Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations
Using the maximum principle for semicontinuous functions, we establish a general ``continuous dependence on the nonlinearities'' estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by \textit{P. E. Souganidis} [J. Differ. Equations 56, 345--390 (1985; Zbl 0506.35020)] for first-order Hamilton-Jacobi equations and a recent result by \textit{B. Cockburn, G. Gripenberg} and \textit{S.-O. Londen} [J. Differ. Equations 170, No. 1, 180--187 (2001; Zbl 0973.35107)] 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) \(L^\infty\) 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.
Hamilton-Jacobi-Bellman partial differential equation
scientific article; zbMATH DE number 1836804
scientific article