Regularity and symmetry results for nonlinear degenerate elliptic equations

TL;DR

This paper establishes regularity, symmetry, and monotonicity properties for solutions of a class of nonlinear degenerate elliptic equations, advancing understanding of their second order regularity and geometric features.

Contribution

It introduces new regularity results and adapts the moving plane method to analyze symmetry and monotonicity in degenerate elliptic equations.

Findings

Solutions exhibit second order regularity under certain conditions.

Positive solutions are symmetric and monotone in convex symmetric domains.

The adapted moving plane method effectively proves symmetry properties.

Abstract

In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.