Lower bounds for the eigenvalues of the fixed vibrating membrane problems (Q759913)

Let \(\Omega\) be a bounded domain in the Euclidean space \(R^ n\). The classical fixed vibrating membrane problem is the following: (1) \(\Delta u=\lambda u\) on \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Delta\) is the Laplacian of \(R^ n\). Let \(\{\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ k\leq...\uparrow \infty \}\) be the eigenvalues of the problem (1) counted with their multiplicities. In this paper is proved that (2) \(\lambda_ k\geq C_ n Vol(\Omega)^{- 2/n}k^{2/n}\delta_ L(\Omega)^{2/n}\). In (2) \(C_ n=4\pi^ 2\omega_ n^{-2/n},\) \(\omega_ n\) is the volume of the unit ball, Vol(\(\Omega)\) is the volume of \(\Omega\) and \(S_ L(\Omega)\) is the lattice packing density of \(\Omega\) in the sense of Rogers. The inequality (2) is the generalization of Polya's one of the results.