Voir la notice de l'article provenant de la source Numdam
We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell–Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.
Keywords: Wentzell eigenvalues, Eigenvalue estimates, Shape optimization, Shape derivatives, Stability, Quantitative isoperimetric inequality
@article{AIHPC_2016__33_2_409_0,
author = {Dambrine, M. and Kateb, D. and Lamboley, J.},
title = {An extremal eigenvalue problem for the {Wentzell{\textendash}Laplace} operator},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {409--450},
publisher = {Elsevier},
volume = {33},
number = {2},
year = {2016},
doi = {10.1016/j.anihpc.2014.11.002},
zbl = {1347.35186},
mrnumber = {3465381},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.002/}
}
TY - JOUR AU - Dambrine, M. AU - Kateb, D. AU - Lamboley, J. TI - An extremal eigenvalue problem for the Wentzell–Laplace operator JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 409 EP - 450 VL - 33 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.002/ DO - 10.1016/j.anihpc.2014.11.002 LA - en ID - AIHPC_2016__33_2_409_0 ER -
%0 Journal Article %A Dambrine, M. %A Kateb, D. %A Lamboley, J. %T An extremal eigenvalue problem for the Wentzell–Laplace operator %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 409-450 %V 33 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.002/ %R 10.1016/j.anihpc.2014.11.002 %G en %F AIHPC_2016__33_2_409_0
Dambrine, M.; Kateb, D.; Lamboley, J. An extremal eigenvalue problem for the Wentzell–Laplace operator. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 409-450. doi: 10.1016/j.anihpc.2014.11.002
Cité par Sources :