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Full Description
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.
Contents
Introduction:
Some motivating questions;
The Cauchy-Kovalevskaya theorem with estimates;
Remarks on the Cauchy-Kovalevskaya theorem;
Zerner's theorem;
The method of globalizing families;
Holmgren's uniqueness theorem;
The continuity method of F. John;
The Bony-Schapira theorem;
Applications of the Bony-Schapira theorem: Part I - Vekua hulls;
Applications of the Bony-Schapira theorem: Part II - Szego'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;
Bibliography;
Index.
