Eigenvalues of the Laplacian on symmetric regions (Q1346278)

The author investigates the effect of domain symmetries on the multiplicity of the eigenvalues in the fixed membrane problem \[ \Delta u + \lambda u = 0 \text{ in } \Omega \subset \mathbb{R}^ n, \quad u = 0 \text{ on } \partial \Omega \] with \(C^ r\) boundary, \(r \geq 3\). He gives a very thorough discussion of how higher symmetry of \(\Omega\) implies higher multiplicity of eigenvalues.