  In this paper, we discuss a class of spectral partition problems with a
measure constraint, for partitions of a given bounded connected open set. We
establish the existence of an optimal open partition, showing that the
corresponding eigenfunctions are locally Lipschitz continuous, and obtain some
qualitative properties for the partition. The proof uses an equivalent weak
formulation that involves a minimization problem of a penalized functional
where the variables are functions rather than domains, suitable deformations,
blowup techniques, and a monotonicity formula.
