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Elliptic Partial Differential Equations with Almost-Real Coefficients

Ariel Barton University of Minnesota, Minneapolis, MN
Available Formats:
Electronic ISBN: 978-0-8218-9875-8
Product Code: MEMO/223/1051.E
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $43.20 Click above image for expanded view Elliptic Partial Differential Equations with Almost-Real Coefficients Ariel Barton University of Minnesota, Minneapolis, MN Available Formats:  Electronic ISBN: 978-0-8218-9875-8 Product Code: MEMO/223/1051.E  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$43.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 2232013; 106 pp
MSC: Primary 35; Secondary 31;

In this monograph the author investigates divergence-form elliptic partial differential equations in two-dimensional Lipschitz domains whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.

He shows that for such operators, the Dirichlet problem with boundary data in $L^q$ can be solved for $q<\infty$ large enough. He also shows that the Neumann and regularity problems with boundary data in $L^p$ can be solved for $p>1$ small enough, and provide an endpoint result at $p=1$.

• Chapters
• 1. Introduction
• 2. Definitions and the Main Theorem
• 3. Useful Theorems
• 4. The Fundamental Solution
• 5. Properties of Layer Potentials
• 6. Boundedness of Layer Potentials
• 7. Invertibility of Layer Potentials and Other Properties
• 8. Uniqueness of Solutions
• 9. Boundary Data in \texorpdfstring{$H^1(\partial V)$}Hardy spaces
• 10. Concluding Remarks
• Requests

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Volume: 2232013; 106 pp
MSC: Primary 35; Secondary 31;

In this monograph the author investigates divergence-form elliptic partial differential equations in two-dimensional Lipschitz domains whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.

He shows that for such operators, the Dirichlet problem with boundary data in $L^q$ can be solved for $q<\infty$ large enough. He also shows that the Neumann and regularity problems with boundary data in $L^p$ can be solved for $p>1$ small enough, and provide an endpoint result at $p=1$.

• Chapters
• 1. Introduction
• 2. Definitions and the Main Theorem
• 3. Useful Theorems
• 4. The Fundamental Solution
• 5. Properties of Layer Potentials
• 6. Boundedness of Layer Potentials
• 7. Invertibility of Layer Potentials and Other Properties
• 8. Uniqueness of Solutions
• 9. Boundary Data in \texorpdfstring{$H^1(\partial V)$}Hardy spaces
• 10. Concluding Remarks
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