The Free Boundary Problem in the Optimization of Composite Membranes

Details The Free Boundary Problem in the Optimization of Composite Membranes The Free Boundary Problem in the Optimization of Composite Membranes

1999-12-15

arxiv.org/abs/math/9912117v1

Mathematics

Analysis of PDEs

20 pages

Scientific paper

This is a continuation of the paper 'Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes' by S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Again, we consider the following eigenvalue optimization problem: Given a bounded domain $\Omega\subset\R^n$ and numbers $\alpha\geq 0$, $A\in [0,|\Omega|]$, find a subset $D\subset\Omega$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-\Delta + \alpha \chi_D$ is as small as possible. In this paper we focus on the study of the free boundary of optimal solutions on general domains.

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