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Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure

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  • © 2023

Overview

Authors:
  1. Pascal Auscher

    1. Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France

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  2. Moritz Egert

    1. Department of Mathematics, TU Darmstadt, Darmstadt, Germany

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  • Winner of the 2022 Ferran Sunyer i Balaguer Prize
  • Presents new results that settle the study of boundary value problems for elliptic systems with block structure
  • Unifies and improves machinery developed over the last two decades

Part of the book series: Progress in Mathematics (PM, volume 346)

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About this book

In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents.  Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established.  The presented uniqueness results are completely new, and the authors also elucidate optimal ranges for problems with fractional regularity data.

The first part of the monograph, which can be read independently, provides optimal ranges of exponents for functional calculus and adapted Hardy spaces for the associated boundary operator.  Methods use and improve, with new results, all the machinery developed over the last two decades to study such problems:  the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangential estimates and square functions, and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions.

Keywords

Table of contents (22 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Introduction and Main Results

    • Pascal Auscher, Moritz Egert
    Pages 1-22

  3. Preliminaries on Function Spaces

    • Pascal Auscher, Moritz Egert
    Pages 23-32

  4. Preliminaries on Operator Theory

    • Pascal Auscher, Moritz Egert
    Pages 33-42

  5. \(\operatorname {H}^p - \operatorname {H}^q\) Bounded Families

    • Pascal Auscher, Moritz Egert
    Pages 43-56

  6. Conservation Properties

    • Pascal Auscher, Moritz Egert
    Pages 57-61

  7. The Four Critical Numbers

    • Pascal Auscher, Moritz Egert
    Pages 63-72

  8. Riesz Transform Estimates: Part I

    • Pascal Auscher, Moritz Egert
    Pages 73-83

  9. Operator-Adapted Spaces

    • Pascal Auscher, Moritz Egert
    Pages 85-109

  10. Identification of Adapted Hardy Spaces

    • Pascal Auscher, Moritz Egert
    Pages 111-140

  11. A Digression: \( \operatorname {H}^\infty \)-Calculus and Analyticity

    • Pascal Auscher, Moritz Egert
    Pages 141-142

  12. Riesz Transform Estimates: Part II

    • Pascal Auscher, Moritz Egert
    Pages 143-146

  13. Critical Numbers for Poisson and Heat Semigroups

    • Pascal Auscher, Moritz Egert
    Pages 147-157

  14. \(\operatorname {L}^p\) Boundedness of the Hodge Projector

    • Pascal Auscher, Moritz Egert
    Pages 159-168

  15. Critical Numbers and Kernel Bounds

    • Pascal Auscher, Moritz Egert
    Pages 169-183

  16. Comparison with the Auscher–Stahlhut Interval

    • Pascal Auscher, Moritz Egert
    Pages 185-187

  17. Basic Properties of Weak Solutions

    • Pascal Auscher, Moritz Egert
    Pages 189-195

  18. Existence in \(\operatorname {H}^p\) Dirichlet and Regularity Problems

    • Pascal Auscher, Moritz Egert
    Pages 197-211

  19. Existence in the Dirichlet Problems with \( \dot {\Lambda }^{\alpha }\)-Data

    • Pascal Auscher, Moritz Egert
    Pages 213-236

  20. Existence in Dirichlet Problems with Fractional Regularity Data

    • Pascal Auscher, Moritz Egert
    Pages 237-252
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Authors and Affiliations

  • Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France

    Pascal Auscher

  • Department of Mathematics, TU Darmstadt, Darmstadt, Germany

    Moritz Egert

About the authors

​Pascal Auscher is professor of Mathematics in the Laboratoire de Mathématiques d’Orsay at the Université Paris-Saclay.  He received his PhD in 1989 at Université Paris-Dauphine under the supervision of Yves Meyer. He is a specialist in harmonic analysis and contributed to the theory of wavelets and to partial differential equations. An outstanding contribution is his participation to the proof of the Kato conjecture in any dimension, which is a starting point for boundary value problems. He has launched a systematic theory of  Hardy spaces associated to operators in relation to tent spaces, which is one core of the present monograph.  He has recently served as director of the national institute for mathematical sciences and interactions (Insmi) at the national center for scientific research (CNRS).Moritz Egert is professor of Mathematics at the Technical University of Darmstadt. He received his PhD in 2015 in Darmstadt under the supervision ofRobert Haller and was subsequently Maître de Conférences in the Laboratoire de Mathématiques d’Orsay at the Université Paris-Saclay. He is a specialist in harmonic analysis and partial differential equations. In his research, he combines methods from harmonic analysis, operator theory and geometric measure theory to study partial differential equations in non-smooth settings. 

Bibliographic Information

  • Book Title: Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure

  • Authors: Pascal Auscher, Moritz Egert

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-29973-5

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-29972-8Published: 28 July 2023

  • Softcover ISBN: 978-3-031-29975-9Due: 28 August 2023

  • eBook ISBN: 978-3-031-29973-5Published: 27 July 2023

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: XIII, 310

  • Number of Illustrations: 12 b/w illustrations, 11 illustrations in colour

  • Topics: Analysis, Abstract Harmonic Analysis, Operator Theory, Functional Analysis

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