Uniqueness for $L_{p}$-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms

TL;DR

This paper establishes the uniqueness of $L_{p}$-viscosity solutions for fully nonlinear uniformly parabolic Isaacs equations with measurable lower order terms, using stability arguments for solutions with certain regularity conditions.

Contribution

It extends the uniqueness results for $L_{p}$-viscosity solutions to Isaacs equations with measurable lower order coefficients and H"older continuous higher-order coefficients.

Findings

Uniqueness of $L_{p}$-viscosity solutions is proven for the considered class of equations.

Stability of maximal and minimal solutions is used to handle measurable lower order terms.

Results apply to fully nonlinear parabolic equations with minimal regularity assumptions.

Abstract

In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be H\"older continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.