Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications

TL;DR

This paper investigates the regularity of solutions to mixed local-nonlocal equations with singular data, establishing interior and boundary H"older regularity, and applies these results to problems with singular nonlinearities.

Contribution

It provides new regularity results for mixed local-nonlocal operators with minimal assumptions and introduces a strong comparison principle for this class of problems.

Findings

Interior gradient H"older regularity for solutions.

Boundary H"older and boundary gradient H"older regularity depending on singularity.

A strong comparison principle for mixed local-nonlocal problems.

Abstract

In this article, we examine the H\"older regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator $\fp + \fqs$ with singular data, under the minimal assumption that $p> sq$. The regularity result is twofold: we establish interior gradient H\"older regularity for locally bounded data and boundary regularity for singular data. We prove both boundary H\"older and boundary gradient H\"older regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.