A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operator

TL;DR

This paper proves local H"older continuity for solutions to a class of anisotropic elliptic equations with singular p-Laplacian behavior, without requiring solutions to be continuous, by establishing a Harnack inequality.

Contribution

It demonstrates that H"older continuity follows from a Harnack inequality even without assuming solution continuity for certain anisotropic elliptic equations.

Findings

Harnack inequality holds without solution continuity

Solutions are H"older continuous under given conditions

Applicable to anisotropic equations with singular p-Laplacian behavior

Abstract

In this brief note we discuss local H\"older continuity for solutions to anisotropic elliptic equations of the type $ \sum_{i=1}^s \partial_{ii} u+ \sum_{i=s+1}^N \partial_i \bigg(A_i(x,u,\nabla u) \bigg) =0,$ for $x \in \Omega \subset \subset \mathbb{R}^N$ and $1\leq s \leq N-1$, where each operator $A_i$ behaves directionally as the singular $p$-Laplacian, $1< p < 2$ and the supercritical condition $p+(N-s)(p-2)>0$ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies H\"older continuity of solutions.