Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why?
This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in “physical” mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.
Graduate students and researchers interested in PDE, especially in holomorphic linear PDE.
Author(s): Dmitry Khavinson; Erik Lundberg
Series: Mathematical Surveys and Monographs, 232
Edition: 1
Publisher: American Mathematical Society
Year: 2018
Language: English
Commentary: Publisher PDF | Published: July 9, 2018 | 2010 Mathematics Subject Classification: Primary 35A20; 31B20; 32A05; 30B40; 14P05.
Pages: 214
City: Providence, Rhode Island
Tags: Potential Theory; Partial Differential Equations; Linear Differential Equations; Holomorphic Functions; Analytic Methods; Higher-Dimensional Theory; Boundary Value; Inverse Problems; Several Complex Variables; Analytic Spaces; Power Series; Series Expansions; Algebraic Geometry
Preface ix
Acknowledgments x
Chapter 1. Introduction: Some Motivating Questions 1
1. Continuation of potentials 1
2. Uniqueness of potentials 2
3. The Schwarz reflection principle 3
4. Szeg˝o’s theorem 3
5. PDE vs. ODE 4
6. Laplacian growth and the inverse potential problem 5
7. Some basic notation 5
Notes 6
Chapter 2. The Cauchy-Kovalevskaya Theorem with Estimates 7
1. Proof of uniqueness 7
2. Proof of existence 8
3. Proofs of accessory lemmas: Fun and useful inequalities 10
Notes 12
Chapter 3. Remarks on the Cauchy-Kovalevskaya Theorem 13
1. The Cauchy problem with holomorphic data 13
2. Transversality of the highest-order derivatives 14
3. The C-K theorem for non-singular hypersurfaces 15
4. The Goursat problem 17
5. Existence of the Riemann function 18
Notes 18
Chapter 4. Zerner’s Theorem 19
1. Real and complex hyperplanes 19
2. Zerner characteristic hypersurfaces 20
3. Proof of Zerner’s theorem 21
4. A corollary: The Delassus-Le Roux theorem 22
Notes 23
Chapter 5. The Method of Globalizing Families 25
1. Globalizing families 25
2. The globalizing principle 25
3. Applications 25
Notes 27
Chapter 6. Holmgren’s Uniqueness Theorem 29
1. A uniqueness result for harmonic functions 29
2. Holmgren’s uniqueness theorem 30
Notes 33
Chapter 7. The Continuity Method of F. John 35
1. A global uniqueness result 35
2. Exercises 36
Notes 37
Chapter 8. The Bony-Schapira Theorem 39
1. Applications of the Bony-Schapira theorem 39
2. Proof of the Bony-Schapira theorem 41
3. Exercises 42
Notes 43
Chapter 9. Applications of the Bony-Schapira Theorem:
Part I - Vekua Hulls 45
1. A uniqueness question for harmonic functions 45
2. A view from Cn: The Vekua hull 48
3. Is the connectivity condition also necessary? 54
Notes 56
Chapter 10. Applications of the Bony-Schapira Theorem:
Part II - Szeg˝o’s Theorem Revisited 57
1. Jacobi polynomial expansions: Generalization of Szeg˝o’s theorem 58
2. Relation to holomorphic PDEs 60
3. Proof of the generalized Szeg˝o theorem 61
4. Nehari’s theorem revisited 64
Notes 70
Chapter 11. The Reflection Principle 73
1. The Schwarz function of a curve 73
2. E. Study’s interpretation of the Schwarz reflection principle 75
3. Failure of the reflection law for other operators 76
Notes 81
Chapter 12. The Reflection Principle (continued) 83
1. The Study relation 83
2. Reflection in higher dimensions 86
3. The even-dimensional case 90
4. The odd-dimensional case 95
Notes 97
Chapter 13. Cauchy Problems and the Schwarz Potential Conjecture 99
1. Analytic continuation of potentials and quadrature domains 101
2. The Schwarz potential conjecture 103
Notes 106
Chapter 14. The Schwarz Potential Conjecture for Spheres 107
Notes 114
Chapter 15. Potential Theory on Ellipsoids: Part I - The Mean Value
Property 115
1. Proof of MacLaurin’s theorem using E. Fischer’s inner product 116
2. The Newtonian potential of an ellipsoid 119
Notes 122
Chapter 16. Potential Theory on Ellipsoids: Part II - There is No Gravity
in the Cavity 123
1. Arbitrary polynomial density 123
2. The standard single layer potential 125
3. Domains of hyperbolicity 127
4. The Schwarz potential conjecture for ellipsoids 128
Notes 130
Chapter 17. Potential Theory on Ellipsoids: Part III - The Dirichlet
Problem 133
1. The Dirichlet problem in an ellipsoid: Polynomial data 133
2. Entire data 134
3. The Khavinson-Shapiro conjectures 136
4. The Brelot-Choquet theorem and harmonic divisors 137
Notes 137
Chapter 18. Singularities Encountered by the Analytic Continuation of
Solutions to the Dirichlet Problem 139
1. The Dirichlet problem: When does entire data imply entire solution? 140
2. When does polynomial data imply polynomial solution? 140
3. The Dirichlet problem and Bergman orthogonal polynomials 142
4. Singularities of the solutions to the Dirichlet problem 142
5. Render’s theorem 144
6. Back to R2: Annihilating measures and closed lightning bolts 146
Notes 149
Chapter 19. An Introduction to J. Leray’s Principle on Propagation of
Singularities through Cn 151
1. Introductory remarks on propagation of singularities 151
2. Local propagation of singularities in Cn: Leray’s principle 154
Notes 165
Chapter 20. Global Propagation of Singularities in Cn 167
1. Global propagation of singularities and Persson equations 167
2. A note on characteristic surfaces for the Laplace operator 176
Notes 178
Chapter 21. Quadrature Domains and Laplacian Growth 181
1. Dynamics of singularities of the Schwarz potential 182
2. Quadrature domains and Richardson’s theorem 183
3. Exact solutions in the plane 185
4. Algebraicity of planar quadrature domains 186
5. Higher-dimensional quadrature domains need not be algebraic 186
Notes 193
Chapter 22. Other Varieties of Quadrature Domains 195
1. Ellipsoids as quadrature domains in the wide sense 195
2. Null quadrature domains 196
3. Arclength quadrature domains 196
4. Lemniscates as quadrature domains for equilibrium measure 197
5. Quadrature domains for other classes of test functions 199
Notes 201
Bibliography 203
Index 213
