Regularity theory of a gradient degenerate Neumann problem

TL;DR

This paper investigates the regularity and comparison principles for a gradient degenerate Neumann problem, extending understanding of free boundary problems with critical scaling, and establishes optimal regularity results in multiple dimensions.

Contribution

It introduces new regularity results for a class of degenerate Neumann problems, including a comparison principle, with implications for homogenization and free boundary analysis.

Findings

Proves $C^{1,1/2}$ regularity in 2D.

Establishes similar regularity in higher dimensions under certain conditions.

Develops a comparison principle for minimal supersolutions.

Abstract

We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) $C^{1,\frac{1}{2}}$ regularity in dimension $d=2$ and we show the same regularity result in $d\geq 3$ conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits.