Stable cones in the Alt-Phillips free boundary problem

TL;DR

The paper addresses the classification of axially symmetric stable global minimizers for the Alt-Phillips free boundary problem in dimensions $n$ where $n$ lies in a range determined by the parameter $\alpha = \frac{2\gamma}{2-\gamma}$ (notably $\gamma\in(0,2/3)$). It develops a general domain-variation framework and then specializes to the Alt-Phillips problem via a $v$-transformation, deriving a first- and second-variation theory that yields a key stability inequality (involving $u^\gamma$ and $|\nabla u|$) and its $v$-form counterpart with $\alpha$. Using this stability, the authors prove a Simons-type radial-derivative estimate for axial symmetry and establish a dimension-dependent rigidity result: any global stable axially symmetric solution is one-dimensional, and its zero set is a half-space in the stated ranges. The work connects the Alt-Phillips problem to the classical Alt-Caffarelli limit as $\alpha\to 0$, enriching the theory of free boundary regularity and cone/classification results through a robust variational framework.

Abstract

In this paper we prove a classification result for axially symmetric one phase minimizers of the Alt-Phillips free boundary problem in dimensions 3, 4, and 5. To accomplish this, we establish a stability inequality that extends the one for the Alt-Caffarelli problem.

Figures (1)

• Figure 1: Visualization of the constraint \ref{['Eq:DimensionConstrain']} on the dimension.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 3.1
• Proposition 3.2
• proof
• Theorem 4.1
• Remark 4.2
• Remark 4.3
• proof : Proof of \ref{['Th:StabilityCondition']}
• Remark 4.4
• Lemma 5.1