Stable and Minimizing Cones in the Alt-Phillips Problem

TL;DR

This paper investigates the structure and minimality of cones in the Alt-Phillips problem for exponents near 1, revealing new minimizers and bifurcations in higher dimensions.

Contribution

It constructs new axially symmetric cones that are global minimizers and analyzes their bifurcation from known solutions when the exponent is close to 1.

Findings

Radial cone is minimizing for $\gamma$ close to 1 in dimension $\ge3$

Constructed a new axially symmetric cone with positive density contact set in dimension $\ge4$

Identified bifurcation phenomena connecting different types of cones near $\gamma=1$

Abstract

We study homogeneous solutions to the Alt-Phillips problem when the exponent $\gamma$ is close to 1. In dimension $d\ge3$, we show that the radial cone is minimizing when $\gamma$ 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 $\gamma=0$. 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 $\gamma=1$. In particular our results show that, when $\gamma<1$ is sufficiently close to 1, there are axis symmetric cones that exhibit the properties of both end point cases $\gamma=0$ and $\gamma=1$.