Optimization of the principal eigenvalue under mixed boundary conditions

We investigate minimization and maximization of the principal eigenvalue of the Laplacian under mixed boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, these optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. We prove existence and uniqueness re- sults, and present some features of the optimizers. In special cases, we prove results of symmetry and results of symmetry breaking for the minimizer.