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

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.