Sharp local boundedness and maximum principle in the infinitely degenerate regime via DeGiorgi iteration

TL;DR

This paper establishes sharp local boundedness and maximum principles for weak solutions to infinitely degenerate elliptic equations, resolving a previously open problem and extending the understanding of degenerate PDEs.

Contribution

It introduces new bounds and principles for degenerate elliptic equations, addressing the 'Moser gap' problem in higher dimensions.

Findings

Local boundedness is sharp in more than two dimensions.

Maximum principles are proven under the same degeneracy conditions.

Answers an open problem from prior research.

Abstract

We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser gap' problem left open in arXiv:1506.09203v5. Finally we obtain a maximum principle for weak solutions under the same condition on the degeneracy.