ABSTRACT |
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain Omega subset of R-n, alpha > 0 and 0 < A < vertical bar Omega vertical bar, find a subset D subset of 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 called as an optimal configuration for the data (Omega, alpha, A). Looking at the well-known extension definition for the fractional Laplacian, in the case s = 1/2 this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to minimize the Steklov eigenvalue. 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-sided unstable obstacle problem. Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration D. On the other hand, we prove that for a convex domain St having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration Omega in the ball case. (C) 2020 Elsevier Inc. All rights reserved. |