Hölder regularity for anisotropic $p$-Laplace equation

Abstract

In this paper, we obtain local Hölder 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 $1<p_1\le p_2\le \cdots \le p_N<\infty$. Under an additional assumption that double truncates of the solution are sub/super solution, we obtain Hölder regularity for the anisotropic equation.

arXiv commentThe proof holds only under a strong assumption and hence the regularity remains open