Cauchy Problem For Differential Operators With Double Characteristics
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Cauchy Problem For Differential Operators With Double Characteristics
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Author : Tatsuo Nishitani
language : en
Publisher: Springer
Release Date : 2017-11-24
Cauchy Problem For Differential Operators With Double Characteristics written by Tatsuo Nishitani and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-24 with Mathematics categories.
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.
Notas E Communica Es De Matem Tica
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Author : Universidade Federal de Pernambuco. Departamento de Matemática
language : en
Publisher:
Release Date : 1965
Notas E Communica Es De Matem Tica written by Universidade Federal de Pernambuco. Departamento de Matemática and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1965 with Differential operators categories.
Cauchy Problem For Noneffectively Hyperbolic Operators
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Author : Tatsuo Nishitani
language : en
Publisher:
Release Date : 2013-06
Cauchy Problem For Noneffectively Hyperbolic Operators written by Tatsuo Nishitani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06 with Mathematics categories.
Annotation At a double characteristic point of a differential operator with real characteristics, the linearization of the Hamilton vector field of the principal symbol is called the Hamilton map and according to either the Hamilton map has non-zero real eigenvalues or not, the operator is said to be effectively hyperbolic or noneffectively hyperbolic. For noneffectively hyperbolic operators, it was proved in the late of 1970s that for the Cauchy problem to be C well posed the subprincipal symbol has to be real and bounded, in modulus, by the sum of modulus of pure imaginary eigenvalues of the Hamilton map. It has been recognized that what is crucial to the C well-posedness is not only the Hamilton map but also the behavior of orbits of the Hamilton flow near the double characteristic manifold and the Hamilton map itself is not enough to determine completely the behavior of orbits of the flow. Strikingly enough, if there is an orbit of the Hamilton flow which lands tangentially on the double characteristic manifold then the Cauchy problem is not C well posed even though the Levi condition is satisfied, only well posed in much smaller function spaces, the Gevrey class of order 1 s 5 and not well posed in the Gevrey class of order s 5. In this lecture, we provide a general picture of the Cauchy problem for noneffectively hyperbolic operators, from the view point that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/not well-posedness of the Cauchy problem, exposing well/not well-posed results of the Cauchy problem with detailed proofs. Book jacket.
Essentials Of Partial Differential Equations
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Author : Marin Marin
language : en
Publisher: Springer
Release Date : 2018-05-09
Essentials Of Partial Differential Equations written by Marin Marin and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-09 with Technology & Engineering categories.
This book offers engineering students an introduction to the theory of partial differential equations and then guiding them through the modern problems in this subject. Divided into two parts, in the first part readers already well-acquainted with problems from the theory of differential and integral equations gain insights into the classical notions and problems, including differential operators, characteristic surfaces, Levi functions, Green’s function, and Green’s formulas. Readers are also instructed in the extended potential theory in its three forms: the volume potential, the surface single-layer potential and the surface double-layer potential. Furthermore, the book presents the main initial boundary value problems associated with elliptic, parabolic and hyperbolic equations. The second part of the book, which is addressed first and foremost to those who are already acquainted with the notions and the results from the first part, introduces readers to modern aspects of the theory of partial differential equations.
The Cauchy Problem For Higher Order Abstract Differential Equations
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Author : Ti-Jun Xiao
language : en
Publisher: Springer
Release Date : 2013-12-11
The Cauchy Problem For Higher Order Abstract Differential Equations written by Ti-Jun Xiao and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-11 with Mathematics categories.
The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.
Theory Of Differential Equations
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Author : I. M. Gel'fand
language : en
Publisher: Academic Press
Release Date : 2014-05-12
Theory Of Differential Equations written by I. M. Gel'fand and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-12 with Mathematics categories.
Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations. This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cauchy problem, systems that are Petrovski?-correct, and generalized eigenfunctions of self-adjoint operators are also reviewed. This text likewise covers the differentiation of functionals of strongly and weakly bounded variation. This volume is beneficial to students and researchers interested in the theory of differential equations.
Uniqueness And Non Uniqueness In The Cauchy Problem
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Author : Zuily
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-21
Uniqueness And Non Uniqueness In The Cauchy Problem written by Zuily and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-21 with Science categories.
Lectures On Cauchy S Problem In Linear Partial Differential Equations
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Author : Jacques Hadamard
language : en
Publisher:
Release Date : 1923
Lectures On Cauchy S Problem In Linear Partial Differential Equations written by Jacques Hadamard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1923 with Cauchy problem categories.
Pseudo Differential Operators
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Author : M. Taylor
language : en
Publisher: Springer
Release Date : 2006-12-08
Pseudo Differential Operators written by M. Taylor and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-08 with Mathematics categories.
These notes are based on the lectures given on partial differential equations at the University of Michigan during the winter semester of 1972, with some extensions. The students to whom these lectures were addressed were assumed to have knowledge of elementary functional analysis, the Fourier transform, distribution theory, and Sobolev spaces, and such tools are used without comment. In this monography, we develop one tool, the calculus of pseudo differential operators, and apply it to several of the main problems of partial differential equations.
Homogenization Of Differential Operators And Integral Functionals
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Author : V.V. Jikov
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Homogenization Of Differential Operators And Integral Functionals written by V.V. Jikov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.