Regularity theory for Second order Integro-PDEs

TL;DR

This paper establishes higher regularity results, including $C^{1,eta}$ and Schauder estimates, for viscosity solutions of non-translation invariant second order integro-PDEs, advancing understanding of their smoothness properties.

Contribution

It provides new $C^{1,eta}$ regularity and Schauder estimates for fully nonlinear, non-translation invariant integro-PDEs, extending previous work to broader classes of equations.

Findings

Established $C^{1,eta}$ regularity for viscosity solutions.

Proved Schauder estimates for convex integro-PDEs.

Extended regularity theory beyond translation-invariant cases.

Abstract

This paper is concerned with higher H\"older regularity for viscosity solutions to non-translation invariant second order integro-PDEs, compared to \cite{mou2018}. We first obtain $C^{1,\alpha}$ regularity estimates for fully nonlinear integro-PDEs. We then prove the Schauder estimates for solutions if the equation is convex.