Kernel Functions and Elliptic Differential Equations in Mathematical Physics

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ISBN: 9780486445533 | 0486445534
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Copyright: 9/1/2005

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This text focuses on the theory of boundary value problems in partial differential equations, which plays a central role in various fields of pure and applied mathematics, theoretical physics, and engineering. Geared toward upper-level undergraduates and graduate students, it discusses a portion of the theory from a unifying point of view and provides a systematic and self-contained introduction to each branch of the applications it employs.

PREFACE

vii

NOTE TO THE READER

Part A Boundary Value Problems for Partial Differential Equations of Elliptic Type

CHAPTER I: THEORY OF HEAT CONDUCTION

1. The differential equation of heat transfer

(2)

2. The special case of steady flow

(4)

3. Point sources and fundamental singularities

(3)

4. Fundamental solutions

(5)

5. Discontinuities

(2)

6. Dirichlet's principle

(3)

7. A modified heat equation

(2)

8. Formal considerations

(3)

CHAPTER II: FLUID DYNAMICS

1. The fundamental equations

(3)

2. Stationary irrotational flow

(2)

3. Incompressible, irrotational, stationary fluid flow

(2)

4. Sources and sinks

(3)

5. Fundamental solutions and flow patterns

(3)

6. Infinite domains

(5)

7. Virtual mass

(3)

8. Dirichlet identities

(9)

9. Virtual mass as an extremum

(3)

10. Variational formulas

(6)

11. Free boundaries, discontinuity surfaces, and virtval mass

(7)

12. Two-dimensional flows

(5)

13. Boundary value problems and fundamental solutions in plane flows

(6)

14. Obstacles in a plane flow

(11)

15. The variation of the Green's function

(5)

16. Lavrentieff's extremum problem for lift

(3)

17. Free boundaries in plane flow

101

(8)

18. The complex kernel function and its applications

109

(10)

19. Axially symmetric motion of an incompressible fluid.

119

(5)

20. Associated partial differential equations

124

(4)

21. Two-dimensional compressible fluid flow

128

(5)

22. Construction of solutions of the differential equations in the hodograph plane

133

(8)

23. The hodograph method and boundary value problems in the physical plane

141

(6)

24. Singularities in the hodograph plane and their application

147

(8)

CHAPTER III. FLECTRO- AND MAGNETOSTATICS

1. The general equations of the electrostatic field

155

(2)

2. Conductor potentials and induction coefficients

157

(3)

3. The energy of the electrostatic field

160

(5)

4. The magnetostatic field

165

(13)

5. The conductor potential; spherical harmonics

178

(10)

6. Polarization of a conducting surface

188

(5)

7. Forces and moments in a conductor surface

193

(5)

8. Orthogonal harmonic functions

198

(8)

CHAPTER IV: ELASTICITY

1. Strain tensor fields

206

(3)

2. Stress tensor fields

209

(2)

3. Stress-strain relations

211

(4)

4. The energy integral and boundary value problems

215

(4)

5. Betti's formula and the fundamental singularity

219

(3)

6. Fundamental solutions

222

(4)

7. Construction of the fundamental solutions in terms of orthonormal solution fields

226

(3)

8. Particular solutions

229

(3)

9. The theory of thin plates

232

(4)

10. Fundamental solutions of the hiharmonic equation

236

(10)

11. Plane strain

246

(12)

Part II Kernel Function Methods in the Theory of Boundary Value Problems

CHAPTER I: PROPERTIES OF SOLUTIONS

1. Introduction

258

(1)

2. Notation and definitions

259

(2)

3. Properties of the solutions of (1.1)

261

(1)

4. Existence theorems for certain solutions

262

(6)

5. Fundamental singularities and fundamental solutions

268

(5)

6. Dirichlet integrals and fundamental functions

273

(2)

CHAPTER II: THE KERNEL FUNCTIONS AND THEIR PROPERTIES

1. Kernel functions

275

(5)

2. Orthonormal systems and construction of the kernel

280

(6)

3. Integral operators and the construction of complete sets of solutions

286

(11)

4. Dirichlet identities

297

(4)

5. Choice of the fundamental singularity

301

(2)

6. The regularity of l (P,Q)

303

(6)

7. An integral equation for the kernel

309

(6)

8. The eigenvalues of the kernel l(P,Q)

315

(11)

9. Series developments and integral equations

326

(9)

CHAPTER III: VARIATIONAL AND COMPARISON THEORY

1. Relations between kernels of different domains

335

(11)

2. Eigenfunctions of the γ-transformation

346

(5)

3. Doubly orthogonal systems

351

(4)

4. An application of the operators to the theory of orthogonal functions

355

(2)

5. Comparison formulas for Green's and Neumann's functions

357

(3)

6. Variational theory

360

(11)

CHAPTER IV: EXISTENCE THEORY

1. Boundary value problem and orthogonal projection

371

(10)

CHAPTER V: DEPENDENCE OF KERNELS ON BOUNDARY CONDITIONS AND THE DIFFERENTIAL EQUATION

1. The positiveness of the fundamental functions

381

(3)

2. Stability of fundamental functions

384

(4)

3. Stability of eigenvalues

388

(4)

4. Construction of a fundamental singularity

392

(3)

CHAPTER VI: GENERALIZATIONS

1. Different types of metrics

395

(7)

2. Further generalizations

402

(2)

List of Symbols Used in Part B

404

(4)

Bibliography Books

408

(4)

Articles

412

(9)

Author Index

421

(4)

Subject Index

425

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Kernel Functions and Elliptic Differential Equations in Mathematical Physics

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Kernel Functions and Elliptic Differential Equations in Mathematical Physics

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