An extremal eigenvalue problem for the Wentzell–Laplace operator

Dambrine, M.; Kateb, D.; Lamboley, J.

doi:10.1016/j.anihpc.2014.11.002

Dambrine, M. ; Kateb, D. ; Lamboley, J.

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 409-450.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2014.11.002

Classification : 35P15, 49K20, 49K40
Mots clés : Wentzell eigenvalues, Eigenvalue estimates, Shape optimization, Shape derivatives, Stability, Quantitative isoperimetric inequality

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     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},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.002/}
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.002/

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