This book provides a coherent, self-contained introduction to central topics of Analytic Partial Differential Equations in the natural geometric setting. The main themes are the analysis in phase-space of analytic PDEs and the Fourier-Bros-Iagolnitzer (FBI) transform of distributions and hyperfunctions, with application to existence and regularity questions. The book begins by establishing the fundamental properties of analytic partial differential equations, starting with the Cauchy-Kovalevskaya theorem, before presenting an integrated overview of the approach to hyperfunctions via analytic functionals, first in Euclidean space and, once the geometric background has been laid out, on analytic manifolds. Further topics include the proof of the Lojaciewicz inequality and the division of distributions by analytic functions, a detailed description of the Frobenius and Nagano foliations, and the Hamilton-Jacobi solutions of involutive systems of eikonalequations. The reader then enters the realm of microlocal analysis, through pseudodifferential calculus, introduced at a basic level, followed by Fourier integral operators, including those with complex phase-functions (à la Sjöstrand). This culminates in an in-depth discussion of the existence and regularity of (distribution or hyperfunction) solutions of analytic differential (and later, pseudodifferential) equations of principal type, exemplifying the usefulness of all the concepts and tools previously introduced. The final three chapters touch on the possible extension of the results to systems of over- (or under-) determined systems of these equations-a cornucopia of open problems.This book provides a unified presentation of a wealth of material that was previously restricted to research articles. In contrast to existing monographs, the approach of the book is analytic rather than algebraic, and tools such as sheaf cohomology, stratification theory of analyticvarieties and symplectic geometry are used sparingly and introduced as required. The first half of the book is mainly pedagogical in intent, accessible to advanced graduate students and postdocs, while the second, more specialized part is intended as a reference for researchers.

Autorentext

François Treves has worked in the field of partial differential equations for 65 years, publishing 17 books and over 150 papers. His well-known contributions to the field include his work with Louis Nirenberg on the local solvability of linear PDEs, his contributions to the theory of locally integrable structures (presented in his book Hypo-analytic Manifolds, Princeton University Press, 1991), his results on the analyticity of the solutions of sum-of-square PDEs (Tartakoff-Treves theorem), his characterization of the conserved quantities of the KDV equation, and his extension of the Ovsyannikov approach to the Cauchy-Kovalevskaya theorem and beyond.

Inhalt

Part 1. Distributions and Analyticity in Euclidean Space.- Chapter 1. Functions and Differential Operators in Euclidean Space.- Chapter 2. Distributions in Euclidean Space.- Chapter 3. Analytic Tools in Distribution Theory.- Chapter 4. Analyticity of Solutions of Linear PDEs. Basic Results.- Chapter 5. The Cauchy-Kovalevskaya Theorem.- Part 2. Hyperfunctions in Euclidean Space.- Chapter 6. Analytic Functionals in Euclidean Space.- Chapter 7. Hyperfunctions in Euclidean Space.- Chapter 8. Hyperdifferential Operators.- Part 3. Geometric Background.- Chapter 9. Elements of Differential Geometry.- Chapter 10. A Primer on Sheaf Cohomology.- Chapter 11. Distributions and Hyperfunctions on a Manifold.- Chapter 12. Lie Algebras of Vector Fields.- Chapter 13. Elements of Symplectic Geometry.- Part 4. Stratification of Analytic Varieties and Division of Distributions by Analytic Functions.- Chapter 14. Analytic Stratifications.- Chapter 15. Division of Distributions by Analytic Functions.- Part 5. Analytic Pseudodifferential Operators and Fourier Integral Operators.- Chapter 16. Elementary Pseudodifferential Calculus in the $C^1$ Class.- Chapter 17. Analytic Pseudodifferential Calculus.- Chapter 18. Fourier Integral Operators.- Part 6. Complex Microlocal Analysis.- Chapter 19. Classical Analytic Formalism.- Chapter 20. Germ Fourier Integral Operators in Complex Space.- Chapter 21. Germ Pseudodifferential Operators in Complex Space.- Chapter 22. Germ FBI Transforms.- Part 7. Analytic Pseudodifferential Operators of Principal Type.- Chapter 23. Analytic PDEs of Principal Type. Local Solvability.- Chapter 24. Analytic PDEs of Principal Type. Regularity of the Solutions.- Chapter 25. Solvability of Constant Vector Fields of Type (1,0).- Chapter 26. Pseudodifferential Solvability and Property $(\Psi)$.- Chapter 27. Poincaré Lemma in Pseudodifferential Tube Structures.- Index.- Bibliography.