Existence and Regularity Results for Some Shape Optimization Problems
We study the existence and regularity of optimal domains for functionals depending on the spectrum of the Dirichlet Laplacian or of more general Schrödinger operators. The domains are subject to perimeter and volume constraints; we also take into ac
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Existence and Regularity Results for Some Shape Optimization Problems
EDIZIONI DELLA NORMALE
19 TESI THESES
tesi di perfezionamento in Matematica sostenuta il 8 novembre 2013 C OMMISSIONE G IUDICATRICE Luigi Ambrosio, Presidente Dorin Bucur Giuseppe Buttazzo Gianni Dal Maso Lorenzo Mazzieri Andrea Carlo Giuseppe Mennucci Edouard Oudet Michel Pierre
Bozhidar Velichkov Laboratoire Jean Kuntzmann (LJK) Universite Joseph Fourier Tour IRMA, BP 53 51 rue des Mathematiques 38041 Grenoble Cedex 9 France Existence and Regularity Results for Some Shape Optimization Problems
Bozhidar Velichkov
Existence and Regularity Results for Some Shape Optimization Problems
c 2015 Scuola Normale Superiore Pisa
ISBN 978-88-7642-527-1 (eBook) ISBN 978-88-7642-526-4 DOI 10.1007/978-88-7642-527-1
Contents
Preface R´esum´e of the main results
ix xiii
1 Introduction and Examples 1.1. Shape optimization problems . . . . . . . . . . . . . . . 1.2. Why quasi-open sets? . . . . . . . . . . . . . . . . . . . 1.3. Compactness and monotonicity assumptions in the shape optimization . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Lipschitz regularity of the state functions . . . . . . . .
1 1 2
2 Shape optimization problems in a box 2.1. Sobolev spaces on metric measure spaces . . . . . . . . 2.2. The strong-γ and weak-γ convergence of energy domains 2.2.1. The weak-γ -convergence of energy sets . . . . . 2.2.2. The strong-γ -convergence of energy sets . . . . 2.2.3. From the weak-γ to the strong-γ -convergence . 2.2.4. Functionals on the class of energy sets . . . . . . 2.3. Capacity, quasi-open sets and quasi-continuous functions 2.3.1. Quasi-open sets and energy sets from a shape optimization point of view . . . . . . . . . . . . . 2.4. Existence of optimal sets in a box . . . . . . . . . . . . 2.4.1. The Buttazzo-Dal Maso Theorem . . . . . . . . 2.4.2. Optimal partition problems . . . . . . . . . . . . 2.4.3. Spectral drop in an isolated box . . . . . . . . . 2.4.4. Optimal periodic sets in the Euclidean space . . 2.4.5. Shape optimization problems on compact manifolds . . . . . . . . . . . . . . . . . . . . . . . 2.4.6. Shape optimization problems in Gaussian spaces 2.4.7. Shape optimization in Carnot-Caratheodory space
13 13 22 23 25 29 33 37
6 9
43 45 46 47 48 50 51 53 54
vi Bozhidar Velichkov
2.4.8. Shape optimization in measure metric spaces . .
56
3 Capacitary measures 59 3.1. Sobolev spaces in Rd . . . . . . . . . . . . . . . . . . . 60 3.1.1. Concentration-compactness principle . . . . . . 61 3.1.2. Capacity, quasi-open sets and quasi-continuous functions . . . . . . . . . . . . . . . . . . . . . 63 3.2. Capacitary measures and the spaces Hμ1 . . . . . . . . . 67 3.3. Torsional rigidity and torsion function . . . . . . . . . . 72 3.4. PDEs involving capacitary measures . . . . . . . . . . . 77 3.4.1. Almost subharmonic functions . . . . . . . . . . 82 3.4.2. Pointwise deSnition, semi-continuity and vanishing at inSnity for solutions of elliptic PDEs . . . 88 92 3.4.3. The set of Sniteness μ of a capacitary
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