Spectral optimization problems with internal constraint

Bucur, Dorin; Buttazzo, Giuseppe; Velichkov, Bozhidar

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 477-495.

We consider spectral optimization problems with internal inclusion constraints, of the form

$$\min \{{\lambda }_k\left(\Omega \right):D\subset \Omega \subset {ℝ}^d,|\Omega |=m\},$$

where the set D is fixed, possibly unbounded, and ${\lambda }_k$ is the k-th eigenvalue of the Dirichlet Laplacian on Ω. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

Classification : 49J45, 49R05, 35P15, 47A75, 35J25
Mots clés : Shape optimization, Capacity, Eigenvalues, Sobolev spaces, Concentration-compactness

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