A corollary to Kodaira-Spencer’s theorem on continuity of eigenvalues

1991 ◽  
Vol 76 (1-2) ◽  
pp. 229-235
Author(s):  
S. Srinivas Rau
Keyword(s):  
Continuity Of Eigenvalues

Continuity of eigenvalues of subordinate processes in domains

2005 ◽  
Vol 252 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Renming Song
Keyword(s):  
Continuity Of Eigenvalues

scholarly journals Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

2021 ◽  
Author(s):  
Alexandre Girouard ◽  
Mikhail Karpukhin ◽  
Jean Lagacé
Keyword(s):  
Upper Bound ◽  
Eigenvalue Problems ◽  
Isoperimetric Problem ◽  
Connected Components ◽  
Shape Optimisation ◽  
Main Theme ◽  
Planar Domains ◽  
Continuity Result ◽  
Steklov Eigenvalue ◽  
Continuity Of Eigenvalues

AbstractWe associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

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