On the negative spectrum of the Robin Laplacian in corner domains

Bruneau, Vincent; Popoff, Nicolas

doi:10.2140/apde.2016.9.1259

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On the negative spectrum of the Robin Laplacian in corner domains

Vincent Bruneau and Nicolas Popoff

Vol. 9 (2016), No. 5, 1259–1283

DOI: 10.2140/apde.2016.9.1259

Abstract

For a bounded corner domain $Ω$ , we consider the attractive Robin Laplacian in $Ω$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the bottom of the spectrum. It allows also the study of the bottom of the essential spectrum on the associated tangent structures given by cones. Then we obtain the asymptotic behavior of the principal eigenvalue for this singular limit in any dimension, with remainder estimates. The same method works for the Schrödinger operator in $ℝ^{n}$ with a strong attractive $δ$ -interaction supported on $\partial Ω$ . Applications to some Ehrling-type estimates and the analysis of the critical temperature of some superconductors are also provided.

Keywords

Robin Laplacian, eigenvalues estimates, corner domains

Mathematical Subject Classification 2010

Primary: 35J10, 35P15, 47F05, 81Q10

Milestones

Received: 18 January 2016

Revised: 30 March 2016

Accepted: 29 April 2016

Published: 29 July 2016

Authors

Vincent Bruneau
Institut de Mathematiques Université de Bordeaux I 351 Cours de La Libération 33405 Talence France

Nicolas Popoff
Institut de Mathematiques Université de Bordeaux I 351 Cours de La Libération 33405 Talence France

Vol. 9, No. 5, 2016