Global gradient estimates for the mixed local and nonlocal problems with measurable nonlinearities

TL;DR

This paper establishes optimal global gradient estimates for solutions to mixed local and nonlocal nonlinear elliptic equations under minimal regularity assumptions and weak boundary conditions.

Contribution

It introduces a minimal regularity framework for mixed local and nonlocal operators and weak geometric boundary assumptions for global gradient estimates.

Findings

Proves global Calderón-Zygmund estimates under minimal regularity.

Identifies weak boundary conditions sufficient for gradient estimates.

Extends theory to measurable nonlinearities in mixed operators.

Abstract

A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calder\'on-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calder\'on-Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical $(-\Delta_p)^1$-Laplace operator with $1<p<\infty$, is measurable in one variable and has a small BMO assumption for the other variables, while the singular kernel associated with the nonlocal $(-\Delta_q)^s$-Laplace operator with $0<s<1<q<\infty$ is merely measurable, and that the boundary of the domain is sufficiently flat in Reifenberg sense, we prove that the gradient of the solution is as integrable as that of the associated non-homogeneous term in the divergence form.