The free boundary problem in the optimization of composite membranes (Q2708674)

The authors study qualitative properties of solutions of an eigenvalue optimization problem that is formulated as follows. Find a domain \(D\) of a prescribed measure, \(D \subset \Omega \subset \mathbb{R}^n\) such that the fundamental (first) eigenvalue of the boundary value problem NEWLINE\[NEWLINE -\Delta u -\alpha \phi_D u=\lambda u , \quad x\in \Omega NEWLINE\]NEWLINE NEWLINE\[NEWLINEu=0, \quad x \in \partial \Omega NEWLINE\]NEWLINE is the least for all \(D\). Here \(\phi_D\) is the characteristic function of \(D\). The work is a continuation of a paper by \textit{S. Chanillo, D. Grieser, M. Imai, K. Kurata} and \textit{I. Ohnishi} [Commun. Math. Phys. 214, No. 2, 315-337 (2000; Zbl 0972.49030)].NEWLINENEWLINEFor the entire collection see [Zbl 0956.00042].