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…
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- Full title :Existence and regularity results for some shape optimization problems / Bozhidar Velichkov.
Summary
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.
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- Publication Date:2015.
Publication Statements
- Publisher(s):Edizioni Della Normale
- Place of Publication:Pisa
- Language:eng
- Publication Date:2015.
The Physical Object
- Description:1 online resource
Edition Identifiers
- ISBN:9788876425271, 8876425276, 8876425268, 9788876425264
- DOI:10.1007/978-88-7642-527-1
- oclc:ocn905543987
- LC Classification:QA402.5
- Dewey Decimal:519.6
Table of Contents
Chapter 4 Subsolutions of shape functionals4.1. Introduction; 4.2. Shape subsolutions for the Dirichlet Energy; 4.3. Interaction between energy subsolutions; 4.3.1. Monotonicity theorems; 4.3.2. The monotonicity factors; 4.3.3. The two-phase monotonicity formula; 4.3.4. Multiphase monotonicity formula; 4.3.5. The common boundary of two subsolutions. Application of the two-phase monotonicity formula.; 4.3.6. Absence of triple points for energy subsolutions. Application of the multiphase monotonicity formula; 4.4. Subsolutions for spectral functionals with measure penalization.
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