Electronic ISBN:  9781470402112 
Product Code:  MEMO/131/622.E 
100 pp 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 131; 1998MSC: 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, pseudodifferential operators, and boundary value problems
 theorems completely explained and proved
 new geometric tools for differential analysis on domains and manifolds
ReadershipGraduate students, research mathematicians, control theorists, engineers and physicists working in boundary value problems for elliptic systems.

Table of Contents

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 1forms and its domain

4. Boutet De Monveltype 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


Request Review Copy

Get Permissions
 Book Details
 Table of Contents

 Request Review Copy
 Get Permissions
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, pseudodifferential 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 1forms and its domain

4. Boutet De Monveltype 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