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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.
List of contents
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.
About the author
¿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.
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"The book under review is both a useful tool and a useful reference for researchers on the topic." (Massimo Lanza de Cristoforis, zbMATH 1542.35146, 2024)
