Hölder gradient regularity for the inhomogeneous normalized $p(x)$-Laplace equation

Abstract

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