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.