The inclusion of Stekloff eigenvalues by boundary data approximation
Let \(G\) be a simply connected plane domain bounded by an analytic curve \(\Gamma\). The following Stekloff eigenvalue problem \[ \Delta u = 0 \quad \text{in} \quad G \quad \text{and} \quad {\partial u \over \partial n} = pu \quad \text{ on } \Gamma \] is considered. Eigenvalue inclusion theorems are derived to obtain two sided bounds for the eigenvalues. The test functions \(u_ *\) must satisfy only the equation \(\Delta u_ * = 0\). The applied test functions build up from the singularity functions \(\ln | x - y_ n |\), where the source points \(y_ n\) lie in the exterior of \(G\). The eigenvalue inclusion theorem is combined with the boundary collocation method. Numerical examples illustrate the advantage of the presented method. Some possibilities of the extension are also mentioned. The inclusion theorems can be interpreted as norm inequalities, too.
scientific article; zbMATH DE number 636030