**Mathematical Surveys and Monographs**

Volume: 232;
2018;
214 pp;
Hardcover

MSC: Primary 35; 31; 32; 30; 14;

**Print ISBN: 978-1-4704-3780-0
Product Code: SURV/232**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4766-3
Product Code: SURV/232.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Linear Holomorphic Partial Differential Equations and Classical Potential Theory

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*Dmitry Khavinson; Erik Lundberg*

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.

#### Readership

Graduate students and researchers interested in PDE, especially in holomorphic linear PDE.

#### Reviews & Endorsements

The book is written in a friendly but rigorous way and is aimed at good graduate students and young analysts.

-- A. Yu Rashkovskii, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Linear Holomorphic Partial Differential Equations and Classical Potential Theory

- Cover Cover11
- Title page i2
- Contents v6
- Preface ix10
- Chapter 1. Introduction: Some Motivating Questions 112
- Chapter 2. The Cauchy-Kovalevskaya Theorem with Estimates 718
- Chapter 3. Remarks on the Cauchy-Kovalevskaya Theorem 1324
- Chapter 4. Zerner’s Theorem 1930
- Chapter 5. The Method of Globalizing Families 2536
- Chapter 6. Holmgren’s Uniqueness Theorem 2940
- Chapter 7. The Continuity Method of F. John 3546
- Chapter 8. The Bony-Schapira Theorem 3950
- Chapter 9. Applications of the Bony-Schapira Theorem: Part I - Vekua Hulls 4556
- Chapter 10. Applications of the Bony-Schapira Theorem: Part II - Szegő’s Theorem Revisited 5768
- Chapter 11. The Reflection Principle 7384
- Chapter 12. The Reflection Principle (continued) 8394
- Chapter 13. Cauchy Problems and the Schwarz Potential Conjecture 99110
- Chapter 14. The Schwarz Potential Conjecture for Spheres 107118
- Chapter 15. Potential Theory on Ellipsoids: Part I - The Mean Value Property 115126
- Chapter 16. Potential Theory on Ellipsoids: Part II - There is No Gravity in the Cavity 123134
- Chapter 17. Potential Theory on Ellipsoids: Part III - The Dirichlet Problem 133144
- Chapter 18. Singularities Encountered by the Analytic Continuation of Solutions to the Dirichlet Problem 139150
- 1. The Dirichlet problem: When does entire data imply entire solution? 140151
- 2. When does polynomial data imply polynomial solution? 140151
- 3. The Dirichlet problem and Bergman orthogonal polynomials 142153
- 4. Singularities of the solutions to the Dirichlet problem 142153
- 5. Render’s theorem 144155
- 6. Back to ℝ²: Annihilating measures and closed lightning bolts 146157
- Notes 149160

- Chapter 19. An Introduction to J. Leray’s Principle on Propagation of Singularities through ℂⁿ 151162
- Chapter 20. Global Propagation of Singularities in ℂⁿ 167178
- Chapter 21. Quadrature Domains and Laplacian Growth 181192
- Chapter 22. Other Varieties of Quadrature Domains 195206
- Bibliography 203214
- Index 213224
- Back Cover Back Cover1226