Windows of given area with minimal heat diffusion

For a bounded Lipschitz domain $\Omega$, we show the existence of a measurable set $D\subset \partial \Omega$ of given area such that the first eigenvalue of the Laplacian with Dirichlet conditions on $D$ and Neumann conditions on $\partial \Omega \setminus D$ becomes minimal. If $\Omega$ is a ball, $D$ will be a spherical cap.