To the theory of viscosity solutions for uniformly parabolic Isaacs equations

TL;DR

This paper develops a theoretical framework connecting solvability in Sobolev spaces to viscosity solutions for uniformly parabolic Isaacs equations, establishing existence, uniqueness, regularity, and convergence rates.

Contribution

It introduces a method to derive viscosity solutions from Sobolev solutions for Isaacs equations, including regularity results and convergence analysis of finite-difference schemes.

Findings

Existence and uniqueness of viscosity solutions established.

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

Finite-difference approximations converge algebraically.

Abstract

We show how a theorem about the solvability in $W^{1,2}_{\infty}$ of special parabolic Isaacs equations can be used to obtain the existence and uniqueness of viscosity solutions of general uniformly nondegenerate parabolic 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 respect to the spatial variables with $\gamma$ slightly less than $1/2$.