The Evans-Krylov theorem for non local fully non linear equations

TL;DR

This paper extends the Evans-Krylov regularity theorem to non-local fully nonlinear equations, providing interior regularity results for solutions of integro-differential Bellman equations, bridging the gap between non-local and classical PDE theory.

Contribution

It establishes an interior regularity result for non-local fully nonlinear equations, generalizing the Evans-Krylov theorem to integro-differential equations.

Findings

Solutions are classically regular under the new conditions.

The regularity result recovers classical Evans-Krylov theorem as the order approaches two.

Provides a framework for understanding solutions of non-local Bellman equations.

Abstract

We prove an interior regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. If we let the order of the equation approach two, we recover the theorem of Evans and Krylov about the regularity of solutions to concave uniformly elliptic partial differential equations.