  We study extrema of the first and the second mixed eigenvalues of the
Laplacian on the disk among some families of Dirichlet-Neumann boundary
conditions. We show that the minimizer of the second eigenvalue among all mixed
boundary conditions lies in a compact 1-parameter family for which an explicit
description is given. Moreover, we prove that among all partitions of the
boundary with bounded number of parts on which Dirichlet and Neumann conditions
are imposed alternately, the first eigenvalue is maximized by the uniformly
distributed partition.
