《KERNEL FUNCTIONS AND ELLIPTIC DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS》

Part ABoundary Value Problems for Partial Differential Equations of Elliptic Type1

CHAPTER Ⅰ：THEORY OF HEAT CONDUCTION1

1．The differential equation of heat transfer1

2．The special case of steady flow3

3．Point sources and fundamental singularities7

4．Fundamental solutions10

5．Discontinuities15

6．Dirichlet＇s principle17

7．A modified heat equation20

8．Formal considerations22

CHAPTER Ⅱ：FLUID DYNAMICS25

1．The fundamental equations25

2．Stationary irrotational flow28

3．Incompressible，irrotational，stationary fluid flow30

4．Sources and sinks32

5．Fundamental solutions and flow patterns35

6．Infinite domains38

7．Virtual mass43

8．Dirichlet identities46

9．Virtual mass as an extremum55

10．Variational formulas58

11．Free boundaries，discontinuity surfaces，and virtual mass64

12．Two-dimensional flows71

13．Boundary value problems and fundamental solutions in plane flows76

14．Obstacles in a plane flow82

15．The variation of the Green＇s function93

16．Lavrentieff＇s extremum problem for lift98

17．Free boundaries in plane flow101

18．The complex kernel function and its applications109

19．Axially symmetric motion of an incompressible fluid119

20．Associated partial differential equations124

21．Two-dimensional compressible fluid flow128

22．Construction of solutions of the differential equations in the hodograph plane133

23．The hodograph method and boundary value problems in the physical plane141

24．Singularities in the hodograph plane and their application147

CHAPTER Ⅲ．ELECTRO-AND MAGNETOSTATICS155

1．The general equations of the electrostatic field155

2．Conductor potentials and induction coefficients157

3．The energy of the electrostatic field160

4．The magnetostatic field165

5．The conductor potential；spherical harmonics178

6．Polarization of a conducting surface188

7．Forces and moments in a conductor surface193

8．Orthogonal harmonic functions198

CHAPTER Ⅳ：ELASTICITY206

1．Strain tensor fields206

2．Stress tensor fields209

3．Stress-strain relations211

4．The energy integral and boundary value problems215

5．Betti＇s formula and the fundamental singularity219

6．Fundamental solutions222

7．Construction of the fundamental solutions in terms of orthonormal solution fields226

8．Particular solutions229

9．The theory of thin plates232

10．Fundamental solutions of the biharmonic equation236

11．Plane strain246

Part BKernel Function Methods in the Theory of Boundary Value Problems258

CHAPTER Ⅰ：PROPERTIES OF SOLUTIONS258

1．Introduction258

2．Notation and definitions259

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

4．Existence theorems for certain solutions262

5．Fundamental singularities and fundamental solutions268

6．Dirichlet integrals and fundamental functions273

CHAPTER Ⅱ：THE KERNEL FUNCTIONS AND THEIR PROPERTIES275

1．Kernel functions275

2．Orthonormal systems and construction of the kernel280

3．Integral operators and the construction of complete sets of solutions286

4．Dirichlet identities297

5．Choice of the fundamental singularity301

6．The regularity of l(P，Q)303

7．An integral equation for the kernel309

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

9．Series developments and integral equations326

CHAPTER Ⅲ：VARIATIONAL AND COMPARISON THEORY335

1．Relations between kernels of different domains335

2．Eigenfunctions of the γ-transformation346

3．Doubly orthogonal systems351

4．An application of the operators to the theory of orthogonal functions355

5．Comparison formulas for Green＇s and Neumann＇s functions357

6．Variational theory360

CHAPTER Ⅳ：EXISTENCE THEORY371

1．Boundary value problem and orthogonal projection371

CHAPTER Ⅴ：DEPENDENCE OF KERNELS ON BOUNDARY CONDITIONS AND THE DIFFERENTIAL EQUATION381

1．The positiveness of the fundamental functions381

2．Stability of fundamental functions384

3．Stability of eigenvalues388

4．Construction of a fundamental singularity392

CHAPTER Ⅵ：GENERALIZATIONS395

1．Different types of metrics395

2．Further generalizations402

List of Symbols Used in Part B404

Bibliography Books408

Articles412

Author Index421

Subject Index425