Jochen Denzler: Windows of given area with minimal heat diffusion Transactions of the AMS 351 (1999), 569-580

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