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E-raamat: Tools and Problems in Partial Differential Equations

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  • Formaat: EPUB+DRM
  • Sari: Universitext
  • Ilmumisaeg: 19-Oct-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030502843
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Tools and Problems in Partial Differential EquationsSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Universitext
  • Ilmumisaeg: 19-Oct-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030502843
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This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations.Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory.

Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject.     

Arvustused

Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in partial differential equations may find this work useful as a summary of analytic theories published in this volume. (Viceniu D. Rdulescu, zbMATH 1461.35001, 2021)

Part I Tools and Problems
1 Elements of Functional Analysis and Distributions
3(16)
1.1 Frechet Spaces
3(2)
1.2 Elements of Functional Analysis
5(7)
1.2.1 Fixed Point Theorems
5(1)
1.2.2 The Banach Isomorphism Theorem
5(1)
1.2.3 The Closed Graph Theorem
5(1)
1.2.4 The Banach-Steinhaus Theorem
6(1)
1.2.5 The Banach-Alaoglu Theorem
6(1)
1.2.6 The Ascoli Theorem
7(1)
1.2.7 The Hahn-Banach Theorem
7(1)
1.2.8 Hilbert Spaces
7(2)
1.2.9 Spectral Theory of Self-Adjoint Compact Operators
9(1)
1.2.10 LP Spaces, 1 ≤ p ≤ + ∞
9(1)
1.2.11 The Holder and Young Inequalities
10(1)
1.2.12 Approximation of the Identity
11(1)
1.3 Elements of Distribution Theory
12(7)
1.3.1 Distributions
12(1)
1.3.2 Tempered Distributions
13(2)
1.3.3 The Fourier Transform
15(1)
1.3.4 The Stationary Phase Method
16(3)
2 Statements of the Problems of Chap. 1
19(6)
3 Functional Spaces
25(16)
3.1 Sobolev Spaces
25(6)
3.1.1 Sobolev Spaces on RJ, d ≥ 1
25(3)
3.1.2 Local Sobolev Spaces Hsloc(Rd)
28(1)
3.1.3 Sobolev Spaces on an Open Subset of Rd
29(2)
3.1.4 Sobolev Spaces on the Torus
31(1)
3.2 The Holder Spaces
31(2)
3.2.1 Holder Spaces of Integer Order
31(1)
3.2.2 Holder Spaces of Fractional Order
32(1)
3.3 Characterization of Sobolev and Holder Spaces in Dyadic Rings
33(2)
3.3.1 Characterization of Sobolev Spaces
34(1)
3.3.2 Characterization of Holder Spaces
34(1)
3.3.3 The Zygmund Spaces
35(1)
3.4 Paraproducts
35(2)
3.5 Some Words on Interpolation
37(2)
3.6 The Hardy-Littlewood-Sobolev Inequality
39(2)
4 Statements of the Problems of Chap. 3
41(20)
5 Microlocal Analysis
61(14)
5.1 Symbol Classes
61(1)
5.1.1 Definition and First Properties
61(1)
5.1.2 Examples
62(1)
5.1.3 Classical Symbols
62(1)
5.2 Pseudo-Differential Operators
62(4)
5.2.1 Definition and First Properties
62(1)
5.2.2 Kernel of a ψDO
63(1)
5.2.3 Image of a ψDO by a Diffeomorphism
63(1)
5.2.4 Symbolic Calculus
64(1)
5.2.5 Action of the ψDO on Sobolev Spaces
65(1)
5.2.6 Garding Inequalities
65(1)
5.3 Invertibility of Elliptic Symbols
66(1)
5.4 Wave Front Set of a Distribution
66(2)
5.4.1 Definition and First Properties
66(1)
5.4.2 Wave Front Set and ψDO
67(1)
5.4.3 The Propagation of Singularities Theorem
67(1)
5.5 Paradifferential Calculus
68(3)
5.5.1 Symbols Classes
68(1)
5.5.2 Paradifferential Operators
69(1)
5.5.3 The Symbolic Calculus
70(1)
5.5.4 Link with the Paraproducts
71(1)
5.6 Microlocal Defect Measures
71(4)
6 Statements of the Problems of Chap. 5
75(26)
7 The Classical Equations
101(24)
7.1 Equations with Analytic Coefficients
101(3)
7.1.1 The Cauchy-Kovalevski Theorem
102(1)
7.1.2 The Holmgren Uniqueness Theorem
103(1)
7.2 The Laplace Equation
104(4)
7.2.1 The Mean Value Property
104(1)
7.2.2 Hypoellipticity: Analytic Hypoellipticity
104(1)
7.2.3 The Maximum Principles
105(1)
7.2.4 The Harnack Inequality
105(1)
7.2.5 The Dirichlet Problem
106(1)
7.2.6 Spectral Theory
106(2)
7.3 The Heat Equation
108(1)
7.3.1 The Maximum Principle
108(1)
7.3.2 The Cauchy Problem
109(1)
7.4 The Wave Equation
109(4)
7.4.1 Homogeneous Cauchy Problem
109(2)
7.4.2 Inhomogencous Cauchy Problem
111(1)
7.4.3 The Mixed Problem
112(1)
7.5 The Schrodinger Equation
113(3)
7.5.1 The Cauchy Problem
113(1)
7.5.2 Properties of the Solution
114(2)
7.6 The Burgers Equation
116(1)
7.7 The Euler Equations
117(5)
7.7.1 The Incompressible Euler Equations
117(4)
7.7.2 The Compressible Euler Equations
121(1)
7.8 The Navier-Stokcs Equations
122(3)
7.8.1 Weak Solutions
122(1)
7.8.2 The Leray Theorem (1934)
122(1)
7.8.3 Strong Solutions: Theorems of Fujita-Kato and Kato
123(2)
8 Statements of the Problems of Chap. 7
125(56)
Part II Solutions of the Problems and Classical Results
9 Solutions of the Problems
181(162)
10 Classical Results
343(10)
10.1 Some Classical Formulas
343(1)
10.1.1 The Leibniz Formula
343(1)
10.1.2 The Taylor Formula with Integral Reminder
343(1)
10.1.3 The Faa-di-Bruno Formula
344(1)
10.2 Elements of Integration
344(4)
10.2.1 Convergence Theorems
344(1)
10.2.2 Change of Variables in Rd
345(1)
10.2.3 Polar Coordinates in Rd
345(1)
10.2.4 The Gauss-Green Formula
346(1)
10.2.5 Integration on a Graph
347(1)
10.3 Elements of Differential Calculus
348(1)
10.4 Elements of Differential Equations
348(2)
10.4.1 The Precise Cauchy-Lipschitz Theorem
348(1)
10.4.2 The Cauchy-Arzela-Peano Theorem
349(1)
10.4.3 Global Theory
349(1)
10.4.4 The Gronwall Inequality
350(1)
10.5 Elements of Holomorphic Functions
350(3)
References 353(2)
Index 355
Thomas Alazard is a senior researcher at CNRS. For several years, he has taught partial differential equations, microlocal analysis and functional analysis at the Ecole Normale Supérieure and the Ecole Normale Supérieure Paris-Saclay. His research focuses on the applications of harmonic analysis and microlocal analysis to the study of nonlinear partial differential equations.Claude Zuily received his PhD from Université Paris-Sud (Orsay), where he was professor of mathematics until 2010. Currently emeritus professor at Université Paris-Saclay, he is the author of several books: Uniqueness and non uniqueness in the Cauchy problem (Birkhäuser 1983), Problèmes de distributions et d'équations aux dérivées partielles (Hermann 1995 and Cassini 2010), Analyse pour l'agrégation (with H. Queffélec) (Dunod 1995), Distributions et équations aux dérivées partielles (Dunod 2002). His primary areas of research are linear and nonlinear partial differentialequations.   
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