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Hodge Theory in the Sobolev Topology for the de Rham Complex

Luigi Fontana Universita di Milano, Milan, Italy
Steven G. Krantz Washington University, St. Louis, MO
Marco M. Peloso Politecnico di Torino, Torino, Italy
Available Formats:
Electronic ISBN: 978-1-4704-0211-2
Product Code: MEMO/131/622.E
100 pp
List Price: $48.00 MAA Member Price:$43.20
AMS Member Price: $28.80 Click above image for expanded view Hodge Theory in the Sobolev Topology for the de Rham Complex Luigi Fontana Universita di Milano, Milan, Italy Steven G. Krantz Washington University, St. Louis, MO Marco M. Peloso Politecnico di Torino, Torino, Italy Available Formats:  Electronic ISBN: 978-1-4704-0211-2 Product Code: MEMO/131/622.E 100 pp  List Price:$48.00 MAA Member Price: $43.20 AMS Member Price:$28.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 1311998
MSC: Primary 35; 58;

In this book, the authors treat the full Hodge theory for the de Rham complex when calculated in the Sobolev topology rather than in the $L^2$ topology. The use of the Sobolev topology strikingly alters the problem from the classical setup and gives rise to a new class of elliptic boundary value problems. The study takes place on both the upper half space and on a smoothly bounded domain.

Features:

• a good introduction to elliptic theory, pseudo-differential operators, and boundary value problems
• theorems completely explained and proved
• new geometric tools for differential analysis on domains and manifolds

Graduate students, research mathematicians, control theorists, engineers and physicists working in boundary value problems for elliptic systems.

• Chapters
• Preliminaries
• 0. Introductory remarks
• 1. Basic notation and definitions
• 2. Formulation of the problem and statement of the main results
• The problem on the half space
• 3. The operator $d$* on 1-forms and its domain
• 4. Boutet De Monvel-type analysis of the boundary value problem
• 5. The explicit solution in the case of functions
• 6. Analysis of the problem on the half space for $q$-forms
• The case of smoothly bounded domains
• 7. Formulation of the problem on a smoothly bounded domain
• 8. A special coordinate system
• 9. The existence theorem
• 10. The regularity theorem in the case of functions
• 11. Estimates for $q$-forms
• 12. The decomposition theorem and conclusions
• 13. Final remarks
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• Get Permissions
Volume: 1311998
MSC: Primary 35; 58;

In this book, the authors treat the full Hodge theory for the de Rham complex when calculated in the Sobolev topology rather than in the $L^2$ topology. The use of the Sobolev topology strikingly alters the problem from the classical setup and gives rise to a new class of elliptic boundary value problems. The study takes place on both the upper half space and on a smoothly bounded domain.

Features:

• a good introduction to elliptic theory, pseudo-differential operators, and boundary value problems
• theorems completely explained and proved
• new geometric tools for differential analysis on domains and manifolds

Graduate students, research mathematicians, control theorists, engineers and physicists working in boundary value problems for elliptic systems.

• Chapters
• Preliminaries
• 0. Introductory remarks
• 1. Basic notation and definitions
• 2. Formulation of the problem and statement of the main results
• The problem on the half space
• 3. The operator $d$* on 1-forms and its domain
• 4. Boutet De Monvel-type analysis of the boundary value problem
• 5. The explicit solution in the case of functions
• 6. Analysis of the problem on the half space for $q$-forms
• The case of smoothly bounded domains
• 7. Formulation of the problem on a smoothly bounded domain
• 8. A special coordinate system
• 9. The existence theorem
• 10. The regularity theorem in the case of functions
• 11. Estimates for $q$-forms
• 12. The decomposition theorem and conclusions
• 13. Final remarks
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