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Concentration of cones in the Alt-Phillips problem

Ovidiu Savin, Hui Yu

TL;DR

The paper analyzes minimizers of the Alt-Phillips energy $\mathcal{E}_\gamma$ for γ near 1 and proves that minimizing cones concentrate around obstacle-problem cones. It exploits a transformed solution to obtain 2-homogeneity, proves a key integral inequality for parabola solutions, and shows stability to deduce that limits are exactly the symmetric cones $P_k$ (radial in a $k$-dimensional subspace and invariant perpendicular to it). As γ→1, minimizers converge, up to rotation, to one of the cones in $\{P_k\}_{k=0}^d$, providing a first instance of concentration-induced symmetry in a free boundary problem. The results sharpen the understanding of singular cone limits and highlight a structured, symmetric limiting behavior governed by the obstacle problem.

Abstract

We study minimizing cones in the Alt-Phillips problem when the exponent γ is close to 1. When γ converges to 1, we show that the cones concentrate around symmetric solutions to the classical obstacle problem. To be precise, the limiting profiles are radial in a subspace and invariant in directions perpendicular to that subspace.

Concentration of cones in the Alt-Phillips problem

TL;DR

The paper analyzes minimizers of the Alt-Phillips energy for γ near 1 and proves that minimizing cones concentrate around obstacle-problem cones. It exploits a transformed solution to obtain 2-homogeneity, proves a key integral inequality for parabola solutions, and shows stability to deduce that limits are exactly the symmetric cones (radial in a -dimensional subspace and invariant perpendicular to it). As γ→1, minimizers converge, up to rotation, to one of the cones in , providing a first instance of concentration-induced symmetry in a free boundary problem. The results sharpen the understanding of singular cone limits and highlight a structured, symmetric limiting behavior governed by the obstacle problem.

Abstract

We study minimizing cones in the Alt-Phillips problem when the exponent γ is close to 1. When γ converges to 1, we show that the cones concentrate around symmetric solutions to the classical obstacle problem. To be precise, the limiting profiles are radial in a subspace and invariant in directions perpendicular to that subspace.

Paper Structure

This paper contains 7 sections, 12 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Alt-Phillips functional
  4. The transformed solution
  5. The obstacle problem
  6. An integral inequality
  7. Concentration around cones with symmetry

Key Result

Theorem 1.1

Given $\varepsilon>0$, there is $\eta>0$, depending only on $\varepsilon$ and the dimension $d$, such that if $u$ is a minimizing cone for the Alt-Phillips problem EqnAP with then, up to a rotation, we have for some $k\in\{0,1,\dots,d\}.$

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3
  • Theorem 2.1: See AP
  • Proposition 2.2: BCPSUWY
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7: See CPSU
  • ...and 11 more