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The Method of Layer Potentials for the Heat Equation in Time-Varying Domains
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Memoirs of the American Mathematical Society
1995; 157 pp; softcover
Volume: 114
ISBN-10: 0-8218-0360-3
ISBN-13: 978-0-8218-0360-8
List Price: US$44 Individual Members: US$26.40
Institutional Members: US\$35.20
Order Code: MEMO/114/545

Recent years have seen renewed interest in the solution of parabolic boundary value problems by the method of layer potentials, a method that has been extraordinarily useful in the solution of elliptic problems. This book develops this method for the heat equation in time-varying domains. In the first chapter, Lewis and Murray show that certain singular integral operators on $$L^p$$ are bounded. In the second chapter, they develop a modification of the David buildup scheme, as well as some extension theorems, to obtain $$L^p$$ boundedness of the double layer heat potential on the boundary of the domains. The third chapter uses the results of the first two, along with a buildup scheme, to show the mutual absolute continuity of parabolic measure and a certain projective Lebesgue measure. Lewis and Murray also obtain $$A_\infty$$ results and discuss the Dirichlet and Neumann problems for a certain subclass of the domains.