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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2020

From Steklov to Neumann via homogenisation

Résumé

We study a new link between the Steklov and Neumann eigenvalues of Eu-clidean domains. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperi-metric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.
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Dates et versions

hal-02969275 , version 1 (16-10-2020)

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Alexandre Girouard, Antoine Henrot, Jean Lagacé. From Steklov to Neumann via homogenisation. Archive for Rational Mechanics and Analysis, 2020, 239, pp.981-1023. ⟨10.1007/s00205-020-01588-2⟩. ⟨hal-02969275⟩
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