Local H\"older Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

TL;DR

This paper proves local boundedness and H"older continuity of solutions to a complex quasilinear elliptic equation involving mixed local and nonlocal operators, variable exponents, and weights, extending classical regularity theory to this new setting.

Contribution

It introduces an analytic approach adapting De Giorgi-Nash-Moser theory to establish regularity for equations with mixed operators, variable exponents, and weights, which was not previously addressed.

Findings

Established local boundedness of solutions.

Proved local H"older continuity of solutions.

Extended regularity theory to mixed local-nonlocal operators with variable exponents.

Abstract

We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad \Omega, $$ where $\Omega \subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_{\Omega} \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^\beta|y|^\beta} \mathrm{d} y $$ Here, $p: \Omega \times \Omega \rightarrow(1, \infty)$ and $s: \Omega \times \Omega \rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{\Omega \times \Omega} p$, and $0 \leq \beta<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights.