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. 2021;239(2):981-1023.
doi: 10.1007/s00205-020-01588-2. Epub 2020 Nov 20.

From Steklov to Neumann via homogenisation

Alexandre Girouard  1 , Antoine Henrot  2 , Jean Lagacé  3
Affiliations

From Steklov to Neumann via homogenisation

Alexandre Girouard et al. Arch Ration Mech Anal. 2021.

Abstract

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. 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 isoperimetric 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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Fig. 1
The domain Ωε
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References

    1. Allaire G. Shape Optimization by the Homogenization Method, Volume 146 of Applied Mathematical Sciences. New York: Springer; 2002.
    1. Amirat Y, Chechkin GA, Gadyl’shin RR. Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary. Zh. Vychisl. Mat. Mat. Fiz. 2006;46(1):102–115.
    1. Arrieta J, Jiménez-Casas Á, Rodríguez-Bernal A. Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary. Rev. Mat. Iberoam. 2008;24(1):183–211. doi: 10.4171/RMI/533. - DOI
    1. Bucur D, Freitas P, Kennedy J. The Robin problem. In: Henrot A, editor. Shape Optimization and Spectral Theory. Warsaw: De Gruyter Open; 2017. pp. 78–119.
    1. Bucur D, Giacomini A, Trebeschi P. L∞ bounds of Steklov eigenfunctions and spectrum stability under domain variations. J. Differ. Equ. 2020;269:11461–11491. doi: 10.1016/j.jde.2020.08.040. - DOI

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