This concise volume presents an overview of equations of mathematical physics and generalized functions. While intended for advanced readers, the accessible introduction and text structure allows beginners to study at their own pace as the material gradually increases in difficulty. The text introduces the concept of generalized Sobolev functions and L. Schwartz distributions briefly in the opening section, gradually approaching a more in-depth study of the “generalized” differential equation (also known as integral equality). In contrast to the traditional presentation of generalized Sobolev functions and L. Schwartz distributions, this volume derives the topology from two natural requirements (which are equivalent to it). The text applies the same approach to the theory of the canonical Maslov operator. It also features illustrative drawings and helpful supplementary reading in the footnotes concerning historical and bibliographic information related to the subject of the book. Additionally, the book devotes a special chapter to the application of the theory of pseudodifferential operators and Sobolev spaces to the inverse magneto/electroencephalography problem.
Explicit numerically realizable formulas related to the Cauchy problem for elliptic equations (including quasilinear ones) and also to the Poincaré--Steklov operators are presented. The book is completed by three additions, which were written by famous mathematicians Yu. V. Egorov, A. B. Antonevich, and S. N. Samborski.
Author(s): A. S. Demidov
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 248
City: Cham, Switzerland
Tags: Sobolev Spaces, Generalized Functions, Pseudo-Differential Operators, Fourier Operators
Preface
Contents
Basic Notation
1 Introduction to Problems of Mathematical Physics
1 Temperature at a Point? No! In Volumes Contracting to the Point
2 The δ-Sequence and the δ-Function
3 Some Spaces of Smooth Functions: Partition of Unity
4 Examples of δ-Sequences
5 On the Laplace Equation
6 On the Heat Equation
7 The Ostrogradsky–Gauss Formula: The Green Formulas and the Green Function
8 The Lebesgue Integrals
9 The Riesz Spaces Lp and Llocp
10 Lloc1(Ω) as the Space of Linear Functionals on C0∞(Ω)
11 Simplest Hyperbolic Equations: Generalized Sobolev Solutions
2 The Spaces D, D and D': Elements of the Distribution Theory (Generalized Functions in the Sense of Schwartz)
12 The Space D of Sobolev Derivatives
13 The Space D# of Generalized Functions
14 Functions Not Locally Integrable as Generalized
15 Generalized Functions with Point Support: The Borel Theorem
16 The space D' of distributions (Schwartz generalized functions)
3 Pseudo-Differential Operators and Fourier Operators
17 Fourier Series and Fourier Transform. The Spaces S and S'
18 The Fourier–Laplace Transform. The Paley–Wiener Theorem
19 Fundamental Solutions. Convolution
20 On the Spaces Hs
21 On Pseudo-Differential Operators
22 On Elliptic Problems
23 The Direct, Inverse, and Central Problems of Magneto-Electroencephalography
24 The Cauchy Problem for Elliptic Equations. Explicit Formulas
25 On the Poincaré–Steklov Operators and Explicit Formulas
26 On the Fourier–Hörmander Operator and the Canonical Maslov Operator
Appendix A. New Approach to the Theory of Generalized Functions (Yu.V.Egorov)
1 Drawbacks of the Theory of Distributions
2 Shock Waves
3 New Definition of Generalized Functions
4 The Weak Equality
Appendix B. Algebras of Mnemonic Functions (A.B. Antonevich)
1 Introduction
2 General Scheme of Construction of Algebras of MnemonicFunctions
3 Algebras of Mnemonic Functions on the Circle
4 Properties of Embedding
5 Compatibility of the Embedding with Multiplication in C∞(T1)
6 Joint Locality and Compatibility with Multiplication of Smooth Functions
7 Analytic Representations of Distributions
8 Examples of Multiplications of Schwartz Distributions
9 Conclusions
Appendix C. Extensions of First-Order Partial Differential Operators(S.N. Samborski)
1 Preliminaries and Notation
2 The Space Cae(X,E)
3 Differentiation in Cae(X,Rm)
4 The Space S(X,E)
5 Differentiation in S(X,R)
6 Equations with Partial Derivatives
7 Compactness in the Space S
References
Index
