On a free boundary problem and minimal surfaces

2017/04/25 by Yong Liu, Kelei Wang, Liu, Yong +3 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1704.07869

openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.

Citations

• Hitchhikerʼs guide to the fractional Sobolev spaces
• Variational problems with two phases and their free boundaries
• Minimal Varieties in Riemannian Manifolds
• A rigidity result for overdetermined elliptic problems in the plane

Cited by

• Parabolic free boundary phase transition and mean curvature flow
• Varifold convergence of free boundary Allen--Cahn equation
• Second order estimates for a free boundary phase transition

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