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An extremal eigenvalue problem for the Wentzell-Laplace operator. (English) Zbl 1347.35186

Let \(d\geq2\) and let \(\Omega\) be a bounded connected open subset of \(\mathbb R^d\) with boundary of class \(C^{3}\) such that \(\int_{\partial\Omega}x=0\) and let \(\Delta_{\tau}\) be the Laplace-Beltrami operator on \(\partial\Omega\), \(\beta \in [0,+\infty[\), \((\lambda_{i,\beta}(\Omega))_{i \in\mathbb N}\) the increasing sequence of eigenvalues of the problem: \(-\Delta u=0\) in \(\Omega\), \(-\beta \Delta_{\tau}u + \partial_{n}u=\lambda u\) on \(\partial\Omega\), where \(n\) denotes the outward unit normal vector to \(\partial\Omega\). Let \(\Lambda[\Omega]\) be the spectral radius of the symmetric and positive semidefinite matrix \(P(\Omega)=(\int_{\partial\Omega} (\delta_{i,j} - n_{i}n_{j}))_{i,j=1,\dots,d}\).
Theorem. \[ \sum_{i=1}^{d} {1 \over \lambda_{i,\beta}(\Omega)} \geq {\int_{\partial\Omega}|x|^{2} \over |\Omega| + \beta \Lambda[\Omega]} \geq {d |B_{1}|^{-1/d}|\Omega|^{1+1/d} \over |\Omega|+ \beta \Lambda[\Omega]}\Big(1+(1+1/d)(2^{-2+1/d}-1/4)\big({|\Omega \Delta B| \over |B|}\big)^2\Big), \] where \(B_{1}\) is the unit ball and \(B\) is the ball of volume \(|\Omega|\) centered in \(0\); equality holds if \(\Omega\) is a ball.
The authors conjecture that the ball maximizes the first non trivial eigenvalue among smooth open sets of given volume and which are homeomorphic to the ball: to support this, they prove that balls are critical domains in any dimension and that they are local maximizers in dimension \(2\) and \(3\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
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[1] Bendali, A.; Lemrabet, K., The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56, 6, 1664-1693, (1996) · Zbl 0869.35068
[2] Betta, M. F.; Brock, F.; Mercaldo, A.; Posteraro, M. R., A weighted isoperimetric inequality and applications to symmetrization, J. Inequal. Appl., 4, 3, 215-240, (1999) · Zbl 1029.26018
[3] Bleecker, D. D., The spectrum of a Riemannian manifold with a unit Killing vector field, Trans. Am. Math. Soc., 275, 1, 409-416, (1983) · Zbl 0506.53021
[4] Bonnaillie-Noël, V.; Dambrine, M.; Hérau, F.; Vial, G., On generalized Ventcel’s type boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42, 2, 931-945, (2010) · Zbl 1209.35035
[5] Brasco, L.; De Philippis, G.; Ruffini, B., Spectral optimization for the Stekloff-Laplacian: the stability issue, J. Funct. Anal., 262, 11, 4675-4710, (2012) · Zbl 1245.35076
[6] Brock, F., An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 81, 1, 69-71, (2001) · Zbl 0971.35055
[7] Caubet, F.; Dambrine, M.; Kateb, D., Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions, Inverse Problems, 29, 11, (2013) · Zbl 1292.65069
[8] Clarke, F. H., Optimization and nonsmooth analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, (1983), John Wiley & Sons, Inc. New York, A Wiley-Interscience Publication · Zbl 0727.90045
[9] Cohen Tannoudji, C.; Diu, B.; Laloe, F., Mécanique quantique, (1997), Hermann Paris
[10] Colbois, B.; Dodziuk, J., Riemannian metrics with large \(\lambda_1\), Proc. Am. Math. Soc., 122, 3, 905-906, (1994) · Zbl 0820.58056
[11] Colbois, B.; Dryden, E. B.; El Soufi, A., Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds, Bull. Lond. Math. Soc., 42, 1, 96-108, (2010) · Zbl 1187.58027
[12] Dambrine, M.; Kateb, D., Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations, J. Math. Anal. Appl., 394, 1, 129-138, (2012) · Zbl 1252.35136
[13] Delfour, M. C.; Zolésio, J.-P., Shapes and geometries, (Analysis, Differential Calculus, and Optimization, Adv. Des. Control, vol. 4, (2001), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA) · Zbl 1002.49029
[14] Desaint, F. R.; Zolésio, J.-P., Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151, 1, 234-269, (1997) · Zbl 0903.58059
[15] Goldstein, G. R., Derivation and physical interpretation of general boundary conditions, Adv. Differ. Equ., 11, 4, 457-480, (2006) · Zbl 1107.35010
[16] Haddar, H.; Joly, P.; Nguyen, H.-M., Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci., 15, 8, 1273-1300, (2005) · Zbl 1084.35102
[17] Henrot, A.; Pierre, M., Variation et optimisation de formes, (Une analyse géométrique, Math. Appl., vol. 48, (2005), Springer Berlin) · Zbl 1098.49001
[18] Hersch, J., Caractérisation variationnelle d’une somme de valeurs propres consécutives; généralisation d’inégalités de Pólya-schiffer et de Weyl, C. R. Acad. Sci. Paris, 252, 1714-1716, (1961) · Zbl 0096.08602
[19] Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B, 270, A1645-A1648, (1970)
[20] Hile, G. N.; Xu, Z. Y., Inequalities for sums of reciprocals of eigenvalues, J. Math. Anal. Appl., 180, 2, 412-430, (1993) · Zbl 0816.47052
[21] Kennedy, J., A Faber-krahn inequality for the Laplacian with generalised Wentzell boundary conditions, J. Evol. Equ., 8, 3, 557-582, (2008) · Zbl 1154.35413
[22] Lemrabet, K.; Teniou, D., Vibrations d’une plaque mince avec raidisseur sur le bord, Maghreb Math. Rev., 2, 1, 27-41, (1992)
[23] Nédélec, J.-C., Acoustic and electromagnetic equations, (Integral Representations for Harmonic Problems, Appl. Math. Sci., vol. 144, (2001), Springer-Verlag New York)
[24] Ortega, J. H.; Zuazua, E., Generic simplicity of the eigenvalues of the Stokes system in two space dimensions, Adv. Differ. Equ., 6, 8, 987-1023, (2001) · Zbl 1223.35252
[25] Stein, E. M.; Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., vol. 32, (1971), Princeton University Press Princeton, NJ · Zbl 0232.42007
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