Continuity in two dimensions for a very degenerate elliptic equation

TL;DR

This paper establishes continuity of solutions for a highly degenerate elliptic equation in two dimensions, providing new regularity results where traditional elliptic theory does not apply.

Contribution

It proves a continuity result for solutions of a degenerate elliptic equation in two dimensions, extending regularity theory to highly degenerate cases.

Findings

Proves continuity of solutions in 2D for degenerate elliptic equations.

Provides applications demonstrating the practical relevance of the regularity result.

Abstract

An elliptic equation div(F(Du)) = f whose ellipticity strongly degenerates for small values of Du (say, F = 0 on B(0,1)) is considered. The aim is to prove regularity for F(Du). The paper proves a continuity result in dimension two and presents some applications.