Comptes Rendus
no. 1
Partial differential equations
A Rellich type theorem for the Helmholtz equation in a conical domain

Presented by: the Editorial Board

Anne-Sophie Bonnet-Ben Dhia1; Sonia Fliss1; Christophe Hazard1; Antoine Tonnoir1
1 POEMS, CNRS/ENSTA-ParisTech/INRIA, 828, boulevard des Maréchaux, 91120 Palaiseau, France
Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 27-32.

We prove that there cannot exist square-integrable nonzero solutions to the Helmholtz equation in an axisymmetric conical domain whose vertex angle is greater than π. This implies in particular the absence of eigenvalues embedded in the essential spectrum of a large class of partial differential operators that coincide with the Laplacian in the conical domain.

On démontre qu'il ne peut exister de solutions non nulles et de carré intégrable de l'équation de Helmholtz dans un domaine conique axisymétrique dont l'angle au sommet est strictement supérieur à π. Ceci implique en particulier l'absence de valeurs propres plongées dans le spectre essentiel pour de nombreux opérateurs qui coïncident avec le laplacien dans le domaine conique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.10.015

Anne-Sophie Bonnet-Ben Dhia 1; Sonia Fliss 1; Christophe Hazard 1; Antoine Tonnoir 1

1 POEMS, CNRS/ENSTA-ParisTech/INRIA, 828, boulevard des Maréchaux, 91120 Palaiseau, France 
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Anne-Sophie Bonnet-Ben Dhia; Sonia Fliss; Christophe Hazard; Antoine Tonnoir. A Rellich type theorem for the Helmholtz equation in a conical domain. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 27-32. doi : 10.1016/j.crma.2015.10.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.015/

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