×

Steklov eigenvalues of reflection-symmetric nearly circular planar domains. (English) Zbl 1425.35229

Summary: We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

MSC:

35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J15 Second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Taylor ME. (2011) Partial differential equations I. New York, NY: Springer.
[2] Girouard A, Polterovich I. (2017) Spectral geometry of the Steklov problem. J. Spectral Theory 7, 321-359. (doi:10.4171/JST) · Zbl 1378.58026
[3] Lipton R. (1998) Optimal fiber configurations for maximum torsional rigidity. Arch. Ration. Mech. Anal. 144, 79-106. (doi:10.1007/s002050050113) · Zbl 0939.74051
[4] Lipton R. (1998) The second Stekloff eigenvalue and energy dissipation inequalities for functionals with surface energy. SIAM J. Math. Anal. 29, 673-680. (doi:10.1137/S0036141096310144) · Zbl 0911.31002
[5] Cakoni F, Colton D, Meng S, Monk P. (2016) Stekloff eigenvalues in inverse scattering. SIAM J. Appl. Math. 76, 1737-1763. (doi:10.1137/16M1058704) · Zbl 1346.35228
[6] Akhmetgaliyev E, Kao CY, Osting B. (2017) Computational methods for extremal Steklov problems. SIAM J. Control Optim. 55, 1226-1240. (doi:10.1137/16M1067263) · Zbl 1432.65164
[7] Bogosel B, Bucur D, Giacomini A. (2017) Optimal shapes maximizing the Steklov eigenvalues. SIAM J. Math. Anal. 49, 1645-1680. (doi:10.1137/16M1075260) · Zbl 1367.49037
[8] Rayleigh JWS. (1945) [1894/96] The theory of sound, vol. 1, 2nd edn. New York, NY: Dover Publications.
[9] Wolf SA, Keller JB. (1994) Range of the first two eigenvalues of the Laplacian. Proc. R. Soc. Lond. A 447, 397-412. (doi:10.1098/rspa.1994.0147) · Zbl 0816.35097
[10] Weinstock R. (1954) Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3, 745-753. (doi:10.1512/iumj.1954.3.53036) · Zbl 0056.09801
[11] Brock F. (2001) An isoperimetric inequality for eigenvalues of the Stekloff problem. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 81, 69-71. (doi:10.1002/(ISSN)1521-4001) · Zbl 0971.35055
[12] Hersch J, Payne LE, Schiffer MM. (1974) Some inequalities for Stekloff eigenvalues. Arch. Ration. Mech. Anal. 57, 99-114. (doi:10.1007/BF00248412) · Zbl 0315.35069
[13] Osting B, Kao CY. (2013) Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues. SIAM J. Sci. Comput. 35, B731-B750. (doi:10.1137/120881865) · Zbl 1273.35196
[14] Osting B, Kao CY. (2014) Minimal convex combinations of three sequential Laplace-Dirichlet eigenvalues. Appl. Math. Optim. 69, 123-139. (doi:10.1007/s00245-013-9219-z) · Zbl 1305.49066
[15] Berger A. (2014) The eigenvalues of the Laplacian with Dirichlet boundary condition in R2 are almost never minimized by discs. Ann. Global Anal. Geom. 47, 285-304. (doi:10.1007/s10455-014-9446-9) · Zbl 1346.35133
[16] Kato T. (1976) Perturbation theory for linear operators, 2nd edn. Berlin, Germany: Springer.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.