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

Abstract

Using the maximum principle for semicontinuous functions [3,4], we prove a general ``continuous dependence on the nonlinearities'' estimate for bounded Holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and Holder continuity results for bounded viscosity solutions of such equations. Our results are general enough to encompass Hamilton-Jacobi-Bellman-Isaacs's equations of zero-sum, two-player stochastic differential games. An immediate consequence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations.

Keywords

• fully nonlinear degenerate elliptic equation
• viscosity solution
• Hamilton-Jacobi-Bellman-Isaacs equation
• continuous dependence estimate
• vanishing viscosity method
• convergence rate