Minimization of eigenvalues by free boundary - free discontinuity methods

a) Under the hypotheses above, problems (P1) and (P2) have at least one solution Ω, which is a bounded set. b) If F is moreover locally Lipschitz, then every solution Ω of (P2) is a bounded set with finite perimeter. c) If in addition F is locally bi-Lipschitz in at least one variable, then every solution Ω of (P1) is a bounded set with finite perimeter. The existence result for (P1) and (P2) under the additional constraint that the competing domains are contained in a prescribed bounded open set was proved by Buttazzo and Dal Maso in [6]. Point a) is proved in [8] and is based on a geometric argument allowing to cut small (unbounded) regions of a set while keeping control on the spectrum. Points b) and c) are a consequence of the analysis of the shape subsolutions by free boundary methods (see [2]) and can be complemented with more information related to the inner density of the optimal shapes. In all cases, the diameter of the optimal sets are controlled.