[Un théorème de type Rellich pour l'équation de Helmholtz dans un domaine conique]
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.
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.
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@article{CRMATH_2016__354_1_27_0,
author = {Bonnet-Ben Dhia, Anne-Sophie and Fliss, Sonia and Hazard, Christophe and Tonnoir, Antoine},
title = {A {Rellich} type theorem for the {Helmholtz} equation in a conical domain},
journal = {Comptes Rendus. Math\'ematique},
pages = {27--32},
publisher = {Elsevier},
volume = {354},
number = {1},
year = {2016},
doi = {10.1016/j.crma.2015.10.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.10.015/}
}
TY - JOUR AU - Bonnet-Ben Dhia, Anne-Sophie AU - Fliss, Sonia AU - Hazard, Christophe AU - Tonnoir, Antoine TI - A Rellich type theorem for the Helmholtz equation in a conical domain JO - Comptes Rendus. Mathématique PY - 2016 SP - 27 EP - 32 VL - 354 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.10.015/ DO - 10.1016/j.crma.2015.10.015 LA - en ID - CRMATH_2016__354_1_27_0 ER -
%0 Journal Article %A Bonnet-Ben Dhia, Anne-Sophie %A Fliss, Sonia %A Hazard, Christophe %A Tonnoir, Antoine %T A Rellich type theorem for the Helmholtz equation in a conical domain %J Comptes Rendus. Mathématique %D 2016 %P 27-32 %V 354 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.10.015/ %R 10.1016/j.crma.2015.10.015 %G en %F CRMATH_2016__354_1_27_0
Bonnet-Ben Dhia, Anne-Sophie; Fliss, Sonia; Hazard, Christophe; Tonnoir, Antoine. A Rellich type theorem for the Helmholtz equation in a conical domain. Comptes Rendus. Mathématique, Tome 354 (2016) no. 1, pp. 27-32. doi : 10.1016/j.crma.2015.10.015. http://www.numdam.org/articles/10.1016/j.crma.2015.10.015/
[1] Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition, SIAM J. Appl. Math., Volume 70 (2009), pp. 677-693
[2] Mathematical analysis of the junction of two acoustic open waveguides, SIAM J. Appl. Math., Volume 71 (2011), pp. 2048-2071
[3] Plane waveguides with corners in the small angle limit, J. Math. Phys., Volume 53 (2012), p. 123529
[4] Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys., Volume 7 (1995), pp. 73-102
[5] On the absence of trapped modes in locally perturbed open waveguides, IMA J. Appl. Math., Volume 80 (2015), pp. 1049-1062
[6] On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM, Control Optim. Calc. Var., Volume 18 (2012), pp. 712-747
[7] Embedded trapped modes in water waves and acoustics, Wave Motion, Volume 45 (2007), pp. 16-29
[8] Über das asymptotische Verhalten der Lösungen von in unendlichen Gebieten, Jahresber. Dtsch. Math.-Ver., Volume 53 (1943), pp. 57-65
[9] Rellich type theorems for unbounded domains, Inverse Probl. Imaging, Volume 8 (2014), pp. 865-883
[10] Absence of eigenvalues of the acoustic propagator in deformed wave guides, Rocky Mt. J. Math., Volume 18 (1988), pp. 495-503
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