Steklov eigenvalues of reflection-symmetric nearly circular planar domains. (English) Zbl 1425.35229

Summary: We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.


35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J15 Second-order elliptic equations
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