## 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

### MSC:

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

Zbl 0505.58036
Full Text:

### References:

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