Analyticity for Solution of Integro-Differential Operators

TL;DR

This paper proves that viscosity solutions to a class of integro-differential equations are locally analytic when the kernel and the function are analytic, extending previous Gevrey class results.

Contribution

It establishes the analyticity of solutions for a specific class of integro-differential equations, advancing the understanding beyond Gevrey class regularity.

Findings

Viscosity solutions are locally analytic under certain kernel conditions.

Analyticity holds when the function f is analytic.

Extends previous results from Gevrey classes to full analyticity.

Abstract

We prove that for a certain class of kernels $K(y)$ that viscosity solutions of the integro-differential equation $$ \int_{\mathbb R^n} (u(x+y) - 2 u(x) + u(x-y)) K(y) dy = f(x,u(x)) $$ are locally analytic if $f$ is an analytic function. This extends the result of Albanese, Fiscella, Valdinoci that such solutions belong to certain Gevrey classes.