H\"older gradient regularity for the inhomogeneous normalized $p(x)$-Laplace equation

TL;DR

This paper establishes local gradient H"older regularity for viscosity solutions to an inhomogeneous normalized $p(x)$-Laplace equation with variable exponent, extending regularity theory to inhomogeneous and variable exponent settings.

Contribution

It proves the gradient H"older regularity for solutions to the inhomogeneous normalized $p(x)$-Laplace equation with Lipschitz continuous exponent and bounded inhomogeneity, a novel extension in the field.

Findings

Proved local gradient H"older regularity of solutions.

Extended regularity results to inhomogeneous equations with variable exponents.

Handled equations with Lipschitz continuous $p(x)$ and bounded inhomogeneity.

Abstract

We prove the local gradient H\"older regularity of viscosity solutions to the inhomogeneous normalized $p(x)$-Laplace equation $$ -\Delta u-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}} = f(x), $$ where $p$ is Lipschitz continuous, $\inf p>1$, and $f$ is continuous and bounded.