Author: Tatsuo Nishitani
Publisher: Springer
ISBN: 3319676121
Category : Mathematics
Languages : en
Pages : 215
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Book Description
Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem. A doubly characteristic point of a differential operator P of order m (i.e. one where Pm = dPm = 0) is effectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is effectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between −Pμj and Pμj , where iμj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
Author: Lars Hörmander
Publisher: Springer
ISBN: 3540269649
Category : Mathematics
Languages : en
Pages : 404
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Book Description
Author received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equations
Author: Vincenzo Ancona
Publisher: CRC Press
ISBN: 9780203911143
Category : Mathematics
Languages : en
Pages : 390
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Book Description
Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, Schrödinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 3540499377
Category : Mathematics
Languages : en
Pages : 537
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Book Description
From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987. "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987.
Author: Sigeru Mizohata
Publisher: Academic Press
ISBN: 148326906X
Category : Mathematics
Languages : en
Pages : 186
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Book Description
Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.
Author: Antonio Bove
Publisher: Springer Science & Business Media
ISBN: 0817645217
Category : Mathematics
Languages : en
Pages : 336
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Book Description
Covers phase space analysis methods, including microlocal analysis, and their applications to physics Treats the linear and nonnlinear aspects of the theory of PDEs Original articles are self-contained with full proofs; survey articles give a quick and direct introduction to selected topics evolving at a fast pace Excellent reference and resource for grad students and researchers in PDEs and related fields
Author: Michael Demuth
Publisher: Birkhäuser
ISBN: 3034890923
Category : Mathematics
Languages : en
Pages : 422
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Book Description
The book contains the contributions to the conference on "Partial Differential Equations" held in Holzhau (Germany) in July 1994, where outstanding specialists from analysis, geometry and mathematical physics reviewed recent progress and new interactions in these areas. Topics of special interest at the conference and which now form the core of this volume are hyperbolic operators, spectral theory for elliptic operators, eta-invariant, singular configura- tions and asymptotics, Bergman-kernel, attractors of non-autonomous evolution equations, pseudo-differential boundary value problems, Mellin pseudo- differential operators, approximation and stability problems for elliptic operators, and operator determinants. In spectral theory adiabatic and semiclassical limits, Dirichlet decoupling and domain perturbations, capacity of obstacles, limiting absorption problems, N-body scattering, and number of bound states are considered. Schrödinger operators are studied with magnetic fields, with random and with many-body potentials, and for nonlinear problems. In semigroup theory the Feller property, errors for product formulas, fractional powers of generators, and functional integration for relativistic semigroups are analyzed.
Author: Michael Ruzhansky
Publisher: Springer Science & Business Media
ISBN: 3034804547
Category : Mathematics
Languages : en
Pages : 327
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Book Description
Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area.
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 364200136X
Category : Mathematics
Languages : en
Pages : 363
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Book Description
From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987 "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987 Honours awarded to Lars Hörmander: Fields Medal 1962, Speaker at International Congress 1970, Wolf Prize 1988, AMS Steele Prize 2006
Author: Stefan Hildebrandt
Publisher: Springer
ISBN: 3540459324
Category : Mathematics
Languages : en
Pages : 316
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Book Description
