Giampiero Esposito

From Ordinary to Partial Differential Equations

£ j Springer

Contents

Part I Ordinary Differential Equations

1 Linear Differential Equations 3 1.1 Fundamental Theorem for First-Order Equations 3 1.2 Differentiability of the Solution 6 1.3 Linear Differential Equations of Second Order 9 1.4 Sturm-Liouville Problems 12 1.5 Singular Points of Linear Differential Equations 16 1.6 Fundamental Properties of the Heun Equation 20

2 Non-linear Equations 25 2.1 First Examples of Non-linear Ordinary Differential

Equations 25 2.2 Non-linear Differential Equations in the Complex Domain.... 30 2.3 Integrals Not Holomorphic at the Origin 33

Part II Linear Elliptic Equations

3 Harmonie Functions 41 3.1 Motivations for the Laplace Equation 41 3.2 Geometiy of the Second Derivatives in the Laplacian 42 3.3 The Three Green Identities 43 3.4 Mean-Value Theorem 47 3.5 The Weak Maximum Principle 50 3.6 Derivative Estimates 51 Appendix 3.a: Frontier of a Set and Manifolds with Boundary 52

4 Mathematical Theory of Surfaces 53 4.1 Quadratic Differential Forms 53 4.2 Invariants and Differential Parameters 54 4.3 Differential Parameters of First Order 56

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Contents

4.4 Equivalence of Quadratic Forms; ChristofFel Formulae 57 4.5 Properties of Christoflfel Symbols . 59 4.6 The Laplacian Viewed as a Differential Parameter

of Order 2 61 4.7 Isothermal Systems 63 4.8 Holomorphic Functions Associated with IsoÜiermal

Systems 65 4.9 Isothermal Parameters 66 4.10 Lie Theorem on the Lines of an Isothermal System 68

Distributions and Sobolev Spaces 71 5.1 The Space D(I2) and Its Strang Dual 1\ 5.2 The Space C1'a(Q) and Its Abstract Completion 73 5.3 The Sobolev Space HlA(Q) 74 5.4 The Spaces C*'a and Hk'a . 75 5.5 The Trace Map for Elements of Hk'a(ü) 76 5.6 The Space HQ'X(Q.) and Its Strang Dual 77 5.7 Sub-spaces of H*-a(RM) 79 5.8 Fundamental Solution and Parametrix of a Linear

Equation 80 Appendix 5.a: Some Basic Concepts of Topology 82

The Caccioppoli-Leray Theorem 85 6.1 Second-Order Linear Elliptic Equations in n Variables 85 6.2 The Leray Lemma and Its Proof 86 6.3 Caccioppoli's Proof of Integral Bounds: Part 1 88 6.4 Caccioppoli's Proof of Integral Bounds: Part 2 93

Advanced Tools for the Caccioppoli-Leray Inequality 103 7.1 The Concept of Weak Solution 103 7.2 Caccioppoli-Leray for Elliptic Systems in Divergence

Form 104 7.3 Legendre versus Legendre-Hadamard Conditions 109 7.4 Uniform, or Strang, or Uniform Strang, or Proper

Ellipticity . 110

Aspects of Spectral Theory 113 8.1 Resolvent Set, Spectrum and Resolvent of a Linear

Operator 113 8.2 Modified Resolvent Set and Modified Resolvent 114 8.3 Eigenvalues and Characteristic Values 116 8.4 Directions of Minimal Growth of the Resolvent 117 8.5 Decay Rate of the Resolvent Along Rays 118 8.6 Strongly Elliptic Boundary-Value Problems,

and an Example 119

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9 Laplace Equation with Mixed Boundary Conditions . 123 9.1 Uniqueness Theorems with Mixed Boundary Conditions 123 9.2 The De Giorgi Family of Solutions 124 9.3 De Giorgi's Clever use of the Characteristic Function

of a Set. 126 9.4 Perimeter of a Set and Reduced Frontier

of Finite-Perimeter Sets 132

10 New Functional Spaces: Morrey and Campanato 135 10.1 Morrey and Campanato Spaces: Definitions and Properties . . . 135 10.2 Functions of Bounded Mean OsciUation, and an Example . . . . 140 10.3 Glancing Backwards: The Spaces w££ and W1* 140 10.4 How Sobolev Discovered His Functional Spaces 142

11 Pseudo-Holomorphic and Polyharmonic Frameworks 157 11.1 Local Theory of Pseudo-Holomorphic Functions 157 11.2 Global Theory of Pseudo-Holomorphic Functions 162 11.3 Upper and Lower Bound for the Increment Ratio. . . 163 11.4 Decomposition Theorem for Biharmonic Functions 165 11.5 Boundary-Value Problems for the Biharmonic Equation 167 11.6 Fundamental Solution of the Biharmonic Equation 169 11.7 Mean-Value Property for Polyharmonic Functions 171

Part III Calculus of Variations

12 The Euler Equations 177 12.1 Statement of the Problem 177 12.2 The Euler Integral Condition . 178 12.3 The Euler Differential Condition. 179 12.4 Variational Problem with Constraints 180

13 Classical Variational Problems 183 13.1 Isoperimetric Problems 183 13.2 Double Integrals and Minimal Surfaces 186 13.3 Minimal Surfaces and Functions of a Complex Variable 188 13.4 The Dirichlet Boundary-Value Problem 194

Part IV Linear and Non-linear Hyperbolic Equations

14 Characteristics and Waves, 1 203 14.1 Systems of Partial Differential Equations 203 14.2 Characteristic Manifolds for First- and Second-Order

Systems 205 14.3 The Concept of Wavelike Propagation 209 14.4 The Concept of Hyperbolic Equation 212

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14.5 Riemann Kernel for a Hyperbolic Equation in 2 Variables.... 214 14.6 Lack of Smooth Cauchy Problem for the Laplace

Equation 218

15 Charactcristics and Waves, 2 221 15.1 Wavelike Propagation for a Generic Normal System 221 15.2 Cauchy's Method for Integrating a First-Order Equation 224 15.3 The Bicharacteristics 229 15.4 Space-Time Manifold; Arc-Length; Geodesics 229

16 Fundamental Solution and Characteristic Conoid 233 16.1 Relation Between Fundamental Solution and Riemann's

Kernel 233 16.2 The Concept of Characteristic Conoid 235 16.3 Fundamental Solutions with an Algebraic Singularity 236 16.4 Geodesic Equations with and Without Reparametrization

Invariance 238

17 How to Build the Fundamental Solution 241 17.1 Hamiltonian Form of Geodesic Equations 241 17.2 The Unique Real-Analytic World Function 244 17.3 Fundamental Solution with Odd Number of Variables 246 17.4 Convergence of the Power Series for U 249

18 Examples of Fundamental Solutions 253 18.1 Even Number of Variables and Logarithmic Term 253 18.2 Smooth Part of the Fundamental Solution 255 18.3 Parametrix of Scalar Wave Equation in Curved

Space-Time 255 18.4 Non-linear Equations for Amplitude and Phase Functions . . . . 257 18.5 Tensor Generalization of the Ermakov-Pinney Equation 259 18.6 Damped Waves 260

19 Linear Systems of Normal Hyperbolic Form 263 19.1 Einstein Equations and Non-linear Theory 263 19.2 Equations Defining the Characteristic Conoid 265 19.3 A Domain of the Characteristic Conoid 267 19.4 Integral Equations for Derivatives of tf and p, 269 19.5 Relations on the Conoid Satisfied by the Unknown

Functions 270 19.6 The Auxiliary Functions ar

s 271 19.7 Integrating Linear Combinations of the Equations 273 19.8 Determination of the Auxiliary Functions ar

s 274 19.9 Evaluation of the ofs 275 19.10 Calculation of a 276

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19.11 Derivatives of the Functions <xrs 278

19.12 Behaviour in the Neighbourhood of the Vertex 280 19.13 Behaviour in the Neighbourhood of Ai = 0 281 19.14 First Derivatives 285 19.15 Reverting to the Functions <fs 286 19.16 Study of a and Its Derivatives 288 19.17 Derivatives of the ofs 292 19.18 Kirchhoff Formulae 293 19.19 Evaluation of the Area and Volume Elements 294 19.20 Limit as r\ -> 0 of the Integral Relations 295 19.21 Reverting to the Kirchhoff Formulae 296 19.22 Summary of the Results 297 19.23 Transformation of Variables 298 19.24 Application of the Results 300 19.25 Linear Systems of Second Order 301

20 Linear System from a Non-linear Hyperbolic System 303 20.1 Non-linear Equations 303 20.2 Differentiation of the Equations (F) 304 20.3 Application of the Results of Chap. 19 306 20.4 Cauchy Data 306 20.5 Summary of Results 308 20.6 Solution of the Cauchy Problem for Non-linear Equations.... 311

21 Cauchy Problem for General Relativity 313 21.1 The Equations of Einstein's Gravity 313 21.2 Vacuum Einstein Equations and Isothermal Coordinates...... 314 21.3 Solution of the Cauchy Problem for the Equations

Gaß = 0 315 21.4 The Solution of Gxß = 0 Verifies the Conditions

of Isothermy 316 21.5 Uniqueness of the Solution 318

22 Causal Structure and Global Hyperbolicity 321 22.1 Causal Structure of Space-Time 321 22.2 Strang Causality 322 22.3 Stable Causality 323 22.4 Global Hyperbolicity 324

Part V Parabolic Equations

23 The Heat Equation 329 23.1 A Summary on Linear Equations in Two Independent

Variables 329 23.2 Fundamental Solution of the Heat Equation 330

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24 The Nash Theorem on Parabolic Equations 335 24.1 The Moment Bound 335 24.2 The G Bound 344 24.3 The Overlap Estimate 350 24.4 Time Continuity 352

Part VI Fuchsian Functions

25 The PoincarS Work on Fuchsian Functions 357 25.1 Properties of Fuchsian Functions 357 25.2 ©-Fuchsian Functions 359 25.3 System of (-Fuchsian Functions 360

26 The Kernel of (Laplacian Plus Exponential) 363 26.1 Motivations for the Analysis 363 26.2 The u Function . 365 26.3 Klein Surfaces 373 26.4 The U Function . 377 Appendix: Vertices in the Theory of Fuchsian Functions 381

Part VII The Riemann (-Function

27 The Functional Equations of Number Theory 387 27.1 The Euler Theorem on Prime Numbers and the (-function.... 387 27.2 T- and (-Function from the Jacobi Function. 389 27.3 The (-function: Its Functional Equation and Its Integral

Representation 392 27.4 Logarithm of the (-Function 393 27.5 The Riemann Hypothesis on Non-trivial Zeros

of the (-Function 400

Part VIII A Window on Modern Theory

28 The Symbol of Pseudo-Differential Operators 405 28.1 From Differential to Pseudo-Differential Operators 405 28.2 The Symbol of Pseudo-Differential Operators

on Manifolds 407 28.3 Geometry Underlying the Symbol Map 408 28.4 Symbol and Leading Symbol as Equivalence Classes. . . . . . . . 413 28.5 A Smooth Linear Equation Without Solution 417 28.6 Solving Pseudo-Differential Equations 422

References 423

Index 429