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 Description
 x, 214 pages : illustrations (some color) ; 27 cm.
 Additional Authors
 Lundberg, Erik,
 Notes
 Includes bibliographical references (pages 203211) and index.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  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 Cn  Global propagation of singularities in Cn  Quadrature domains and Laplacian growth  Other varieties of quadrature domains.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.