The inclusion of free membrane eigenvalues (Q1347026)

Eigenvalues of the free membrane are considered. The inclusion of eigenvalues \(\mu\) of the Neumann eigenvalue problem \(\Delta u + \mu u = 0\) in \(G\), \((\partial/\partial n) u = 0\) on \(\Gamma\), is studied (\(G\) is a two-dimensional and bounded domain with an analytic or strictly polygonal boundary \(\Gamma\), \(n\) is directed into the exterior). Two- sided bounds from a given approximate eigenpair \((u_ *, \mu_ *)\) can be computed by the developed method. These bounds depend on the quadratic forms \(\int_ G(\Delta u_ * + \mu_ * u_ *)^ 2 df\) and \(\oint_ \Gamma({\partial \over \partial n} u_ *)^ 2 ds\) only. Two new inclusion theorems are proved for the eigenvalues of the Neumann eigenvalue problem without the constraints by \textit{J. R. Kuttler} and \textit{V. G. Sigillito} [Estimating eigenvalues with a posteriori/a priori inequalities. (1985; Zbl 0634.65100)]. As an example close eigenvalue bounds are computed for an \(L\)-shaped polygon consisting of three unit squares.