Degenerate elliptic equations with $\Phi$-admissible weights

TL;DR

This paper establishes regularity results for degenerate elliptic equations with weights governed by a generalized Orlicz framework, extending classical theories to weaker inequality assumptions.

Contribution

It introduces a novel regularity theory for degenerate elliptic equations with $\Phi$-admissible weights, utilizing a modified DeGiorgi iteration under weaker inequalities.

Findings

Proves local boundedness of weak solutions.

Shows continuity of solutions under weighted Orlicz-Sobolev inequalities.

Extends classical regularity results to broader weighted settings.

Abstract

We develop regularity theory for degenerate elliptic equations with the degeneracy controlled by a weight. More precisely, we show local boundedness and continuity of weak solutions under the assumption of a weighted Orlicz-Sobolev and Poincar\'{e} inequalities. The proof relies on a modified DeGiorgi iteration scheme, developed in arXiv:1608.01630 and arXiv:1703.00774. The Orlicz-Sobolev inequality we assume here is much weaker than the classical $(2\sigma,2)$ Sobolev inequality with $\sigma>1$ which is typically used in the DeGiorgi or Moser iteration.