Classification of global solutions of a free boundary problem in the plane

TL;DR

This paper classifies specific homogeneous solutions to a differential equation related to free boundary problems in the plane, extending understanding to cases with negative exponents and employing specialized ODE techniques.

Contribution

It provides a comprehensive classification of solutions for a free boundary problem with negative exponents, a novel extension in the field.

Findings

Classification of all nontrivial homogeneous solutions

Extension of free boundary problem analysis to negative exponents

Development of bespoke ODE results for the classification

Abstract

We classify nontrivial, nonnegative, positively homogeneous solutions of the equation \begin{equation*} \Delta u=\gamma u^{\gamma-1} \end{equation*} in the plane. The problem is motivated by the analysis of the classical Alt-Phillips free boundary problem, but considered here with negative exponents $\gamma$. The proof relies on several bespoke results for ordinary differential equations.