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Steklov eigenvalues and quasiconformal maps of simply connected planar domains. (English) Zbl 1333.35274

Summary: We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74K15 Membranes
30C62 Quasiconformal mappings in the complex plane
74H45 Vibrations in dynamical problems in solid mechanics
35P15 Estimates of eigenvalues in context of PDEs
35B45 A priori estimates in context of PDEs

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SyFi; FEniCS
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References:

[1] Aissen, M.I, A set function defined for convex plane domaines, Pac. J. Math., 8, 383-399, (1958) · Zbl 0084.18603
[2] Auchmuty., G., Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim., 25, 321-348, (2004) · Zbl 1072.35133
[3] BandleC.: Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston, 1980 · Zbl 1011.35100
[4] Brock., F., An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z. Angew. Math. Mech., 81, 69-71, (2001) · Zbl 0971.35055
[5] Bucur, D.; Buttazzo, G.; Henrot., A., Minimization of \({λ_{2}(Ω)}\) with a perimeter constraint, Indiana Univ. Math. J., 58, 2709-2728, (2009) · Zbl 1186.49032
[6] Colbois, B.; El Soufi, A.; Girouard, A., Isoperimetric control of the Steklov spectrum, J. Funct. Anal., 261, 1384-1399, (2011) · Zbl 1235.58020
[7] Dittmar, B., Stekloffsche eigenwerte und konforme abbildungen, Z. Anal. Anwendungen, 7, 149-163, (1988) · Zbl 0651.30011
[8] Dittmar, B., Sums of reciprocal Stekloff eigenvalues, Math. Nachr., 268, 44-49, (2004) · Zbl 1054.35041
[9] Dittmar, B.: Sums of Reciprocal Eigenvalues, Complex Analysis and Potential Theory, pp. 54-65. World Sci. Publ., Hackensack, 2007 · Zbl 1158.35066
[10] Edward, J., An inequality for Steklov eigenvalues for planar domains, Z. Angew. Math. Phys., 45, 493-496, (1994) · Zbl 0868.35078
[11] Escobar, J. F., The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal., 150, 544-556, (1997) · Zbl 0888.58066
[12] Fraser, A.; Schoen, R., The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226, 4011-4030, (2011) · Zbl 1215.53052
[13] Garau, E. M.; Morin, P., Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Numer. Anal., 31, 914-946, (2011) · Zbl 1225.65107
[14] A., Girouard; I., Polterovich, Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci., 33, 501-516, (2010) · Zbl 1186.35121
[15] Girouard, A.; Polterovich, I., On the hersch-payne-schiffer estimates for the eigenvalues of the Steklov problem, Funct. Anal. Appl., 44, 106-117, (2010) · Zbl 1217.35125
[16] Girouard, A.; Polterovich, I., Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., 19, 77-85, (2012) · Zbl 1257.58019
[17] Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem. J. Spectr. Theory (to appear). arXiv:1411.6567 · Zbl 1375.49056
[18] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988 (reprint of the 1952 edition) · Zbl 1319.35130
[19] Harrell, E. M.; Hermi, L., Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Anal., 254, 3173-3191, (2008) · Zbl 1147.35062
[20] Hassannezhad, A., Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. Funct. Anal., 261, 3419-3436, (2011) · Zbl 1232.58023
[21] Henrot, A.; Oudet, E., Le stade ne minimise pas \({λ_2}\) parmi LES ouverts convexes du plan, C. R. Acad. Sci. Paris Sér. I Math., 332, 417-422, (2001) · Zbl 1011.35100
[22] Henrot, A.; Philippin, G. A., Safoui some isoperimetric inequalities with application to the Stekloff problem, J. Convex Anal., 15, 581-592, (2008) · Zbl 1155.26019
[23] Hersch, J.; Payne, L.E., Extremal principles and isoperimetric inequalities for some mixed problems of stekloff’s type, Z. Angew. Math. Phys., 19, 802-817, (1968) · Zbl 0165.12603
[24] Hersch, J.; Payne, L. E.; Schiffer, M. M., Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57, 99-114, (1975) · Zbl 0315.35069
[25] Ilias, S.; Makhoul, O., A reilly inequality for the first Steklov eigenvalue, Differ. Geom. Appl., 29, 699-708, (2011) · Zbl 1222.35131
[26] Jammes, P.: Une inégalité de Cheeger pour le spectre de Steklov. Ann. Inst. Fourier (to appear) · Zbl 1346.58011
[27] Kulczycki, T., Kwaśnicki, M., Siudeja, B.: On the shape of the fundamental sloshing mode in axisymmetric containers (preprint). arXiv:1411.2572 · Zbl 0888.58066
[28] Kuttler, J. R.; Sigillito, V. G., Lower bounds for Stekloff and free membrane eigenvalues, SIAM Rev., 10, 368-370, (1968) · Zbl 0159.16104
[29] Kuznetsov, N.; Kulczycki, T.; Kwaśnicki, M.; Nazarov, A.; Poborchi, S.; Polterovich, I.; Siudeja, B., The legacy of vladimir andreevich Steklov, Notices Am. Math. Soc., 61, 9-22, (2014) · Zbl 1322.01050
[30] Lamberti, P.D., Provenzano, L.: Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues. In: Proceedings of the 9th ISAAC Congress, Kraków (2013). arXiv:1410.0517 · Zbl 1325.35124
[31] Laugesen, R. S.; Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mapping, J. Funct. Anal., 155, 64-108, (1998) · Zbl 0917.47018
[32] Laugesen, R. S.; Siudeja, B. A., Sharp spectral bounds on starlike domains, J. Spectr. Theory, 4, 309-347, (2014) · Zbl 1296.35099
[33] Laugesen, R. S.; Siudeja, B. A., Magnetic spectral bounds on starlike plane domains, ESAIM Control Optim. Calc. Var., 21, 670-689, (2015) · Zbl 1319.35130
[34] Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, New York, 1973 (translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126) · Zbl 0267.30016
[35] Li, P.; Yau, S.T., On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88, 309-318, (1983) · Zbl 0554.35029
[36] Logg, A., Mardal, K.-A., Wells, G.N. (eds). Automated solution of differential equations by the finite element method. In: The FEniCS book, Lecture Notes in Computational Science and Engineering, vol. 84. Springer, Heidelberg, 2012 · Zbl 1247.65105
[37] Payne, L. E., Isoperimetric inequalities and their applications, SIAM Rev., 9, 453-488, (1967) · Zbl 0154.12602
[38] Payne, L. E., Some isoperimetric inequalities for harmonic functions, SIAM J. Math. Anal., 1, 354-359, (1970) · Zbl 0199.16902
[39] Pólya, G.; Schiffer, M., Convexity of functionals by transplantation, J. Anal. Math., 3, 245-346, (1954) · Zbl 0056.32701
[40] Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton, 1951
[41] Siudeja, B.: Generalized tight \(p\)-frames and spectral bounds for Laplace-like operators. arXiv:1409.7409 · Zbl 1403.35188
[42] Steklov, V., Sur LES problèmes fondamentaux de la physique mathématique, Ann. Sci. Ecole Norm. Sup., 19, 455-490, (1902)
[43] Sylvester, J., Uhlmann, G.: The Dirichlet to Neumann map and applications, Inverse problems in partial differential equations (Arcata, CA, 1989), pp. 101-139. SIAM, Philadelphia, 1990 · Zbl 0868.35078
[44] Weinstock, R., Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3, 745-753, (1954) · Zbl 0056.09801
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