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  • Book
  • Open Access
  • © 2023

Regularity of the One-phase Free Boundaries

Authors:

  1. Bozhidar Velichkov

    1. Department of Mathematics, University of Pisa, Pisa, Italy

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  • This book is open access, which means that you have free and unlimited access

  • Winner of the 2019 Book Prize of the Unione Matematica Italiana

  • Freely available to all readers, this book is open access

  • Provides a comprehensive and self-contained Introduction to the free boundary regularity theory

Part of the book series: Lecture Notes of the Unione Matematica Italiana (UMILN, volume 28)

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Softcover Book USD 49.99
Price excludes VAT (USA)
  • ISBN: 978-3-031-13237-7
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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiii
    PDF

  2. Introduction and Main Results

    • Bozhidar Velichkov
    Pages 1-22Open Access

  3. Existence of Solutions, Qualitative Properties and Examples

    • Bozhidar Velichkov
    Pages 23-38Open Access

  4. Lipschitz Continuity of the Minimizers

    • Bozhidar Velichkov
    Pages 39-57Open Access

  5. Non-degeneracy of the Local Minimizers

    • Bozhidar Velichkov
    Pages 59-64Open Access

  6. Measure and Dimension of the Free Boundary

    • Bozhidar Velichkov
    Pages 65-72Open Access

  7. Blow-Up Sequences and Blow-Up Limits

    • Bozhidar Velichkov
    Pages 73-88Open Access

  8. Improvement of Flatness

    • Bozhidar Velichkov
    Pages 89-109Open Access

  9. Regularity of the Flat Free Boundaries

    • Bozhidar Velichkov
    Pages 111-123Open Access

  10. The Weiss Monotonicity Formula and Its Consequences

    • Bozhidar Velichkov
    Pages 125-147Open Access

  11. Dimension of the Singular Set

    • Bozhidar Velichkov
    Pages 149-161Open Access

  12. Regularity of the Free Boundary for Measure Constrained Minimizers

    • Bozhidar Velichkov
    Pages 163-187Open Access

  14. Back Matter

    Pages 223-247
    PDF
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About this book

This open access book is an introduction to the regularity theory for free boundary problems. The focus is on the one-phase Bernoulli problem, which is of particular interest as it deeply influenced the development of the modern free boundary regularity theory and is still an object of intensive research. 

The exposition is organized around four main theorems, which are dedicated to the one-phase functional in its simplest form. Many of the methods and the techniques presented here are very recent and were developed in the context of different free boundary problems. We also give the detailed proofs of several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories.

This book is aimed at graduate students and researches and is accessible to anyone with a moderate level of knowledge of elliptical PDEs.
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Keywords

  • Open Access
  • Free Boundary Problems
  • Regularity
  • One-phase Problem
  • Bernoulli Free Boundary Problem
  • Alt-Caffarelli
  • Monotonicity Formulas
  • Epiperimetric Inequality
  • Partial Differential Equations
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Authors and Affiliations

  • Department of Mathematics, University of Pisa, Pisa, Italy

    Bozhidar Velichkov

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About the author

Bozhidar Velichkov is working in the fields of Calculus of Variations and Partial Differential Equations, in particular, his research is focused on the regularity and the local structure of the solutions to free boundary problems. He has several important contributions to the theory of the vectorial free boundary problems and developed new tools as the epiperimetric and the log-epiperimetric inequalities for free boundary problems.


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Bibliographic Information

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Softcover Book USD 49.99
Price excludes VAT (USA)
  • ISBN: 978-3-031-13237-7
  • Dispatched in 3 to 5 business days
  • Exclusive offer for individuals only
  • Free shipping worldwide
    See shipping information.
  • Tax calculation will be finalised during checkout
Learn about institutional subscriptions