2020/04/22 by Alexandre Girouard, Girouard, Alexandre, Mikhail Karpukhin +3
Computer Science · Engineering · #35P15 (Primary) 35B27 #58C40 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2004.10784
openalex publication_date 2020/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We associate a sequence of variational eigenvalues to any Radon measure on a\ncompact Riemannian manifold. For particular choices of measures, we recover the\nLaplace, Steklov and other classical eigenvalue problems. In the first part of\nthe paper we study the properties variational eigenvalues and establish a\ngeneral continuity result, which shows for a sequence of measures converging in\nthe dual of an appropriate Sobolev space, that the associated eigenvalues\nconverge as well. The second part of the paper is devoted to various\napplications to shape optimization. The main theme is studying sharp\nisoperimetric inequalities for Steklov eigenvalues without any assumption on\nthe number of connected components of the boundary. In particular, we solve the\nisoperimetric problem for each Steklov eigenvalue of planar domains: the best\nupper bound for the k-th perimeter-normalised Steklov eigenvalue is\n8\πk, which is the best upper bound for the k-th area-normalised\neigenvalue of the Laplacian on the sphere. The proof involves realising a\nweighted Neumann problem as a limit of Steklov problems on perforated domains.\nFor k = 1, the number of connected boundary components of a maximizing\nsequence must tend to infinity, and we provide a quantitative lower bound on\nthe number of connected components. A surprising consequence of our analysis is\nthat any maximizing sequence of planar domains with fixed perimeter must\ncollapse to a point.\n