H\"older regularity for anisotropic $p$-Laplace equation

TL;DR

This paper establishes local H"older regularity for bounded weak solutions to an anisotropic p-Laplace equation with variable exponents, under certain additional assumptions, extending regularity theory to anisotropic settings.

Contribution

It provides the first H"older regularity results for solutions to anisotropic p-Laplace equations with variable exponents under specific truncation conditions.

Findings

Proved local H"older continuity of solutions.

Extended regularity results to anisotropic equations with variable exponents.

Identified conditions under which solutions exhibit regularity.

Abstract

In this paper, we obtain local H\"older regularity for bounded, weak solutions to the anisotropic $p$-Laplace equation whose prototype structure is given by $$ \sum_{i=1}^N (|u_{x_i}|^{p_i-2}u_{x_i})_{x_i}=0,$$ where $1 < p_1 \leq p_2 \leq \cdots \leq p_N < \infty$. Under an additional assumption that double truncates of the solution are sub/super solution, we obtain H\"older regularity for the anisotropic equation.