On some rescaled shape optimization problems
Giuseppe Buttazzo, Alfred Wagner · Advances in Calculus of Variations · 2010
We consider Cheeger-like shape optimization problems of the form Min{|Ω| α J (Ω) : Ω ⊂ D } where D is a given bounded domain and α is above the natural scaling. We show the existence of a solution and analyze as J (Ω) the particular cases of the compliance functional C (Ω) and of the first eigenvalue λ 1 (Ω) of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.