On the area-preserving Willmore flow of small bubbles sliding on a domain's boundary

2022/03/24 by Jan-Henrik Metsch, Metsch, Jan-Henrik · 1 citation
Computer Science · Mathematics · #47J07 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 53E40 #Secondary: 35G31 #math.AP #math.DG #msc:35G31 #msc:47J07 #msc:53E40

paper · pdf · doi:10.48550/arxiv.2203.12956

Preprint, 53 pages

arxiv created 2022/03/24 · openalex publication_date 2022/03/24 · arxiv updated 2022/03/25 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We consider the area-preserving Willmore evolution of surfaces that are close to a half-sphere with a small radius, sliding on the boundary S of a domain while meeting it orthogonally. We prove that the flow exists for all times and keeps a 'half-spherical' shape. Additionally, we investigate the asymptotic behaviour of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.

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