# Smoothness and stability in the Alt–Phillips problem

## TL;DR

This paper investigates the Alt–Phillips free boundary problem for negative exponents, proving smoothness and deriving new stability conditions.

## Contribution

A unified proof of smoothness and a new stability condition for the Alt–Phillips problem with negative exponents.

## Key findings

- Smoothness of $C^{1,\alpha}$-regular free boundaries is proven using Schauder estimates for degenerate quasilinear PDEs.
- A new stability condition is derived, ruling out nontrivial axially symmetric stable cones in low dimensions.
- A variational criterion for stability recovers the one for minimal surfaces as $\gamma \rightarrow -2$.

## Abstract

We study the one-phase Alt–Phillips free boundary problem, focusing on the case of negative exponents \documentclass[12pt]{minimal}
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\begin{document}$$\gamma \in (-2,0)$$\end{document}γ∈(-2,0). The goal of this paper is twofold. On the one hand, we prove smoothness of \documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\alpha }$$\end{document}C1,α-regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt–Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt–Phillips problem, which recovers the one for minimal surfaces in the singular limit as \documentclass[12pt]{minimal}
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\begin{document}$$\gamma \rightarrow -2$$\end{document}γ→-2.