Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

TL;DR

This paper introduces a novel method to analyze singularities near cross points in an unstable free boundary problem, revealing geometric regularity of the zero level set in two dimensions.

Contribution

The study provides a new approach for understanding singularities at supercharacteristic points, showing regularity of the zero level set for all solutions in two dimensions.

Findings

Near unbounded second derivatives, the zero level set forms two $C^1$-curves meeting at right angles.

The results apply to all solutions, not just minimal ones.

The method addresses points where classical theory fails due to incompatible scalings.

Abstract

We introduce a new method for the analysis of singularities in the unstable problem $$\Delta u = -\chi_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of $u$ are unbounded, the level set $\{u=0\}$ consists of two $C^1$-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.