On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue (Q2719189)

Let \(\Omega\) be a domain in the Euclidean space and let \(B\) be a ball in \(\Omega\). Consider the operator \(-\Delta+ \alpha\mathbf{1}_B\). The perturbation is an obstacle for \(\alpha= \infty\), a soft obstacle if \(\alpha> 0\), and a well for \(\alpha< 0\). The lowest eigenvalue for all these situations is nondegenerate. It is studied where \(B\) has to be placed in \(\Omega\) such the \(\lambda\) becomes maximal or minimal. The qualitative result is that \(\lambda\) is minimal if \(B\) is near the boundary of \(\Omega\), and \(\lambda\) becomes maximal if \(B\) is in the interior of \(\Omega\). A series of very illustrating examples is given.