1982), 177–191.
113. , Viscosity solutions for weakly coupled systems of first order PDEs, J. Math. Anal.
Appl. 131 (1988), 180–193.
114. S. Lenhart and N. Yamada, Perron’s method for viscosity solutions associated with piecewise-
deterministic processes, Funkcial. Ekvac. (to appear).
115. , Viscosity solutions associated with switching game for piecewise-deterministic
processes, preprint.
116. P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math-
ematics 69, Pitman, Boston, MA, 1982.
117. , On the Hamilton-Jacobi-Bellman equations, Acta Appl. 1 (1983), 17–41.
118. , Some recent results in the optimal control of diffusion processes: Stochastic analy-
sis, Proceedings of the Taniguchi International Symposium on Stochastic Analysis (Katata
and Kyoto 1982), Kinokuniya, Tokyo, 1984.
119. , Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations.
Part 1: The dynamic programming principle and applications and Part 2: Viscosity so-
lutions and uniqueness, Comm. Partial Differential Equations 8 (1983), 1101–1174 and
1229–1276.
120. , A remark on the Bony maximum principle, Proc. Amer. Math. Soc. 88 (1983),
503–508.
121. , Neumann type boundary conditions for Hamilton-Jacobi equations, Duke J. Math.
52 (1985), 793–820.
122. , Regularizing effects for first-order Hamilton-Jacobi equations, Applic. Anal. 20
(1985), 283–308.
123. , Equations de Hamilton-Jacobi et solutions de viscosite`, Colloque De Giori, Paris,
1983; Pitman, London, 1985.
124. , Some properties of the viscosity semigroups of Hamilton-Jacobi equations, Non-
linear Differential Equations and Applications, Pitman, London, 1988.
125. , Viscosity solutions of fully nonlinear second-order equations and optimal stochas-
tic control in infinite dimensions. Part I: The case of bounded stochastic evolutions, Acta
Math. 161 (1988), 243–278.
126. , Viscosity solutions of fully nonlinear second-order equations and optimal stochas-
tic control in infinite dimensions. Part II: Optimal Control of Zakai’s Equation, Stochas-
tic Partial Differential Equations and Applications II, Lecture Notes in Math., vol. 1390,
Springer, Berlin, 1989. (G. DuPrato and L. Tubaro, eds.), Proceedings of the International
Conference on Infinite Dimensional Stochastic Differential Equations, Trento, Springer,
Berlin, 1988.

68 M. G. CRANDALL, HITOSHI ISHII, AND PIERRE-LOUIS LIONS
127. , Viscosity solutions of fully nonlinear second-order equations and optimal stochas-
tic control in infinite dimensions. Part III: Uniqueness of viscosity solutions of general
second order equations, J. Funct. Anal. 86 (1989), 1–18.
128. P. L. Lions, G. Papanicolau, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi
equations, preprint.
129. P. L. Lions and B. Perthame, Remarks on Hamilton-Jacobi-Equations with measurable
time-dependent Hamiltonians, Nonlinear Anal. Theory Methods Appl. 11 (1987), 613–
622.
130. P. L. Lions and M. Nisio, A uniqueness result for the semigroup associated with the
Hamilton-Jacobi-Bellman operator, Proc. Japan Acad. 58 (1982), 273–276.
131. P. L. Lions and Rochet, Hopf formula and multitime Hamilton-Jacobi equations, Proc.
Amer. Math. Soc. 90 (1980), 79–84.
132. P. L. Lions and P. E. Souganidis, Viscosity solutions of second-order equations, stochas-
tic control and stochastic differential games, Stochastic Differential Systems, Stochastic
Control Theory and Applications (W. H. Fleming and P. L. Lions, eds.), IMA Vol. Math.
Appl., vol. 10, Springer, Berlin, 1988.
133. , Convergence of MUSCL type methods for scalar conservation laws, C. R. Acad.
Sci. Paris Se´r. A 311 (1990), 259–264.
134. P. L. Lions, P. E. Souganidis, and J. Vazquez, The relation between the porous medium
equation and the eikonal equation in several space dimensions, Rev. Math. Ibero. 3 (1987),
275–310.
135. F. Mignot, Controˆle optimal dans les ine´quations variationelles elliptiques, J. Funct. Anal.
22 (1976), 130–185.
136. R. Newcomb, Existence and correspondence of value functions and viscosity solutions of
Hamilton-Jacobi equations, Univ. of Wisconsin-Madison, 1988.
137. D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second-order fully non-
linear parabolic equations with discontinuous time-dependence, Differential Integral Equa-
tions 3 (1990), 77–91.
138. , Existence and uniqueness of unbounded viscosity solutions of parabolic equations
with discontinuous time-dependence, preprint.
139. O. A. Ole˘ınik and E. V. Radekvic, Second order equations with nonnegative characteristic
form, American Mathematical Society, Providence, RI, 1973.
140. S. Osher and J. Sethian, Fronts propagating with curvature dependent speed : algorithms
based on Hamilton-Jacobi formulations, J. Comp. Physics 79 (1988), 12–49.
141. B. Perthame, Perturbed dynamical systems with an attracting singularity and weak vis-
cosity limits in H. J. equations, Trans. Amer. Math. Soc. 317 (1990), 723–747.
142. B. Perthame and R. Sanders, The Neumann problem for fully nonlinear second order
singular perturbation problems, SIAM J. Math. Anal. 19 (1988), 295–311.
143. C. Pucci, Su una limitazone per soluzioni di equazioni ellittche, Boll. Un. Mat. Ital. (6)
21 (1966), 228–233.
144. E. Rouy and A. Tourin, A viscosity solution approach to shape—from shading, preprint.
145. W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987.
146. M. Sato, Comparison principle for singular degenerate elliptic equations on unbounded
domains, Proc. Japan Acad. Ser. A 66 (1990), 252–256.
147. N. Yamada, Viscosity solutions for a system of inequalities with bilateral obstacles, Funk-
cial. Ekvac. 30 (1987), 417–425.
148. A. Sayah, Equations d’Hamilton-Jacobi du permier ordre avec termes inte´gro differentiels.
Partres I and II, Comm. Partial Differential Equations 16 (1991), 1057–1221.
149. M. Soner, Optimal control with state-space constraint I, SIAM J. Control Optim. 24 (1986),
552–562.
150. , Optimal control with state-space constraint II, SIAM J. Control Optim. 24 (1986),
1110–1122.
151. , On the Hamilton-Jacobi-Bellman equations in Banach spaces, J. Optim. Theory
Appl. 57 (1988), 121–141.
152. , Motion of a set by the curvature of its mean boundary, preprint.
153. P. E. Souganidis, Max-Min representations and product formulas for the viscosity solutions
of Hamilton-Jacobi equations with applications to differential games, Nonlin. Anal. Theory
Methods Appl. 9 (1985), 217–257.

USER’S GUIDE TO VISCOSITY SOLUTIONS 69
154. , Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equa-
tions 56 (1985), 345–390.
155. , Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J.
Differential Equations 59 (1985), 1–43.
156. , A remark about viscosity solutions at the boundary, Proc. Amer. Math. Soc. 96
(1986), 323–330.
157. D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear
terms, J. Math. Anal. Appl. 163 (1992), 345–392.
158. N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of
second order elliptic equations, Rev. Mat. Ibero 4 (1988), 453–468.
159. , Ho¨lder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc.
Edinburgh Sect. A 108 (1988), 57–65.
160. , On the twice differentiability of viscosity solutions of nonlinear elliptic equations,
Bull. Austral. Math. Soc. 39 (1989), 443–447.
161. J. L. Vila and T. Zariphopolou, Optimal consumption and portfolio choice with borrowing
constraints, preprint.
162. T. Zariphopolou, Investment-consumption model with transaction costs and Markov-Chain
parameters, SIAM J. Control Optim. 30, (to appear, 1992).
Department of Mathematics, University of California, Santa Barbara, Califor-
nia 93106
Department of Mathematics, Chuo University, Bunkyo-ku, Tokyo 112, Japan
Ceremade, Universite´ Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris
Cedex 16, France