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000726192 019__ $$a914434612
000726192 020__ $$a9788876425271$$qelectronic book
000726192 020__ $$a8876425276$$qelectronic book
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000726192 0247_ $$a10.1007/978-88-7642-527-1$$2doi
000726192 035__ $$aSP(OCoLC)ocn905543987
000726192 035__ $$aSP(OCoLC)905543987$$z(OCoLC)914434612
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000726192 050_4 $$aQA402.5
000726192 08204 $$a519.6$$223
000726192 1001_ $$aVelichkov, Bozhidar,$$eauthor.
000726192 24510 $$aExistence and regularity results for some shape optimization problems$$h[electronic resource] /$$cBozhidar Velichkov.
000726192 264_1 $$aPisa :$$bEdizioni della Normale,$$c2015.
000726192 300__ $$a1 online resource.
000726192 336__ $$atext$$btxt$$2rdacontent
000726192 337__ $$acomputer$$bc$$2rdamedia
000726192 338__ $$aonline resource$$bcr$$2rdacarrier
000726192 4901_ $$aTesi ;$$v19
000726192 504__ $$aIncludes bibliographical references.
000726192 5050_ $$aCover; Title Page; Copyright Page; Table of Contents; Preface; Résumé of the main results; Chapter 1 Introduction and Examples; 1.1. Shape optimization problems; 1.2. Why quasi-open sets?; 1.3. Compactness and monotonicity assumptions in the shape optimization; 1.4. Lipschitz regularity of the state functions; Chapter 2 Shape optimization problems in a box; 2.1. Sobolev spaces on metric measure spaces; 2.2. The strong-γ and weak-γ convergence of energy domains; 2.2.1. The weak-γ -convergence of energy sets; 2.2.2. The strong-γ -convergence of energy sets
000726192 5058_ $$a2.2.3. From the weak-γ to the strong-γ -convergence2.2.4. 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
000726192 5058_ $$a2.4.7. Shape optimization in Carnot-Caratheodory space2.4.8. Shape optimization in measure metric spaces; Chapter 3 Capacitary measures; 3.1. Sobolev spaces in Rd; 3.1.1. Concentration-compactness principle; 3.1.2. Capacity, quasi-open sets and quasi-continuous functions; 3.2. Capacitary measures and the spaces H1μ; 3.3. Torsional rigidity and torsion function; 3.4. PDEs involving capacitary measures; 3.4.1. Almost subharmonic functions; 3.4.2. Pointwise definition, semi-continuity and vanishing at infinity for solutions of elliptic PDEs
000726192 5058_ $$a3.4.3. The set of finiteness Ωμ of a capacitary measure3.4.4. The resolvent associated to a capacitary measure μ; 3.4.5. Eigenvalues and eigenfunctions of the operator -Δ + μ; 3.4.6. Uniform approximation with solutions of boundary value problems; 3.5. The γ -convergence of capacitary measures; 3.5.1. Completeness of the γ -distance; 3.5.2. The γ -convergence of measures and the convergence of the resolvents Rμ; 3.7. Concentration-compactness principle for capacitary measures; 3.7.1. The γ -distance between comparable measures; 3.7.2. The concentration-compactness principle
000726192 5058_ $$aChapter 4 Subsolutions of shape functionals4.1. Introduction; 4.2. Shape subsolutions for the Dirichlet Energy; 4.3. Interaction between energy subsolutions; 4.3.1. Monotonicity theorems; 4.3.2. The monotonicity factors; 4.3.3. The two-phase monotonicity formula; 4.3.4. Multiphase monotonicity formula; 4.3.5. The common boundary of two subsolutions. Application of the two-phase monotonicity formula.; 4.3.6. Absence of triple points for energy subsolutions. Application of the multiphase monotonicity formula; 4.4. Subsolutions for spectral functionals with measure penalization
000726192 506__ $$aAccess limited to authorized users.
000726192 520__ $$aWe study the existence and regularity of optimal domains for functionals depending on the spectrum of the Dirichlet Laplacian or of more general Schrödinger operators. The domains are subject to perimeter and volume constraints; we also take into account the possible presence of geometric obstacles. We investigate the properties of the optimal sets and of the optimal state functions. In particular, we prove that the eigenfunctions are Lipschitz continuous up to the boundary and that the optimal sets subject to the perimeter constraint have regular free boundary. We also consider spectral optimization problems in non-Euclidean settings and optimization problems for potentials and measures, as well as multiphase and optimal partition problems.
000726192 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed March 26, 2015).
000726192 650_0 $$aMathematical optimization.
000726192 650_0 $$aFunction spaces.
000726192 77608 $$iPrint version:$$z9788876425264
000726192 830_0 $$aTesi (Pisa. Italy) ;$$v19.
000726192 852__ $$bebk
000726192 85640 $$3SpringerLink$$uhttps://univsouthin.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-88-7642-527-1$$zOnline Access$$91397441.1
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