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The Method of Layer Potentials for the Heat Equation in Time-Varying Domains

John L. Lewis University of Kentucky
Margaret A. M. Murray Virginia Polytech Institute & State University
Available Formats:
Electronic ISBN: 978-1-4704-0124-5
Product Code: MEMO/114/545.E
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $28.20 Click above image for expanded view The Method of Layer Potentials for the Heat Equation in Time-Varying Domains John L. Lewis University of Kentucky Margaret A. M. Murray Virginia Polytech Institute & State University Available Formats:  Electronic ISBN: 978-1-4704-0124-5 Product Code: MEMO/114/545.E  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$28.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1141995; 157 pp
MSC: Primary 42; 35;

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.

Researchers and graduate students studying harmonic analysis and partial differential equations.

• Chapters
• I. Singular integrals
• II. The David buildup scheme
• III. Absolute continuity and Dirichlet-Neumann problems
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Volume: 1141995; 157 pp
MSC: Primary 42; 35;

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.