eBook ISBN:  9781470405052 
Product Code:  MEMO/192/899.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470405052 
Product Code:  MEMO/192/899.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 192; 2008; 72 ppMSC: Primary 58; Secondary 46
The authors study Sobolev classes of weakly differentiable mappings \(f:{\mathbb X}\rightarrow {\mathbb Y}\) between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in \({\mathcal W}^{1,n}({\mathbb X}\, ,\, {\mathbb Y})\,\), \(n=\mbox{dim}\, {\mathbb X}\). The central themes being discussed are:
 smooth approximation of those mappings
 integrability of the Jacobian determinant
The approximation problem in the category of Sobolev spaces between manifolds \({\mathcal W}^{1,p}({\mathbb X}\, ,\, {\mathbb Y})\), \(1\leqslant p \leqslant n\), has been recently settled. However, the point of the results is that the authors make no topological restrictions on manifolds \({\mathbb X}\) and \({\mathbb Y}\). They characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of their approach is that they were able to detect tiny sets on which the mappings are continuous. These sets give rise to the socalled weblike structure of \({\mathbb X}\) associated with the given mapping \(f: {\mathbb X}\rightarrow {\mathbb Y}\).
The integrability theory of Jacobians in a manifold setting is really different than one might a priori expect based on the results in the Euclidean space. To the authors' surprise, the case when the target manifold \({\mathbb Y}\) admits only trivial cohomology groups \(H^\ell ({\mathbb Y})\), \(1\leqslant \ell <n= \mbox{dim}\, {\mathbb Y}\), like \(n\)sphere, is more difficult than the nontrivial case in which \({\mathbb Y}\) has at least one nonzero \(\ell\)cohomology. The necessity of topological constraints on the target manifold is a new phenomenon in the theory of Jacobians.

Table of Contents

Chapters

1. Introduction

2. Preliminaries concerning manifolds

3. Examples

4. Some classes of functions

5. Smooth approximation

6. $\mathcal {L}^1$estimates of the Jacobian

7. $\mathcal {H}^1$estimates

8. Degree theory

9. Mappings of finite distortion


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The authors study Sobolev classes of weakly differentiable mappings \(f:{\mathbb X}\rightarrow {\mathbb Y}\) between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in \({\mathcal W}^{1,n}({\mathbb X}\, ,\, {\mathbb Y})\,\), \(n=\mbox{dim}\, {\mathbb X}\). The central themes being discussed are:
 smooth approximation of those mappings
 integrability of the Jacobian determinant
The approximation problem in the category of Sobolev spaces between manifolds \({\mathcal W}^{1,p}({\mathbb X}\, ,\, {\mathbb Y})\), \(1\leqslant p \leqslant n\), has been recently settled. However, the point of the results is that the authors make no topological restrictions on manifolds \({\mathbb X}\) and \({\mathbb Y}\). They characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of their approach is that they were able to detect tiny sets on which the mappings are continuous. These sets give rise to the socalled weblike structure of \({\mathbb X}\) associated with the given mapping \(f: {\mathbb X}\rightarrow {\mathbb Y}\).
The integrability theory of Jacobians in a manifold setting is really different than one might a priori expect based on the results in the Euclidean space. To the authors' surprise, the case when the target manifold \({\mathbb Y}\) admits only trivial cohomology groups \(H^\ell ({\mathbb Y})\), \(1\leqslant \ell <n= \mbox{dim}\, {\mathbb Y}\), like \(n\)sphere, is more difficult than the nontrivial case in which \({\mathbb Y}\) has at least one nonzero \(\ell\)cohomology. The necessity of topological constraints on the target manifold is a new phenomenon in the theory of Jacobians.

Chapters

1. Introduction

2. Preliminaries concerning manifolds

3. Examples

4. Some classes of functions

5. Smooth approximation

6. $\mathcal {L}^1$estimates of the Jacobian

7. $\mathcal {H}^1$estimates

8. Degree theory

9. Mappings of finite distortion