Reflection groups and the eigenvalue problems of vibrating membranes with mixed boundary conditions. (English) Zbl 0552.35014

Let (M,g) be an n-dimensional space form of constant curvature (the Euclidean space \(R^ n\), the standard sphere \(S^ n\) or the hyperbolic space \(H^ n)\). Let \(\Omega\) be a bounded domain in M with an appropriately regular boundary \(\partial \Omega\). This paper refers to three different boundary value eigenvalue problems for the non-negative Laplacian of (M,g). \(S_ 1(\Omega)\), \(S_ 2(\Omega)\), \(S_ 3(\Omega)\) denote the spectra of these problems respectively. Two domains \(\Omega\), \({\tilde \Omega}\) are called congruent in the space form (M,g) if there exists an isometry \(\phi\) of (M,g) such that \(\phi (\Omega)={\tilde \Omega}.\)
In an earlier paper [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 441- 456 (1982; Zbl 0505.58036)] the author proved that there exist two domains \(\Omega\), \({\tilde \Omega}\) in x such that \(S_ 1(\Omega)=S_ 1({\tilde \Omega})\) and \(S_ 2(\Omega)=S_ 2({\tilde \Omega})\), but \(\Omega\) and \({\tilde \Omega}\) are not congruent in X, where \(X\equiv R^ n\) or \(X\equiv S^{n-1}\), \(n\geq 4.\)
In this paper the author gives the proof of the following: Let (M,g) be simply connected with \(n\geq 4\). Then there exist two domains \(\Omega\), \({\tilde \Omega}\) in (M,g) such that \(S_ 1(\Omega)=S_ 1({\tilde \Omega})\), \(S_ 2(\Omega)=S_ 2({\tilde \Omega})\) and \(S_ 3(\Omega)=S_ 3({\tilde \Omega})\) for each real number \(\rho\), but \(\Omega\) and \({\tilde \Omega}\) are not congruent in (M,g).
Reviewer: I.Ecsedi


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P05 General topics in linear spectral theory for PDEs


Zbl 0505.58036
Full Text: DOI


[1] C. BANDLE, Isoperimetrc Inequalities and Applications, Pitman, Boston, London, Melbourne, 1980. · Zbl 0436.35063
[2] P. BEJRARD AND G. BESSON, Spectres et groupes cristallographiques, II: domaine spherique, Ann. Inst. Fourier, Grenoble, 30 (3) (1980), 237-248. · Zbl 0426.35073
[3] N. BOURBAKI, Groupes et algebres de Lie, Chapitres 4 a 6, Hermann, Paris, 1968 · Zbl 0186.33001
[4] M. KAC, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23 · Zbl 0139.05603
[5] S. KOBAYASHI AND K. NoMizu, Foundations of Differential Geometry, I, Interscienc Publ., New York, London, Sydney, 1963. · Zbl 0119.37502
[6] H. PRUFER, Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Func tionnen, Math. Ann. 95 (1926), 499-518. · JFM 52.0455.01
[7] H. URAKAWA, Bounded domains which are isospectral but not congruent, Ann. scient Ecol. Norm. Sup. 15 (3) (1982), 441-456. · Zbl 0505.58036
[8] K. YOSIDA, Method of Solving Differential Equations (in Japanese), 2nd ed., Iwanami, Tokyo, 1978. · Zbl 1098.81716
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.