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We consider the following eigenvalue optimization in the composite membraneproblem with fractional Laplacian: given a bounded domain $\Omega\subset\mathbb{R}^n$, $\alpha>0$ and $0<A<|\Omega|$, find a subset $D\subset \Omega$of area $A$ such that the first Dirichlet eigenvalue of the operator$(-\Delta)^s+\alpha \chi_D$ is as small as possible. The solution $D$ is calledas an optimal configuration for the data $(\Omega,\alpha,A)$. Looking at thewell-known extension definition for the fractional Laplacian, in the case$s=1/2$ this is essentially the composite membrane problem for which the massis concentrated at the boundary as one is trying to maximize the Stekloveigenvalue.  We prove existence of solutions and study properties of optimal configuration$D$. This is a free boundary problem which could be formulated as a two-sidedunstable obstacle problem.  Moreover, we show that for some rotationally symmetric domains (thin annuli),the optimal configuration is not rotational symmetric, which implies thenon-uniqueness of the optimal configuration $D$. On the other hand, we provethat for a convex domain $\Omega$ having reflection symmetries, the optimalconfiguration possesses the same symmetries, which implies uniqueness of theoptimal configuration $D$ in the ball case.

Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian