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Geometric Analysis of PDE and Several Complex Variables
This volume is dedicated to Francois Treves, who made substantial contributions to the geometric side of the theory of partial differential equations (PDEs) and several complex variables. One of his best-known contributions, reflected in many of the articles here, is the study of hypo-analytic structures. An international group of well-known mathematicians contributed to the volume. Articles generally reflect the interaction of geometry and analysis that is typical of Treves's work, such as the study of the special types of partial differential equations that arise in conjunction with CR-manifolds, symplectic geometry, or special families of vector fields. There are many topics in analysis and PDEs covered here, unified by their connections to geometry. The material is suitable for graduate students and research mathematicians interested in geometric analysis of PDEs and several complex variables.
Analytic Partial Differential Equations
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 ...
Multidimensional Complex Analysis and Partial Differential Equations
This collection of papers by outstanding contributors in analysis, partial differential equations and several complex variables is dedicated to Professor Treves in honour of his 65th birthday. There are five excellent survey articles covering analytic singularities, holomorphically nondegenerate algebraic hypersurfaces, analyticity of CR mappings, removable singularities of vector fields and local solvability for systems of vector fields. The other papers are original research contributions on topics such as Klein-Gordon and Dirac equations, Toeplitz operators, elliptic structures, complexification of Lie groups, and pseudo-differential operators.
Basic Linear Partial Differential Equations
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories. The four-part treatment covers the basic examples of linear partial differential equations and their fundamental solutions; the Cauchy problem; boundary value problems; and mixed problems and evolution equations. Nearly 400 exercises appear throughout the text, several containing detailed information that enables readers to reconstruct the proofs.
Topological Vector Spaces, Distributions and Kernels
Extending beyond the boundaries of Hilbert and Banach space theory, this text focuses on key aspects of functional analysis, particularly in regard to solving partial differential equations. 1967 edition.
Linear Partial Differential Equations with Constant Coefficients
Existence and approximation theorems for general differential operators -- General L2 estimates -- Fundamental solutions -- The approximation theorem -- Existence theorems for differential operators with constant coefficients -- Convexity with respect to a differential polynomial -- Interior regularity of solutions -- Partial hypoellipticity -- Existence and approximation theorems in spaces of analytic functions -- Appendix A. Semi-algebraic sets -- Appendix B. On uniqueness in the Cauchy problem -- Appendix C. Some formulas of non-commutative algebra.
Locally Convex Spaces and Linear Partial Differential Equations
It is hardly an exaggeration to say that, if the study of general topolog ical vector spaces is justified at all, it is because of the needs of distribu tion and Linear PDE * theories (to which one may add the theory of convolution in spaces of hoi om orphic functions). The theorems based on TVS ** theory are generally of the "foundation" type: they will often be statements of equivalence between, say, the existence - or the approx imability -of solutions to an equation Pu = v, and certain more "formal" properties of the differential operator P, for example that P be elliptic or hyperboJic, together with properties of the manifold X on which P is defined. The latter are generally geometric o...
Hypo-Analytic Structures (PMS-40), Volume 40
In Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports. The contents of this book consist of many results accumulated in the last decade ...
