×

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues. (English. Russian original) Zbl 1217.35125

Funct. Anal. Appl. 44, No. 2, 106-117 (2010); translation from Funkts. Anal. Prilozh. 44, No. 2, 33-47 (2010).
The Hersch-Payne-Schiffer inequalities for the Steklov eigenvalues are studied in this paper. If \(\Omega\) is a simply connected bounded planar domain with Lipschitz boundary \(\partial\Omega\) and \(p\geq 0\) with \(p\in L^\infty(\partial\Omega)\), we consider the problem \(\Delta u = 0\) in \(\Omega\), \(\partial u/\partial\nu= \sigma p u\) on \(\partial\Omega\) (\(\nu\) unit outward normal vector) with \(M=\int_{\partial\Omega} p\).
The main result ( Theorem 1.3.1) states that if \(\sigma(\Omega)\) is the \(n\)th eigenvalue, there exists a family \(\Omega_\varepsilon\) of such domains verifying \(\lim\sigma_n(\Omega_\varepsilon)M(\Omega_\varepsilon)= 2\pi n\), \(n= 2,3,\dots\) and \(\lim\sigma_n(\Omega_\varepsilon) \sigma_{n+ 1}(\Omega_\varepsilon) M(\Omega_\varepsilon)^2= 4\pi^2 n^2\). These domains degenerate into a union of \(n\) identical disks when \(\varepsilon\) goes to \(0\). This implies that the inequalities are sharp for all \(n> 1\). It is also proved that \(\sigma_2(\Omega)M(\Omega)< 4\pi\) (Theorem 1.3.6) by using 2 the Riemann mapping theorem.

MSC:

35P05 General topics in linear spectral theory for PDEs
49R05 Variational methods for eigenvalues of operators
35J25 Boundary value problems for second-order elliptic equations
47A75 Eigenvalue problems for linear operators
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] M. S. Ashbaugh and R. D. Benguria, ”Isoperimetric inequalities for eigenvalues of the Laplacian,” in: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math., vol. 76, Part 1, Amer. Math. Soc., Providence, RI, 2007, 105–139. · Zbl 1221.35261
[2] C. Bandle, ”Über des Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete,” Z. Angew. Math. Phys., 19 (1968), 627–237. · Zbl 0157.42802
[3] C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 1980. · Zbl 0436.35063
[4] F. Brock, ”An isoperimetric inequality for eigenvalues of the Stekloff problem,” Z. Angew. Math. Mech., 81:1 (2001), 69–71. · Zbl 0971.35055
[5] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser, Boston, MA, 2005. · Zbl 1117.49001
[6] D. Bucur and A. Henrot, ”Minimization of the third eigenvalue of the Dirichlet Laplacian,” Proc. Roy. Soc. London, Ser. A, 456:1996 (2000), 985–996. · Zbl 0974.35082
[7] R. Courant, ”Beweis des Satzes, da und gegebener Spannung die kreisförmige den tiefsten Grundton besitzt,” Math. Z., 1:2–3 (1918), 321–328. · JFM 46.0740.02
[8] M. Delfour and J.-P. Zolěsio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control, vol. 4, SIAM, Philadelphia, 2001. · Zbl 1002.49029
[9] Z. Ding, ”A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Amer. Math. Soc., 124:2 (1996), 591–600. · Zbl 0841.46021
[10] B. Dittmar, ”Sums of reciprocal Stekloff eigenvalues,” Math. Nachr., 268 (2004), 44–49. · Zbl 1054.35041
[11] J. Edward, ”An inequality for Steklov eigenvalues for planar domains,” Z. Angew. Math. Phys., 45:3 (1994), 493–496. · Zbl 0868.35078
[12] G. Faber, ”Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt,” Sitzungberichte der mathematischphysikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München Jahrgang, 1923, 169–172. · JFM 49.0342.03
[13] D. W. Fox and J. P. Kuttler, ”Sloshing frequencies,” Z. Angew. Math. Phys., 34:5 (1983), 668–696. · Zbl 0539.76022
[14] A. Girouard, N. Nadirashvili, and I. Polterovich, ”Maximization of the second positive Neumann eigenvalue for planar domains,” J. Differential Geom., 83:3 (2009), 637–662. · Zbl 1186.35120
[15] R. Hempel, L. Seco, and B. Simon, ”The essential spectrum of Neumann Laplacians on some bounded singular domains,” J. Funct. Anal., 102:2 (1991), 448–483. · Zbl 0741.35043
[16] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Birkhäuser, Basel, 2006. · Zbl 1109.35081
[17] A. Henrot and M. Pierre, Variation et optimisation de formes, Springer-Verlag, Berlin, 2005. · Zbl 1098.49001
[18] A. Henrot, G. Philippin, and A. Safoui, Some isoperimetric inequalities with application to the Stekloff problem, http://arxiv.org/abs/0803.4242 . · Zbl 1155.26019
[19] J. Hersch, ”Quatre propriétés isopérimétriques de membranes sphériques homog‘enes,” C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1645–A1648. · Zbl 0224.73083
[20] J. Hersch, L. E. Payne, and M. M. Schiffer, ”Some inequalities for Stekloff eigenvalues,” Arch. Rat. Mech. Anal., 57 (1974), 99–114. · Zbl 0315.35069
[21] S. Jimbo and Y. Morita, ”Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels,” Comm. Partial Differential Equations, 17:3–4 (1992), 523–552. · Zbl 0766.35029
[22] E. Krahn, ”Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises,” Math. Ann., 94:1 (1925), 97–100. · JFM 51.0356.05
[23] E. Krahn, ”Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen,” Acta Comm. Unic. Dorpat, A9 (1926), 1–44. · JFM 52.0510.03
[24] J. R. Kuttler and V. G. Sigillito, ”An inequality of a Stekloff eigenvalue by the method of defect,” Proc. Amer. Math. Soc., 20 (1969), 357–360. · Zbl 0176.09901
[25] N. Nadirashvili, ”Isoperimetric inequality for the second eigenvalue of a sphere,” J. Differential Geom., 61:2 (2002), 335–340. · Zbl 1071.58024
[26] L. Payne, ”Isoperimetric inequalities and their applications,” SIAM Rev., 9:3 (1967), 453–488. · Zbl 0154.12602
[27] J. W. S. Rayleigh, The Theory of Sound, vol. 1, McMillan, London, 1877. · JFM 15.0848.02
[28] W. Stekloff, ”Sur les problèmes fondamentaux de la physique mathématique,” Ann. Sci. Ecole Norm. Sup., 19 (1902), 455–490. · JFM 33.0800.01
[29] G. Szego, ”Inequalities for certain eigenvalues of a membrane of given area,” J. Rational Mech. Anal., 3 (1954), 343–356. · Zbl 0055.08802
[30] M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. · Zbl 0869.35003
[31] G. Uhlmann and J. Sylvester, ”The Dirichlet to Neumann map and applications,” in: Inverse Problems in Partial Differential Equations (Arcata, CA, 1989), SIAM, Philadelphia, PA, 1990, 101–139.
[32] H. F. Weinberger, ”An isoperimetric inequality for the N-dimensional free membrane problem,” J. Rational Mech. Anal., 5 (1956), 633–636. · Zbl 0071.09902
[33] R. Weinstock, ”Inequalities for a classical eigenvalue problem,” J. Rational Mech. Anal., 3 (1954), 745–753. · Zbl 0056.09801
[34] A. Wolf and J. Keller, ”Range of the first two eigenvalues of the Laplacian,” Proc. Roy. Soc. London, Ser. A, 447 (1994), 397–412. · Zbl 0816.35097
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.