Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

2022/11/01 by Daniela De Silva, Ovidiu Savin, De Silva, Daniela +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2211.00553

openalex publication_date 2022/11/01 · openalex created_date 2022/11/07 · openalex updated_date 2026/07/28

Abstract

We investigate the rigidity of global minimizers u ≥ 0 of the Alt-Phillips functional involving negative power potentials ∫_Ω(|∇ u|² + u^-γ χ_\ugt;0\) dx, γ∈ (0,2), when the exponent γ is close to the extremes of the admissible values. In particular we show that global minimizers in ℝⁿ are one-dimensional if γ is close to 2 and n ≤ 7, or if γ is close to 0 and n ≤ 4.

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• Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem
• Smoothness and stability in the Alt-Phillips problem

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