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 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.

Provides a detailed and self-contained introduction to the recent results and techniques in shape optimization Presents new techniques concerning the regularity of the optimal sets Self-contained exposition requiring only basic knowledge of Sobolev spaces and BV functions Includes a self-contained and simplified introduction to the existence theory introduced by Buttazzo and Dal Maso in the 90s

Inhalt

1. Introduction and examples.- 2. Shape optimization problems in a box.- 3. Capacitary measures.- 4. Subsolutions of shape functionals.- 5. Shape supersolutions and quasi-minimizers.- 6. Spectral optimization problems in R^d.- 7. Shape optimization problems for graphs.- Bibliography.