The spectral drop problem

We consider spectral optimization problems of the form $\min\Big\{λ_1(Ω;D):\ Ω\subset D,\ |Ω|=1\Big\},$ where $D$ is a given subset of the Euclidean space $\mathbb{R}^d$. Here $λ_1(Ω;D)$ is the first eigenvalue of the Laplace operator $-Δ$ with Dirichlet conditions on $\partialΩ\cap D$ and Neumann or Robin conditions on $\partialΩ\cap\partial D$. The equivalent variational formulation $λ_1(Ω;D)=\min\left\{\int_Ω|\nabla u|^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{ on }\partialΩ\cap D,\ \|u\|_{L^2(Ω)}=1\right\}$ reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.

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