Stable and Minimizing Cones in the Alt-Phillips Problem
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909510547800064 |
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| author | Savin, Ovidiu Yu, Hui |
| author_facet | Savin, Ovidiu Yu, Hui |
| contents | 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 $γ=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable and Minimizing Cones in the Alt-Phillips Problem Savin, Ovidiu Yu, Hui Analysis of PDEs 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 $γ=1$. |
| title | Stable and Minimizing Cones in the Alt-Phillips Problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.18192 |