Preprint 2001-004

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

Espen R. Jakobsen and Kenneth Hvistendahl Karlsen

Abstract: Using the maximum principle for semicontinuous functions \cite{CrIs:MaxPrinc,CrIsLi:UserGuide}, we establish here 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 Souganidis \cite{eSou} for first order Hamilton-Jacobi equations and a recent result by Cockburn, Gripenperg, and Londen \cite{CockGripenLonden} 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 H\"{o}lder regularity estimates.
(iii) A rate of convergence for the vanishing viscosity method.
Finally, we illustrate the 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. In \cite{JKR:2ndOrderSplit}, the basic result given herein is used to derive an explicit rate of convergence for certain numerical approximations.

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Espen R. Jakobsen, <>
Kenneth Hvistendahl Karlsen, <>
Publishing information:
UCLA Computational and Applied Mathematics Report
Submitted by:
<> February 1 2001.

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