To the theory of viscosity solutions for uniformly elliptic Isaacs equations

TL;DR

This paper develops a theoretical framework for the existence, uniqueness, and regularity of viscosity solutions to uniformly elliptic Isaacs equations, including convergence rates of finite-difference approximations.

Contribution

It introduces a method leveraging solvability in $C^{1,1}$ to establish key properties of viscosity solutions for general Isaacs equations.

Findings

Existence and uniqueness of viscosity solutions proven.

Established $C^{1+ ext{chi}}$ regularity of solutions.

Finite-difference schemes converge at an algebraic rate.

Abstract

We show how a theorem about solvability in $C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the $C^{1+\chi}$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in $C^{\gamma}$ with $\gamma$ slightly less than $1/2$.