On H²-gradient Flows for the Willmore Energy

2017/03/19 by H. Ralph Schumacher, Schumacher, Henrik
Computer Science · Mathematics · #49Q10 #65N30 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1703.06469

openalex publication_date 2017/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the concept of H²-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class W^2,p from a compact, n-dimensional manifold into Euclidean space, provided that p ≥ 2 and p>n. We also discuss why this is not true for Sobolev class H²=W^2,2. In the case of equality constraints, we provide sufficient conditions for the existence of the projected H²-gradient flow and demonstrate its usability for optimization with several numerical examples.

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