Existence of $C^\alpha$ solutions to integro-PDEs

TL;DR

This paper proves the existence of $C^{eta}$ viscosity solutions for certain second order integro-PDEs, establishing a weak Harnack inequality and using it to demonstrate regularity and existence of solutions.

Contribution

It introduces a weak Harnack inequality for non-translation invariant integro-PDEs and applies it to prove solution existence and regularity.

Findings

Established a weak Harnack inequality for the class of integro-PDEs.

Proved Hölder regularity of solutions.

Demonstrated existence of $C^{eta}$ solutions.

Abstract

This paper is concerned with existence of a $C^{\alpha}$ viscosity solution of a second order non-translation invariant integro-PDE. We first obtain a weak Harnack inequality for such integro-PDE. We then use the weak Harnack inequality to prove H\"older regularity and existence of solutions of the integro-PDEs.