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

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.