Existence and Regularity Results for Some Shape Optimization Problems

✍ Scribed by Bozhidar Velichkov

• Publisher: Edizioni della Normale
• Year: 2015
• Tongue: English
• Leaves: 362
• Series: Publications of the Scuola Normale Superiore
• Category: Library

✦ Synopsis

​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 account the possible presence of geometric obstacles. We investigate the properties of the optimal sets and of the optimal state functions. In particular, we prove that the eigenfunctions are Lipschitz continuous up to the boundary and that the optimal sets subject to the perimeter constraint have regular free boundary. We also consider spectral optimization problems in non-Euclidean settings and optimization problems for potentials and measures, as well as multiphase and optimal partition problems

✦ Table of Contents

Front Matter....Pages i-xvi
Introduction and Examples....Pages 1-12
Shape optimization problems in a box....Pages 13-58
Capacitary measures....Pages 59-136
Subsolutions of shape functionals....Pages 137-201
Shape supersolutions and quasi-minimizers....Pages 203-257
Spectral optimization problems in ℝ d ....Pages 259-306
Appendix: Shape optimization problems for graphs....Pages 307-335
Back Matter....Pages 337-349

✦ Subjects

Mathematical optimization;Mathematics