|
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) |
|
|
|
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 |
|
|
|
|
|
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) |
|
|
|
47 | (3) |
|
3.5 The Weak Maximum Principle |
|
|
50 | (1) |
|
|
|
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) |
|
|
|
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 |
|
|
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
306 | (2) |
|
|
|
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) |
|
|
|
322 | (1) |
|
|
|
323 | (1) |
|
22.4 Global Hyperbolicity |
|
|
324 | (5) |
|
Part V Parabolic Equations |
|
|
|
|
|
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) |
|
|
|
335 | (9) |
|
|
|
344 | (6) |
|
24.3 The Overlap Estimate |
|
|
350 | (2) |
|
|
|
352 | (5) |
|
Part VI Fuchsian Functions |
|
|
|
25 The Poincare Work on Fuchsian Functions |
|
|
357 | (6) |
|
25.1 Properties of Fuchsian Functions |
|
|
357 | (2) |
|
|
|
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) |
|
|
|
365 | (8) |
|
|
|
373 | (4) |
|
|
|
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 | |
Lisainfo e-raamatute kohta