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Part A Boundary Value Problems for Partial Differential Equations of Elliptic Type 1

CHAPTER Ⅰ：THEORY OF HEAT CONDUCTION 1

1．The differential equation of heat transfer 1

2．The special case of steady flow 3

3．Point sources and fundamental singularities 7

4．Fundamental solutions 10

5．Discontinuities 15

6．Dirichlet＇s principle 17

7．A modified heat equation 20

8．Formal considerations 22

CHAPTER Ⅱ：FLUID DYNAMICS 25

1．The fundamental equations 25

2．Stationary irrotational flow 28

3．Incompressible，irrotational，stationary fluid flow 30

4．Sources and sinks 32

5．Fundamental solutions and flow patterns 35

6．Infinite domains 38

7．Virtual mass 43

8．Dirichlet identities 46

9．Virtual mass as an extremum 55

10．Variational formulas 58

11．Free boundaries，discontinuity surfaces，and virtual mass 64

12．Two-dimensional flows 71

13．Boundary value problems and fundamental solutions in plane flows 76

14．Obstacles in a plane flow 82

15．The variation of the Green＇s function 93

16．Lavrentieff＇s extremum problem for lift 98

17．Free boundaries in plane flow 101

18．The complex kernel function and its applications 109

19．Axially symmetric motion of an incompressible fluid 119

20．Associated partial differential equations 124

21．Two-dimensional compressible fluid flow 128

22．Construction of solutions of the differential equations in the hodograph plane 133

23．The hodograph method and boundary value problems in the physical plane 141

24．Singularities in the hodograph plane and their application 147

CHAPTER Ⅲ．ELECTRO-AND MAGNETOSTATICS 155

1．The general equations of the electrostatic field 155

2．Conductor potentials and induction coefficients 157

3．The energy of the electrostatic field 160

4．The magnetostatic field 165

5．The conductor potential；spherical harmonics 178

6．Polarization of a conducting surface 188

7．Forces and moments in a conductor surface 193

8．Orthogonal harmonic functions 198

CHAPTER Ⅳ：ELASTICITY 206

1．Strain tensor fields 206

2．Stress tensor fields 209

3．Stress-strain relations 211

4．The energy integral and boundary value problems 215

5．Betti＇s formula and the fundamental singularity 219

6．Fundamental solutions 222

7．Construction of the fundamental solutions in terms of orthonormal solution fields 226

8．Particular solutions 229

9．The theory of thin plates 232

10．Fundamental solutions of the biharmonic equation 236

11．Plane strain 246

Part B Kernel Function Methods in the Theory of Boundary Value Problems 258

CHAPTER Ⅰ：PROPERTIES OF SOLUTIONS 258

1．Introduction 258

2．Notation and definitions 259

3．Properties of the solutions of(1．1) 261

4．Existence theorems for certain solutions 262

5．Fundamental singularities and fundamental solutions 268

6．Dirichlet integrals and fundamental functions 273

CHAPTER Ⅱ：THE KERNEL FUNCTIONS AND THEIR PROPERTIES 275

1．Kernel functions 275

2．Orthonormal systems and construction of the kernel 280

3．Integral operators and the construction of complete sets of solutions 286

4．Dirichlet identities 297

5．Choice of the fundamental singularity 301

6．The regularity of l(P，Q) 303

7．An integral equation for the kernel 309

8．The eigenvalues of the kernel l(P，Q) 315

9．Series developments and integral equations 326

CHAPTER Ⅲ：VARIATIONAL AND COMPARISON THEORY 335

1．Relations between kernels of different domains 335

2．Eigenfunctions of the γ-transformation 346

3．Doubly orthogonal systems 351

4．An application of the operators to the theory of orthogonal functions 355

5．Comparison formulas for Green＇s and Neumann＇s functions 357

6．Variational theory 360

CHAPTER Ⅳ：EXISTENCE THEORY 371

1．Boundary value problem and orthogonal projection 371

CHAPTER Ⅴ：DEPENDENCE OF KERNELS ON BOUNDARY CONDITIONS AND THE DIFFERENTIAL EQUATION 381

1．The positiveness of the fundamental functions 381

2．Stability of fundamental functions 384

3．Stability of eigenvalues 388

4．Construction of a fundamental singularity 392

CHAPTER Ⅵ：GENERALIZATIONS 395

1．Different types of metrics 395

2．Further generalizations 402

List of Symbols Used in Part B 404

Bibliography Books 408

Articles 412

Author Index 421

Subject Index 425