eBook ISBN:  9781470427498 
Product Code:  MEMO/239/1133.E 
List Price:  $100.00 
MAA Member Price:  $90.00 
AMS Member Price:  $60.00 
eBook ISBN:  9781470427498 
Product Code:  MEMO/239/1133.E 
List Price:  $100.00 
MAA Member Price:  $90.00 
AMS Member Price:  $60.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 239; 2015; 209 ppMSC: Primary 35; 93; 26
Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics.
This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.

Table of Contents

Chapters

1. Introduction

1. Weakly degenerate operators with Dirichlet boundary conditions

2. Controllability and inverse source problems: Notation and main results

3. Global Carleman estimates for weakly degenerate operators

4. Some Hardytype inequalities (proof of Lemma )

5. Asymptotic properties of elements of $H^2 (\Omega ) \cap H^1 _{A,0}(\Omega )$

6. Proof of the topological lemma

7. Outlines of the proof of Theorems and

8. Step 1: computation of the scalar product on subdomains (proof of Lemmas and )

9. Step 2: a first estimate of the scalar product: proof of Lemmas , , and

10. Step 3: the limits as $\Omega ^\delta \to \Omega $ (proof of Lemmas and )

11. Step 4: partial Carleman estimate (proof of Lemmas and )

12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas –)

13. Step 6: global Carleman estimate (proof of Lemmas , and )

14. Proof of observability and controllability results

15. Application to some inverse source problems: proof of Theorems and

2. Strongly degenerate operators with Neumann boundary conditions

16. Controllability and inverse source problems: notation and main results

17. Global Carleman estimates for strongly degenerate operators

18. Hardytype inequalities: proof of Lemma and applications

19. Global Carleman estimates in the strongly degenerate case: proof of Theorem

20. Proof of Theorem (observability inequality)

21. Lack of null controllability when $\alpha \geq 2$: proof of Proposition

22. Explosion of the controllability cost as $\alpha \to 2^$ in space dimension $1$: proof of Proposition

3. Some open problems

23. Some open problems


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Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics.
This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.

Chapters

1. Introduction

1. Weakly degenerate operators with Dirichlet boundary conditions

2. Controllability and inverse source problems: Notation and main results

3. Global Carleman estimates for weakly degenerate operators

4. Some Hardytype inequalities (proof of Lemma )

5. Asymptotic properties of elements of $H^2 (\Omega ) \cap H^1 _{A,0}(\Omega )$

6. Proof of the topological lemma

7. Outlines of the proof of Theorems and

8. Step 1: computation of the scalar product on subdomains (proof of Lemmas and )

9. Step 2: a first estimate of the scalar product: proof of Lemmas , , and

10. Step 3: the limits as $\Omega ^\delta \to \Omega $ (proof of Lemmas and )

11. Step 4: partial Carleman estimate (proof of Lemmas and )

12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas –)

13. Step 6: global Carleman estimate (proof of Lemmas , and )

14. Proof of observability and controllability results

15. Application to some inverse source problems: proof of Theorems and

2. Strongly degenerate operators with Neumann boundary conditions

16. Controllability and inverse source problems: notation and main results

17. Global Carleman estimates for strongly degenerate operators

18. Hardytype inequalities: proof of Lemma and applications

19. Global Carleman estimates in the strongly degenerate case: proof of Theorem

20. Proof of Theorem (observability inequality)

21. Lack of null controllability when $\alpha \geq 2$: proof of Proposition

22. Explosion of the controllability cost as $\alpha \to 2^$ in space dimension $1$: proof of Proposition

3. Some open problems

23. Some open problems