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.
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