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Existence and Regularity Results for Some Shape Optimization Problems

  • Book
  • © 2015

Overview

Authors:
  1. Bozhidar Velichkov

    1. Laboratoire Jean Kuntzmann (LJK), Universite Joseph Fourier, Grenoble Cedex 9, France

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  • 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

Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 19)

Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction and Examples

    • Bozhidar Velichkov
    Pages 1-12

  3. Shape optimization problems in a box

    • Bozhidar Velichkov
    Pages 13-58

  4. Capacitary measures

    • Bozhidar Velichkov
    Pages 59-136

  5. Subsolutions of shape functionals

    • Bozhidar Velichkov
    Pages 137-201

  6. Shape supersolutions and quasi-minimizers

    • Bozhidar Velichkov
    Pages 203-257

  7. Spectral optimization problems in ℝd

    • Bozhidar Velichkov
    Pages 259-306

  8. Appendix: Shape optimization problems for graphs

    • Bozhidar Velichkov
    Pages 307-335

  9. Back Matter

    Pages 337-349
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Keywords

About this book

​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. 

Authors and Affiliations

  • Laboratoire Jean Kuntzmann (LJK), Universite Joseph Fourier, Grenoble Cedex 9, France

    Bozhidar Velichkov

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