  We consider spectral optimization problems of the form
$$\min\Big\{\lambda_1(\Omega;D):\ \Omega\subset D,\ |\Omega|=1\Big\},$$ 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 $$\lambda_1(\Omega;D)=\min\left\{\int_\Omega|\nabla
u|^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{
on }\partial\Omega\cap D,\ \|u\|_{L^2(\Omega)}=1\right\}$$ 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.
