Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations

Abstract

Using the maximum principle for semicontinuous functions (Differential Integral Equations 3 (1990), 1001-1014; Bull. Amer. Math. Soc. (N.S) 27 (1992), 1-67), we establish a general "continuous dependence on the non-linearities" estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations 56 (1985), 345-390) for first-order Hamilton-Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations 170 (2001), 180-187) for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)-(iii) on the Hamilton-Jacobi-Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here. © 2002 Elsevier Science (USA).