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Hardcover ISBN:  9781470437800 
Product Code:  SURV/232 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470447663 
Product Code:  SURV/232.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470437800 
eBook ISBN:  9781470447663 
Product Code:  SURV/232.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 232; 2018; 214 ppMSC: Primary 35; 31; 32; 30; 14
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.
ReadershipGraduate students and researchers interested in PDE, especially in holomorphic linear PDE.

Table of Contents

Chapters

Introduction: Some motivating questions

The CauchyKovalevskaya theorem with estimates

Remarks on the CauchyKovalevskaya theorem

Zerner’s theorem

The method of globalizing families

Holmgren’s uniqueness theorem

The continuity method of F. John

The BonySchapira theorem

Applications of the BonySchapira theorem: Part I  Vekua hulls

Applications of the BonySchapira theorem: Part II  Szegő’s theorem revisited

The reflection principle

The reflection principle (continued)

Cauchy problems and the Schwarz potential conjecture

The Schwarz potential conjecture for spheres

Potential theory on ellipsoids: Part I  The mean value property

Potential theory on ellipsoids: Part II  There is no gravity in the cavity

Potential theory on ellipsoids: Part III  The Dirichlet problem

Singularities encountered by the analytic continuation of solutions to the Dirichlet problem

An introduction to J. Leray’s principle on propagation of singularities through $\mathbb {C}^n$

Global propagation of singularities in $\mathbb {C}^n$

Quadrature domains and Laplacian growth

Other varieties of quadrature domains


Additional Material

Reviews

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


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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.

Chapters

Introduction: Some motivating questions

The CauchyKovalevskaya theorem with estimates

Remarks on the CauchyKovalevskaya theorem

Zerner’s theorem

The method of globalizing families

Holmgren’s uniqueness theorem

The continuity method of F. John

The BonySchapira theorem

Applications of the BonySchapira theorem: Part I  Vekua hulls

Applications of the BonySchapira theorem: Part II  Szegő’s theorem revisited

The reflection principle

The reflection principle (continued)

Cauchy problems and the Schwarz potential conjecture

The Schwarz potential conjecture for spheres

Potential theory on ellipsoids: Part I  The mean value property

Potential theory on ellipsoids: Part II  There is no gravity in the cavity

Potential theory on ellipsoids: Part III  The Dirichlet problem

Singularities encountered by the analytic continuation of solutions to the Dirichlet problem

An introduction to J. Leray’s principle on propagation of singularities through $\mathbb {C}^n$

Global propagation of singularities in $\mathbb {C}^n$

Quadrature domains and Laplacian growth

Other varieties of quadrature domains

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