Vol. 9, No. 5, 2016

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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
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 n with a strong attractive δ-interaction supported on 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