Stable and Minimizing Cones in the Alt-Phillips Problem
Ovidiu Savin, Hui Yu
TL;DR
The paper analyzes homogeneous minimizers of the Alt-Phillips functional $\,\mathcal{E}_\gamma$ for gamma near one, showing that in dimensions $d\ge3$ the radial cone is minimizing and, in dimensions $d\ge4$, an axially symmetric cone with a positive-density contact set exists as a global minimizer. The axial cone $u_{\text{as}}$ is constructed via a transformed equation that yields a pair of matched regions near the equator and near the south pole, followed by a careful gluing to obtain a global solution; foliations around both cones are then built to certify minimality. The work reveals a dichotomy between cusp-like and cone-like contact-set behavior as $\gamma\to1^{-}$ and demonstrates that both phenomena can occur for gamma sufficiently close to one. Overall, the results advance understanding of singular minimizing cones in the Alt-Phillips problem and illuminate how near-endpoint behaviors ($\gamma\approx0$ and $\gamma\approx1$) interact under axial symmetry.
Abstract
We study homogeneous solutions to the Alt-Phillips problem when the exponent $γ$ is close to 1. In dimension $d\ge3$, we show that the radial cone is minimizing when $γ$ is close to 1. In dimension $d \ge 4$, we construct an axially symmetric cone whose contact set has with positive density. We show that it is a global minimizer. It is analogous to the De Silva-Jerison \cite{DJ} cone for the Alt-Caffarelli functional which corresponds to exponent $γ=0$. The cone we construct bifurcates from another minimizing cone whose contact set has zero density, obtained as the trivial extension of the radial solution. This second cone is analogous to a quadratic polynomial solution in the classical obstacle problem which corresponds to exponent $γ=1$. In particular our results show that, when $γ<1$ is sufficiently close to 1, there are axis symmetric cones that exhibit the properties of both end point cases $γ=0$ and $γ=1$.