In this post, we will see the book *Potential Theory and Its Application to Basic Problems of Mathematical Physics* by N. M. Günter.

# About the book

The present book is the translation of N. M. Günter’s monograph

“La théorie du potentiel et ses applications aux problèmes fondamentaux de la physique mathématique”published in Paris in 1934. The work arose from a special seminar on potential theory held by the author during the early twenties at Leningrad University.Potential theory and problems of mathematical physics thereto related have been focal points of the mathematician’s interest since the beginning of the 19th century. At first the properties of the different potentials were not subjected to rigorous investigation, and there were thus various unfounded results in applying potential theory to boundary value problems of mathematical physics. On the other hand, up to the end of the 19th century there were no definite, deep-seated results on the properties of the solutions of these problems near the boundary.

The well-known work of Lyapunov

“Sur certaines questions qui se rattachent au problème de Dirichlet”(1898) was of fundamental importance in overcoming these deficiencies. To a certain extent, Günter’s book is closely connected with the results of this work. His objective was a precise and more detailed study of the properties of various potentials and the boundary-value problems of mathematical physics involving them.

The book was translated from Russian by John R. Schulenberger and was published in 1967.

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# Contents

## Chapter I LEMMATA 1

§1. On the Boundary of Regions 1

§2. On Functions Defined in the Interior of a Region 7

§3. The Hugoniot-Hadamard Theorem 10

§4. A Finite Covering of a Surface 14

§5. The Formulas of Ostrogradskii and Stokes 15

§6. A Remark on the Integration of Unbounded Functions 19

§7. On Harmonic Functions 24

§8. Green’s Identities 26

§9. Gauss’ Integral 34

§10. Another Proof of Gauss’ Formulas 37

Appendix to Chapter I 40

## Chapter II POTENTIAL THEORY 41

§1. The Potential of a Simple Layer 41

§2. Continuity of the Potential of the Simple Layer 44

§3. Three Theorems on the Potential of the Double Layer 48

§4. On the Normal Derivative of the Potential of the Simple Layer 58

§5. The Continuity of the Normal Derivative of the Potential of the Simple Layer 61

§6. A Theorem on the Normal Derivative of the Potential of the Simple Layer 64

§7. On the Derivatives of the Potential of the Simple Layer 67

§8. The Derivatives of the Potential of a Simple Layer with Differentiable Density 69

§9. The Normal Derivative of the Potential of a Double Layer 71

§10. The Derivatives of the Potential of a Double Layer with Differentiable Density 73

§11. On the Convergence of Certain Integrals 77

§12. On the Newtonian Potential 78

§13. On the First Derivatives of the Newtonian Potential 81

§14. On the Existence of the Second Derivatives of the Newtonian Potential 85

§15. The Theorem of Poisson 90

§16. On the Continuity of the Second Derivatives of the Newtonian Potential 93

§17. The Derivatives of the Newtonian Potential with Differentiable Density 95

§18. The Function Classes H(𝓁,𝐴,𝜆) and the SurfacesLk 97

§19. The Potential of the Simple and Double Layer for a Surface L_{k} 102

§20. The Newtonian Potential in a Region Bounded by a Surface L_{k} 105

§21. The Values on a Surface Lk of the Potential of a Double Layer and of the Normal Derivative of the Potential of a Simple Layer 106

§22. Remarks on the Potentials of Class C^{(k)} 107

§23. The Potential of the Simple and Double Layer with Summable Density 108

§24. The Newtonian Potential with Summable Density 117

## Chapter III THE NEUMANN PROBLEM AND THE ROBIN PROBLEM 122

§1. The Neumann Problem 122

§2. Replacement of Problem A by Another Problem 124

§3. The Formal Solution of Equation (B) 126

§4. Investigation of the Iterated Kernels 128

§5. The Actual Solution of Equation (B) 132

§6. A Lemma 133

§7. Proof of the Theorems in §5 139

§8. The Necessary Condition that 𝜁 = 1 Not Be a Pole 142

§9. The Sufficiency of the Conditions Found 145

§10. The Solution of the Inner Neumann Problem 149

§11. The Solution of the Outer Neumann Problem for the Case (𝑬) and for the Ordinary Case 152

§12. The Eigenfunctions Corresponding to the Pole 𝜁 = 1 in the Ordinary Case and the Robin Problem

§13. The Eigenfunction Corresponding to the Pole 𝜁 = —1 in Case (𝑱) and the Robin Problem for This Case 157

§14. The Eigenfunction Corresponding to the Pole 𝜁 = —1 in Case (𝑬) and the Robin Problem for This Case 160

§15. The Pole 𝜁 = —1 in the Case (𝑱) 163

§16. The Outer Neumann Problem for the Case (𝑱) 167

§17. The Eigenfunctions Corresponding to the Pole 𝜁 = —1 in the Case (𝑱) 167

§18. A Remark on the Question of Whether the Solution of the Neumann Problem Belongs to the Class H(𝓁,𝐴,𝜆) 170

§19. On the Uniqueness of the Solution of the Neumann Problem 174

## Chapter IV THE DIRICHLET PROBLEM 178

§1. The Statement of the Dirichlet Problem 178

§2. Replacing Problem A by Another Problem 179

§3. The Formal Solution of Problem C 180

§4.The Actual Solution of Problem C 182

§5. Some Remarks on the Kernel K_{n} (1,0) 184

§6. Proof of the Assertions Made in 185

§7. Two Lemmas Concerning an Integral Equation with Kernel K_{n} (1,0) 190

§8. Two Lemmas on the Potential of the Double Layer 193

§9. Consequences of the Lemmas of §8 198

§10. The Solution of the Inner Dirichlet Problem for Case (𝑬) and the Ordinary Case 199

§11. Investigation of the Pole 𝜁 = 1 in Case (𝑬) and in the Ordinary Case 199

§12. Interpretation of Conditions (42) 202

§13. The Solution of the Outer Problem for Case (𝑬) 203 §14. The Case (𝑱). The Conditions that 𝜁 = —1 Not Be a Pole 205

§15. The Solution of the Inner Problem for the Case (/) Assuming the Validity of Conditions (53); The Meaning of These Conditions 207

§16. The Solution of the Inner Dirichlet Problem for the Case 𝑱 208

§17. The Outer Problem for the Case 𝑱 210

§18. A Remark on the Question of Whether the Solution of the Dirichlet Problem Belongs to the Class H(𝓁,𝐴,𝜆) 213

## Chapter V GREEN’S FUNCTIONS AND THEIR APPLICATIONS 216

§1. Green’s Function and Its Principal Properties 216

§2. The Solution of the Dirichlet Problem for a Special Case 219

§3. A Lemma Due to Lyapunov 220

§4. The Solution of the Dirichlet Problem in the General Case 222

§5. The Function of F. Neumann and Its Properties 225

§6. The Solution of the Neumann Problem 230

§7. The Problem of Stationary Temperature 231

§8. Green’s Function in the Problem of Stationary Temperature 237

§9.Green’s Function and Poisson’s Equation 239

§10. Problems Involving the Equation 𝛥u= Lu + K 247

§11. Lemma 252

§12. Remarks on the Poles of the Solution of the Integral Equation (67) 255

§13. Closedness of the Sequence of Eigenfunctions in a Special Function Space 257

§14. The Closedness of the Sequence of Eigenfunctions 260

§15. On Expansion in Terms of Eigenfunctions 263

§16. The Functions of A. Korn 266

§17. Integration of the Wave Equation 269

§18. On the Heat Problem 276

§19. A Remark on Problems Related to the Laplace Operator 280

§20. A Remark on the Solution of the Poisson Equation and the Eigenfunctions 281

## APPENDIX 285

§I. The Theorem of Lyapunov on the First Derivatives of the Simple-Layer Potential with H-Continuous Density 285

§II. The Theorems of Lyapunov on the Normal Derivative of the Potential of the Double Layer 297

§III. A Theorem on the Second Derivatives of the Newtonian Potential 306

§IV. The Direct Values of the Double Layer Potential and of the Normal Derivative of the Simple-Layer Potential on a Surface L_{k} 312

A SHORT BIOGRAPHY 327

INDEX 337

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