*Preprint*

**Inserted:** 12 may 2014

**Last Updated:** 13 may 2014

**Year:** 2014

**Abstract:**

We consider spectral optimization problems of the form

$\min\Big\{\lambda_1(\Omega;D):\ \Omega\subset D,\ \vert\Omega\vert=1\Big\},$

where $D$ is a given subset of the Euclidean space $\mathbb{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lambda_1(\Omega;D)=\min\left\{\int_\Omega\vert\nabla u\vert^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{ on }\partial\Omega\cap D,\ \int_\Omega u^2\,dx=1\right\}$

reminds the classical drop problems, where the first eigenvalue is replaced by the perimeter in $D$. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.

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