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E-raamat: From Ordinary to Partial Differential Equations

  • Formaat: PDF+DRM
  • Sari: UNITEXT 106
  • Ilmumisaeg: 23-Jun-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319575445
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From Ordinary to Partial Differential EquationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: UNITEXT 106
  • Ilmumisaeg: 23-Jun-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319575445
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This book is addressed to mathematics and physics students who want to develop an interdisciplinary view of mathematics, from the age of Riemann, Poincaré and Darboux to basic tools of modern mathematics. It enables them to acquire the sensibility necessary for the formulation and solution of difficult problems, with an emphasis on concepts, rigour and creativity. It consists of eight self-contained parts: ordinary differential equations; linear elliptic equations; calculus of variations; linear and non-linear hyperbolic equations; parabolic equations; Fuchsian functions and non-linear equations; the functional equations of number theory; pseudo-differential operators and pseudo-differential equations. The author leads readers through the original papers and introduces new concepts, with a selection of topics and examples that are of high pedagogical value.

Arvustused

The selection of material is quite interesting and the book could be useful for graduate students and specialists conducting research in related areas. (Svitlana P. Rogovchenko, zbMath 1419.00002, 2019) This is a comprehensive textbook, making it well suited for beginner to intermediate graduate-level courses in partial differential equations. This book would benefit from the addition of exercise problems, but this fact does not detract from its many merits. In the end, this text is well organized and offers a clear conceptual framework to approach proofs. Summing Up: Recommended. Graduate students, researchers,and faculty. (V. K. Chellamuthu, Choice, Vol. 55 (9), May, 2018)

Part I Ordinary Differential Equations
1 Linear Differential Equations
3(22)
1.1 Fundamental Theorem for First-Order Equations
3(3)
1.2 Differentiability of the Solution
6(3)
1.3 Linear Differential Equations of Second Order
9(3)
1.4 Sturm-Liouville Problems
12(4)
1.5 Singular Points of Linear Differential Equations
16(4)
1.6 Fundamental Properties of the Heun Equation
20(5)
2 Non-linear Equations
25(16)
2.1 First Examples of Non-linear Ordinary Differential Equations
25(5)
2.2 Non-linear Differential Equations in the Complex Domain
30(3)
2.3 Integrals Not Holomorphic at the Origin
33(8)
Part II Linear Elliptic Equations
3 Harmonic Functions
41(12)
3.1 Motivations for the Laplace Equation
41(1)
3.2 Geometry of the Second Derivatives in the Laplacian
42(1)
3.3 The Three Green Identities
43(4)
3.4 Mean-Value Theorem
47(3)
3.5 The Weak Maximum Principle
50(1)
3.6 Derivative Estimates
51(1)
Appendix 3.a Frontier of a Set and Manifolds with Boundary
52(1)
4 Mathematical Theory of Surfaces
53(18)
4.1 Quadratic Differential Forms
53(1)
4.2 Invariants and Differential Parameters
54(2)
4.3 Differential Parameters of First Order
56(1)
4.4 Equivalence of Quadratic Forms; Christoffel Formulae
57(2)
4.5 Properties of Christoffel Symbols
59(2)
4.6 The Laplacian Viewed as a Differential Parameter of Order 2
61(2)
4.7 Isothermal Systems
63(2)
4.8 Holomorphic Functions Associated with Isothermal Systems
65(1)
4.9 Isothermal Parameters
66(2)
4.10 Lie Theorem on the Lines of an Isothermal System
68(3)
5 Distributions and Sobolev Spaces
71(14)
5.1 The Space D(Ω) and Its Strong Dual
71(2)
5.2 The Space C1,α(Ω) and Its Abstract Completion
73(1)
5.3 The Sobolev Space α(Ω)
74(1)
5.4 The Spaces Ck,α and Hk,α
75(1)
5.5 The Trace Map for Elements of Hk,α(Ω)
76(1)
5.6 The Space Hk,α(Ω) and Its Strong Dual
77(2)
5.7 Sub-spaces of Hk,α(Rn)
79(1)
5.8 Fundamental Solution and Parametrix of a Linear Equation
80(2)
Appendix 5.a Some Basic Concepts of Topology
82(3)
6 The Caccioppoli-Leray Theorem
85(18)
6.1 Second-Order Linear Elliptic Equations in n Variables
85(1)
6.2 The Leray Lemma and Its Proof
86(2)
6.3 Caccioppoli's Proof of Integral Bounds: Part 1
88(5)
6.4 Caccioppoli's Proof of Integral Bounds: Part 2
93(10)
7 Advanced Tools for the Caccioppoli-Leray Inequality
103(10)
7.1 The Concept of Weak Solution
103(1)
7.2 Caccioppoli-Leray for Elliptic Systems in Divergence Form
104(5)
7.3 Legendre versus Legendre-Hadamard Conditions
109(1)
7.4 Uniform, or Strong, or Uniform Strong, or Proper Ellipticity
110(3)
8 Aspects of Spectral Theory
113(10)
8.1 Resolvent Set, Spectrum and Resolvent of a Linear Operator
113(1)
8.2 Modified Resolvent Set and Modified Resolvent
114(2)
8.3 Eigenvalues and Characteristic Values
116(1)
8.4 Directions of Minimal Growth of the Resolvent
117(1)
8.5 Decay Rate of the Resolvent Along Rays
118(1)
8.6 Strongly Elliptic Boundary-Value Problems, and an Example
119(4)
9 Laplace Equation with Mixed Boundary Conditions
123(12)
9.1 Uniqueness Theorems with Mixed Boundary Conditions
123(1)
9.2 The De Giorgi Family of Solutions
124(2)
9.3 De Giorgi's Clever use of the Characteristic Function of a Set
126(6)
9.4 Perimeter of a Set and Reduced Frontier of Finite-Perimeter Sets
132(3)
10 New Functional Spaces: Morrey and Campanato
135(22)
10.1 Morrey and Campanato Spaces: Definitions and Properties
135(5)
10.2 Functions of Bounded Mean Oscillation, and an Example
140(1)
10.3 Glancing Backwards: The Spaces W1, pLoc and W1, p
140(2)
10.4 How Sobolev Discovered His Functional Spaces
142(15)
11 Pseudo-Holomorphic and Poly harmonic Frameworks
157(20)
11.1 Local Theory of Pseudo-Holomorphic Functions
157(5)
11.2 Global Theory of Pseudo-Holomorphic Functions
162(1)
11.3 Upper and Lower Bound for the Increment Ratio
163(2)
11.4 Decomposition Theorem for Biharmonic Functions
165(2)
11.5 Boundary-Value Problems for the Biharmonic Equation
167(2)
11.6 Fundamental Solution of the Biharmonic Equation
169(2)
11.7 Mean-Value Property for Polyharmonic Functions
171(6)
Part III Calculus of Variations
12 The Euler Equations
177(6)
12.1 Statement of the Problem
177(1)
12.2 The Euler Integral Condition
178(1)
12.3 The Euler Differential Condition
179(1)
12.4 Variational Problem with Constraints
180(3)
13 Classical Variational Problems
183(20)
13.1 Isoperimetric Problems
183(3)
13.2 Double Integrals and Minimal Surfaces
186(2)
13.3 Minimal Surfaces and Functions of a Complex Variable
188(6)
13.4 The Dirichlet Boundary-Value Problem
194(9)
Part IV Linear and Non-linear Hyperbolic Equations
14 Characteristics and Waves, 1
203(18)
14.1 Systems of Partial Differential Equations
203(2)
14.2 Characteristic Manifolds for First- and Second-Order Systems
205(4)
14.3 The Concept of Wavelike Propagation
209(3)
14.4 The Concept of Hyperbolic Equation
212(2)
14.5 Riemann Kernel for a Hyperbolic Equation in 2 Variables
214(4)
14.6 Lack of Smooth Cauchy Problem for the Laplace Equation
218(3)
15 Characteristics and Waves, 2
221(12)
15.1 Wavelike Propagation for a Generic Normal System
221(3)
15.2 Cauchy's Method for Integrating a First-Order Equation
224(5)
15.3 The Bicharacteristics
229(1)
15.4 Space-Time Manifold; Arc-Length; Geodesies
229(4)
16 Fundamental Solution and Characteristic Conoid
233(8)
16.1 Relation Between Fundamental Solution and Riemann's Kernel
233(2)
16.2 The Concept of Characteristic Conoid
235(1)
16.3 Fundamental Solutions with an Algebraic Singularity
236(2)
16.4 Geodesic Equations with and Without Reparametrization Invariance
238(3)
17 How to Build the Fundamental Solution
241(12)
17.1 Hamiltonian Form of Geodesic Equations
241(3)
17.2 The Unique Real-Analytic World Function
244(2)
17.3 Fundamental Solution with Odd Number of Variables
246(3)
17.4 Convergence of the Power Series for U
249(4)
18 Examples of Fundamental Solutions
253(10)
18.1 Even Number of Variables and Logarithmic Term
253(2)
18.2 Smooth Part of the Fundamental Solution
255(1)
18.3 Parametrix of Scalar Wave Equation in Curved Space-Time
255(2)
18.4 Non-linear Equations for Amplitude and Phase Functions
257(2)
18.5 Tensor Generalization of the Ermakov-Pinney Equation
259(1)
18.6 Damped Waves
260(3)
19 Linear Systems of Normal Hyperbolic Form
263(40)
19.1 Einstein Equations and Non-linear Theory
263(2)
19.2 Equations Defining the Characteristic Conoid
265(2)
19.3 A Domain of the Characteristic Conoid
267(2)
19.4 Integral Equations for Derivatives of xi and pi
269(1)
19.5 Relations on the Conoid Satisfied by the Unknown Functions
270(1)
19.6 The Auxiliary Functions σrs
271(2)
19.7 Integrating Linear Combinations of the Equations
273(1)
19.8 Determination of the Auxiliary Functions σrs
274(1)
19.9 Evaluation of the ωrs
275(1)
19.10 Calculation of σ
276(2)
19.11 Derivatives of the Functions σrs
278(2)
19.12 Behaviour in the Neighbourhood of the Vertex
280(1)
19.13 Behaviour in the Neighbourhood of λ1 = 0
281(4)
19.14 First Derivatives
285(1)
19.15 Reverting to the Functions σrs
286(2)
19.16 Study of σ and Its Derivatives
288(4)
19.17 Derivatives of the σrs
292(1)
19.18 Kirchhoff Formulae
293(1)
19.19 Evaluation of the Area and Volume Elements
294(1)
19.20 Limit as η → 0 of the Integral Relations
295(1)
19.21 Reverting to the Kirchhoff Formulae
296(1)
19.22 Summary of the Results
297(1)
19.23 Transformation of Variables
298(2)
19.24 Application of the Results
300(1)
19.25 Linear Systems of Second Order
301(2)
20 Linear System from a Non-linear Hyperbolic System
303(10)
20.1 Non-linear Equations
303(1)
20.2 Differentiation of the Equations (F)
304(2)
20.3 Application of the Results of Chap. 19
306(1)
20.4 Cauchy Data
306(2)
20.5 Summary of Results
308(3)
20.6 Solution of the Cauchy Problem for Non-linear Equations
311(2)
21 Cauchy Problem for General Relativity
313(8)
21.1 The Equations of Einstein's Gravity
313(1)
21.2 Vacuum Einstein Equations and Isothermal Coordinates
314(1)
21.3 Solution of the Cauchy Problem for the Equations Gαβ = 0
315(1)
21.4 The Solution of Gαβ = 0 Verifies the Conditions of Isothermy
316(2)
21.5 Uniqueness of the Solution
318(3)
22 Causal Structure and Global Hyperbolicity
321(8)
22.1 Causal Structure of Space-Time
321(1)
22.2 Strong Causality
322(1)
22.3 Stable Causality
323(1)
22.4 Global Hyperbolicity
324(5)
Part V Parabolic Equations
23 The Heat Equation
329(6)
23.1 A Summary on Linear Equations in Two Independent Variables
329(1)
23.2 Fundamental Solution of the Heat Equation
330(5)
24 The Nash Theorem on Parabolic Equations
335(22)
24.1 The Moment Bound
335(9)
24.2 The G Bound
344(6)
24.3 The Overlap Estimate
350(2)
24.4 Time Continuity
352(5)
Part VI Fuchsian Functions
25 The Poincare Work on Fuchsian Functions
357(6)
25.1 Properties of Fuchsian Functions
357(2)
25.2 Fuchsian Functions
359(1)
25.3 System of ζ-Fuchsian Functions
360(3)
26 The Kernel of (Laplacian Plus Exponential)
363(24)
26.1 Motivations for the Analysis
363(2)
26.2 The u Function
365(8)
26.3 Klein Surfaces
373(4)
26.4 The U Function
377(4)
Appendix: Vertices in the Theory of Fuchsian Functions
381(6)
Part VII The Riemann ζ-Function
27 The Functional Equations of Number Theory
387(18)
27.1 The Euler Theorem on Prime Numbers and the ζ-function
387(2)
27.2 Γ- and ζ-Function from the Jacobi Function
389(3)
27.3 The ζ-function: Its Functional Equation and Its Integral Representation
392(1)
27.4 Logarithm of the ζ-Function
393(7)
27.5 The Riemann Hypothesis on Non-trivial Zeros of the ζ-Function
400(5)
Part VIII A Window on Modern Theory
28 The Symbol of Pseudo-Differential Operators
405(18)
28.1 From Differential to Pseudo-Differential Operators
405(2)
28.2 The Symbol of Pseudo-Differential Operators on Manifolds
407(1)
28.3 Geometry Underlying the Symbol Map
408(5)
28.4 Symbol and Leading Symbol as Equivalence Classes
413(4)
28.5 A Smooth Linear Equation Without Solution
417(5)
28.6 Solving Pseudo-Differential Equations
422(1)
References 423(6)
Index 429
Prof. Giampiero Esposito (1962) obtained an honours (cum laude) degree in Physics from Naples University in 1986, and was a St. John's Benefactor's Scholar at DAMTP in Cambridge (UK) from 1987 to 1991, where he received the  J.T. Knight Prize Essay award in 1989 and obtained his Ph.D. degree. He was elected to INFN and ICTP post-doctoral positions at Naples and Trieste, respectively, and has been an INFN Research Fellow at Naples (position with tenure) since 1993, and INFN Primo Ricercatore since 2007.





His original contributions are mainly devoted to quantum gravity and quantum field theory on manifolds with boundary (one-loop conformal anomalies, mixed and diff-invariant boundary conditions for Euclidean quantum gravity, heat-kernel asymptotics, Casimir effect and measurement of variations of zero-point energy), spontaneous symmetry breaking in the early universe, accelerated expansion of the universe, singularity avoidance in quantum cosmology, and scattering fromsingular potentials in quantum mechanics.
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