Stationary level surfaces and Liouville-type theorems characterizing hyperplanes

Abstract

We consider an entire graph S : xN+1 = f(x), x ∈ ℝN in ℝN+1 of a continuous real function f over ℝN with N ≥ 1. Let Ω be an unbounded domain in ℝN+1 with boundary ∂Ω = S. Consider nonlinear diffusion equations of the form ∂tU = Δϕ(U) containing the heat equation ∂tU = ΔU. Let U = U(X,t) = U(x,xN+1,t) be the solution of either the initial-boundary value problem over Ω where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial datum is the characteristic function of the set ℝN+1∖Ω. The problem we consider is to characterize S in such a way that there exists a stationary level surface of U in Ω. We introduce a new class of entire graphs S and, by using the sliding method due to Berestycki, Caffarelli, and Nirenberg, we show that S ∈ A must be a hyperplane if there exists a stationary level surface of U in Ω. This is an improvement of the previous result (Magnanini and Sakaguchi in J. Differ. Equ. 252:236–257, 2012, Theorem 2.3 and Remark 2.4). Next, we consider the heat equation in particular and we introduce the class B of entire graphs S of functions f such that {|f(x) − f(y)| : |x − y|≤ 1} is bounded. With the help of the theory of viscosity solutions, we show that S ∈ B must be a hyperplane if there exists a stationary isothermic surface of U in Ω. This is a considerable improvement of the previous result (Magnanini and Sakaguchi in J. Differ. Equ. 248:1112–1119, 2010, Theorem 1.1, case (ii)). Related to the problem, we consider a class W of Weingarten hypersurfaces in ℝN+1 with N≥1. Then we show that, if S belongs to W in the viscosity sense and S satisfies some natural geometric condition, then S ∈ B must be a hyperplane. This is also a considerable improvement of the previous result (Sakaguchi in Discrete Contin. Dyn. Syst., Ser. S 4:887–895, 2011, Theorem 1.1).