Regularity for solutions of non local parabolic equations II

TL;DR

This paper establishes boundary regularity and interior estimates for solutions to nonlocal parabolic equations, extending results to non-translation invariant operators and highlighting the impact of boundary data regularity.

Contribution

It provides boundary regularity and compactness results for nonlocal parabolic equations with non-translation invariant operators, advancing the understanding of their solution regularity.

Findings

Proves boundary regularity for nonlocal parabolic equations.

Establishes $C^{1,eta}$ interior estimates under certain conditions.

Extends regularity results to non-translation invariant operators.

Abstract

We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form $u_t-Iu=f$, where the operator $I$ is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain $C^{1,\alpha}$ interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.