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

ISBN-13: 9783031299728 / Angielski / Twarda / 2023

Pascal Auscher;Moritz Egert

Opis

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.

Spis treści

Chapter. 1. Introduction and main results

Chapter. 2. Preliminaries on function spaces

Chapter. 3. Preliminaries on operator theory

Chapter. 4. Hp - Hq bounded families

Chapter. 5. Conservation properties

Chapter. 6. The four critical numbers

Chapter. 7. Riesz transform estimates: Part I

Chapter. 8. Operator-adapted spaces

Chapter. 9. Identification of adapted Hardy spaces

Chapter. 10. A digression: H -calculus and analyticity

Chapter. 11. Riesz transform estimates: Part II

Chapter. 12. Critical numbers for Poisson and heat semigroups

Chapter. 13. Lp boundedness of the Hodge projector

Chapter. 14. Critical numbers and kernel bounds

Chapter. 15. Comparison with the Auscher–Stahlhut interval

Chapter. 16. Basic properties of weak solutions

Chapter. 17. Existence in Hp Dirichlet and Regularity problems

Chapter. 18. Existence in the Dirichlet problems with data

Chapter. 19. Existence in Dirichlet problems with fractional regularity data

Chapter. 20. Single layer operators for L and estimates for L-1

Chapter. 21. Uniqueness in regularity and Dirichlet problems

Chapter. 22. The Neumann problem

Appendix A. Non-tangential maximal functions and traces

Appendix B. The Lp-realization of a sectorial operator in L2

References

Index

O autorze

​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 of Robert 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.