Global Carleman Estimates for Degenerate Parabolic Operators with Applications.
Cannarsa, P.
Global Carleman Estimates for Degenerate Parabolic Operators with Applications. - 1st ed. - 1 online resource (225 pages) - Memoirs of the American Mathematical Society Series ; v.239 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Stochastic invariance -- 1.2. Laminar flow -- 1.3. Budyko-Sellers climate models -- 1.4. Fleming-Viot gene frequency model -- 1.5. Null controllability in one space dimension -- Part 1 . Weakly degenerate operators with Dirichlet boundary conditions -- Chapter 2. Controllability and inverse source problems: Notation and main results -- 2.1. Notation and assumptions -- 2.1.1. Geometric assumptions and properties of domains -- 2.1.2. Assumptions on degeneracy -- 2.2. Statement of the controllability problem and main results -- 2.3. Statement of the inverse source problems and main results -- 2.3.1. First inverse source problem -- 2.3.2. Second inverse source problem -- 2.4. Estimates with respect to the degeneracy parameter \a -- Chapter 3. Global Carleman estimates for weakly degenerate operators -- 3.1. Function spaces and well-posedness -- 3.1.1. Function spaces -- 3.1.2. Trace theory and integration-by-parts formula -- 3.1.3. Regularity results -- 3.1.4. Well-posedness -- 3.2. Observability: inequality and cost -- 3.3. Fundamental tools -- 3.3.1. Improved Hardy type inequalities -- 3.3.2. Topological lemmas -- 3.4. Global Carleman estimates for weakly degenerate operators -- 3.4.1. Statement of the global Carleman estimate -- 3.4.2. Comparison with the literature on Carleman estimates -- 3.4.3. Additional remarks -- 3.5. Extensions -- 3.5.1. Degenerate parabolic operators with lower order terms -- 3.5.2. Weakened geometric assumptions -- Chapter 4. Some Hardy-type inequalities (proof of Lemma 3.18) -- 4.1. Hardy-type inequalities in space dimension 1 -- 4.1.1. The well known Hardy-type inequality in space dimension 1 -- 4.1.2. An improved version: proof of Lemma 3.16 -- 4.2. Hardy-type inequalities in space dimension 2 -- 4.2.1. Extension of the classical Hardy inequality under Hyp. 2.4. 4.2.2. Consequence: proof of Lemma 3.18 under Hyp. 2.4. -- 4.2.3. Consequence: proof of Lemma 3.18 under Hyp. 2.2. -- Chapter 5. Asymptotic properties of elements of ²(Ø)∩ ¹_(Ø) -- 5.1. Asymptotic behavior near the boundary of the elements of ¹_(Ø) and ²_(Ø)∩ ¹_(Ø) under Hyp. 2.4 -- 5.1.1. Statement of the main asymptotic properties -- 5.1.2. Proof of Lemma 5.2 -- 5.1.3. Proof of Lemma 5.1 -- 5.2. Asymptotic properties under Hyp. 2.2 -- Chapter 6. Proof of the topological lemma 3.21 -- 6.1. Preliminary Lemma -- 6.2. Proof of Lemma 6.1 -- 6.3. Proof of Lemma 3.21 -- Chapter 7. Outlines of the proof of Theorems 3.23 and 3.26 -- 7.1. Outlines of the proof of Theorems 3.23 and 3.26 under Hyp. 2.4 -- 7.1.1. Under Hyp. 2.4: choice of the weight functions and objectives -- 7.1.2. Step 1 (under Hyp. 2.4): computation of the scalar product on subdomains Ø^ -- 7.1.3. Step 2 (under Hyp. 2.4): an estimate of the scalar product on subdomains Ø^ -- 7.1.4. Step 3 (under Hyp. 2.4): the limits as Ø^→Ø -- 7.1.5. Step 4 (under Hyp. 2.4): partial Carleman estimate -- 7.1.6. Step 5 (under Hyp. 2.4): from the partial to the global Carleman estimate -- 7.1.7. Step 6 (under Hyp. 2.4): global Carleman estimates -- 7.2. Generalization: main changes under Hyp. 2.2 -- 7.2.1. The choice of the weight functions under Hyp. 2.2 -- 7.2.2. Step 1 (under Hyp. 2.2): computation of the scalar product on subdomains -- 7.2.3. Step 2 (under Hyp. 2.2): estimates for the distributed terms -- 7.2.4. Step 3 (under Hyp. 2.2): the limits Ø^→Ø -- 7.2.5. Step 4 (under Hyp. 2.2): partial Carleman estimate -- 7.2.6. Steps 5 and 6 (under Hyp. 2.2): from the partial to the global Carleman estimate -- Chapter 8. Step 1: computation of the scalar product on subdomains (proof of Lemmas 7.1 and 7.16) -- 8.1. The scalar product under Hyp. 2.4. 8.2. The scalar product under Hyp. 2.2 -- Chapter 9. Step 2: a first estimate of the scalar product: proof of Lemmas 7.2, 7.4, 7.18 and 7.19 -- 9.1. A first estimate of the scalar product under Hyp. 2.4: proof of Lemmas 7.2 and 7.4 -- 9.1.1. Estimate of the first order terms ₁^: proof of Lemma 7.2 -- 9.1.2. Estimate of the zero order term ₀^: proof of Lemma 7.4 -- 9.2. A first estimate of the scalar product under Hyp. 2.2: proof of Lemmas 7.18 and 7.19 -- 9.2.1. A general result about the asymptotic behaviour near the boundary: proof of Lemma 7.17 -- 9.2.2. First consequence: estimate of the first order term ₁^ (proof of Lemma 7.18) -- 9.2.3. Second consequence: estimate of the zero order term ₀^ (proof of Lemma 7.19) -- Chapter 10. Step 3: the limits as Ø^→Ø (proof of Lemmas 7.5 and 7.20) -- 10.1. The limits as Ø^→Ø under Hyp. 2.4 (proof of Lemma 7.5) -- 10.1.1. Statement of the convergence results -- 10.1.2. Convergence of the distributed terms: proof of Lemma 10.1 -- 10.1.3. Convergence of the boundary term: proof of Lemma 10.2 -- 10.1.4. An identity of interpolation type -- 10.2. The limits as Ø^→Ø under Hyp. 2.2 (proof of Lemma 7.20) -- 10.2.1. Statement of the convergence results -- 10.2.2. Ideas of the proof of Lemmas 10.7 and 10.8 -- Chapter 11. Step 4: partial Carleman estimate (proof of Lemmas 7.6 and 7.21) -- 11.1. The partial Carleman estimate under Hyp. 2.4 (proof of Lemma 7.6) -- 11.1.1. The consequence of the estimate of the scalar product given in Lemma 7.5 -- 11.1.2. Some adapted Hardy-type inequalities -- 11.1.3. Consequence: proof of Lemma 7.6 -- 11.2. The partial Carleman estimate under Hyp. 2.2 (proof of Lemma 7.21) -- 11.2.1. The consequence of the estimate of the scalar product given in Lemma 7.20 -- 11.2.2. Proof of Lemma 7.21 -- 11.2.3. Uniform estimates when \a→1⁻ (under Hyp. 2.12). Chapter 12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas 7.9-7.11) -- 12.1. Estimate of the zero order term: proof of Lemma 7.8 -- 12.2. Estimate of the first order spatial derivatives: proof of Lemma 7.9 -- 12.2.1. The non uniform estimate -- 12.2.2. The uniform estimate -- 12.3. Estimate of the second order spatial derivatives: proof of Lemma 7.10 -- 12.4. Estimate of the first order time derivative: proof of Lemma 7.11 -- Chapter 13. Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14 and 7.15) -- 13.1. The global Carleman estimate for : proof of Lemma 7.12 -- 13.2. The first global Carleman estimate for : proof of Lemma 7.14 -- 13.2.1. Zero order term estimates. -- 13.2.2. First order spatial derivatives estimates. -- 13.2.3. First order time derivative estimate. -- 13.2.4. Second order spatial derivatives estimates. -- 13.2.5. Conclusion: proof of Lemma 7.14. -- 13.3. The second global Carleman estimate for : proof of Lemma 7.15 -- Chapter 14. Proof of observability and controllability results -- 14.1. Proof of Theorem 3.13 -- 14.2. Equivalence between null controllability and observability -- Chapter 15. Application to some inverse source problems: proof of Theorems 2.9 and 2.11 -- 15.1. Proof of Theorem 2.9 -- 15.2. Proof of Theorem 2.11 -- Part 2 . Strongly degenerate operators with Neumann boundary conditions -- Chapter 16. Controllability and inverse source problems: notation and main results -- 16.1. Notation and assumptions -- 16.1.1. Geometric assumptions and properties of the domain -- 16.1.2. Assumptions on degeneracy -- 16.2. Statement of the controllability problem and main results -- 16.2.1. A null controllability result for \a∈[1,2). -- 16.2.2. Counterexample for \a∈[2,+∞). -- 16.2.3. Explosion of the controllability cost as \a→2⁻ in space dimension 1. 16.3. Statement of the inverse source problems and main results -- 16.3.1. First inverse source problem -- 16.3.2. Second inverse source problem -- Chapter 17. Global Carleman estimates for strongly degenerate operators -- 17.1. Functional spaces and well-posedness -- 17.1.1. Function spaces -- 17.1.2. Normal trace theory and integration-by-parts formula -- 17.1.3. Regularity results -- 17.1.4. Well-posedness -- 17.2. Observability: inequality and cost -- 17.3. Global Carleman estimates for strongly degenerate operators -- 17.4. Fundamental tools -- 17.5. Some extensions -- 17.5.1. Global Carleman estimate for a more general degenerate parabolic equation -- 17.5.2. Weakened geometric assumptions -- Chapter 18. Hardy-type inequalities: proof of Lemma 17.10 and applications -- 18.1. Some Hardy-type inequalities in dimension 1 -- 18.1.1. The classical Hardy inequality when \a∈(1,2) -- 18.1.2. A first extension of the classical Hardy inequality -- 18.2. Proof of Lemma 17.10 -- 18.2.1. Proof of Lemma 17.10, part (i), under Hypothesis 17.9. -- 18.2.2. Proof of Lemma 17.10, part (ii), under Hypothesis 17.9. -- 18.2.3. Proof of Lemma 17.10 under Hypothesis 16.1. -- 18.3. Some Hardy-type inequalities adapted to our problem -- 18.3.1. The natural extension of Lemma 17.10. -- 18.3.2. Consequence of Lemma 18.3: another Hardy-type inequalities. -- 18.3.3. Consequences of Lemma 18.4. -- Chapter 19. Global Carleman estimates in the strongly degenerate case: proof of Theorem 17.7 -- 19.1. Outlines of the proof of Theorem 17.7 -- 19.2. Proof of Theorem 17.7 under Hyp. 17.9 -- 19.2.1. Steps 1 and 2 (under Hyp. 17.9): computation and estimate of the scalar product on subdomains -- 19.2.2. Step 3 (under Hyp. 17.9): the limits as Ø^→Ø -- 19.2.3. Step 4 (under Hyp. 17.9): partial Carleman estimate. 19.2.4. Steps 5 and 6 (under Hyp. 17.9): from the partial to the global Carleman estimate (proof of Theorem 17.7).
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.
9781470427498
Elliptic operators.
Parabolic operators.
Carleman theorem.
Electronic books.
QA329.4 .C36 2015
515/.3534
Global Carleman Estimates for Degenerate Parabolic Operators with Applications. - 1st ed. - 1 online resource (225 pages) - Memoirs of the American Mathematical Society Series ; v.239 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Stochastic invariance -- 1.2. Laminar flow -- 1.3. Budyko-Sellers climate models -- 1.4. Fleming-Viot gene frequency model -- 1.5. Null controllability in one space dimension -- Part 1 . Weakly degenerate operators with Dirichlet boundary conditions -- Chapter 2. Controllability and inverse source problems: Notation and main results -- 2.1. Notation and assumptions -- 2.1.1. Geometric assumptions and properties of domains -- 2.1.2. Assumptions on degeneracy -- 2.2. Statement of the controllability problem and main results -- 2.3. Statement of the inverse source problems and main results -- 2.3.1. First inverse source problem -- 2.3.2. Second inverse source problem -- 2.4. Estimates with respect to the degeneracy parameter \a -- Chapter 3. Global Carleman estimates for weakly degenerate operators -- 3.1. Function spaces and well-posedness -- 3.1.1. Function spaces -- 3.1.2. Trace theory and integration-by-parts formula -- 3.1.3. Regularity results -- 3.1.4. Well-posedness -- 3.2. Observability: inequality and cost -- 3.3. Fundamental tools -- 3.3.1. Improved Hardy type inequalities -- 3.3.2. Topological lemmas -- 3.4. Global Carleman estimates for weakly degenerate operators -- 3.4.1. Statement of the global Carleman estimate -- 3.4.2. Comparison with the literature on Carleman estimates -- 3.4.3. Additional remarks -- 3.5. Extensions -- 3.5.1. Degenerate parabolic operators with lower order terms -- 3.5.2. Weakened geometric assumptions -- Chapter 4. Some Hardy-type inequalities (proof of Lemma 3.18) -- 4.1. Hardy-type inequalities in space dimension 1 -- 4.1.1. The well known Hardy-type inequality in space dimension 1 -- 4.1.2. An improved version: proof of Lemma 3.16 -- 4.2. Hardy-type inequalities in space dimension 2 -- 4.2.1. Extension of the classical Hardy inequality under Hyp. 2.4. 4.2.2. Consequence: proof of Lemma 3.18 under Hyp. 2.4. -- 4.2.3. Consequence: proof of Lemma 3.18 under Hyp. 2.2. -- Chapter 5. Asymptotic properties of elements of ²(Ø)∩ ¹_(Ø) -- 5.1. Asymptotic behavior near the boundary of the elements of ¹_(Ø) and ²_(Ø)∩ ¹_(Ø) under Hyp. 2.4 -- 5.1.1. Statement of the main asymptotic properties -- 5.1.2. Proof of Lemma 5.2 -- 5.1.3. Proof of Lemma 5.1 -- 5.2. Asymptotic properties under Hyp. 2.2 -- Chapter 6. Proof of the topological lemma 3.21 -- 6.1. Preliminary Lemma -- 6.2. Proof of Lemma 6.1 -- 6.3. Proof of Lemma 3.21 -- Chapter 7. Outlines of the proof of Theorems 3.23 and 3.26 -- 7.1. Outlines of the proof of Theorems 3.23 and 3.26 under Hyp. 2.4 -- 7.1.1. Under Hyp. 2.4: choice of the weight functions and objectives -- 7.1.2. Step 1 (under Hyp. 2.4): computation of the scalar product on subdomains Ø^ -- 7.1.3. Step 2 (under Hyp. 2.4): an estimate of the scalar product on subdomains Ø^ -- 7.1.4. Step 3 (under Hyp. 2.4): the limits as Ø^→Ø -- 7.1.5. Step 4 (under Hyp. 2.4): partial Carleman estimate -- 7.1.6. Step 5 (under Hyp. 2.4): from the partial to the global Carleman estimate -- 7.1.7. Step 6 (under Hyp. 2.4): global Carleman estimates -- 7.2. Generalization: main changes under Hyp. 2.2 -- 7.2.1. The choice of the weight functions under Hyp. 2.2 -- 7.2.2. Step 1 (under Hyp. 2.2): computation of the scalar product on subdomains -- 7.2.3. Step 2 (under Hyp. 2.2): estimates for the distributed terms -- 7.2.4. Step 3 (under Hyp. 2.2): the limits Ø^→Ø -- 7.2.5. Step 4 (under Hyp. 2.2): partial Carleman estimate -- 7.2.6. Steps 5 and 6 (under Hyp. 2.2): from the partial to the global Carleman estimate -- Chapter 8. Step 1: computation of the scalar product on subdomains (proof of Lemmas 7.1 and 7.16) -- 8.1. The scalar product under Hyp. 2.4. 8.2. The scalar product under Hyp. 2.2 -- Chapter 9. Step 2: a first estimate of the scalar product: proof of Lemmas 7.2, 7.4, 7.18 and 7.19 -- 9.1. A first estimate of the scalar product under Hyp. 2.4: proof of Lemmas 7.2 and 7.4 -- 9.1.1. Estimate of the first order terms ₁^: proof of Lemma 7.2 -- 9.1.2. Estimate of the zero order term ₀^: proof of Lemma 7.4 -- 9.2. A first estimate of the scalar product under Hyp. 2.2: proof of Lemmas 7.18 and 7.19 -- 9.2.1. A general result about the asymptotic behaviour near the boundary: proof of Lemma 7.17 -- 9.2.2. First consequence: estimate of the first order term ₁^ (proof of Lemma 7.18) -- 9.2.3. Second consequence: estimate of the zero order term ₀^ (proof of Lemma 7.19) -- Chapter 10. Step 3: the limits as Ø^→Ø (proof of Lemmas 7.5 and 7.20) -- 10.1. The limits as Ø^→Ø under Hyp. 2.4 (proof of Lemma 7.5) -- 10.1.1. Statement of the convergence results -- 10.1.2. Convergence of the distributed terms: proof of Lemma 10.1 -- 10.1.3. Convergence of the boundary term: proof of Lemma 10.2 -- 10.1.4. An identity of interpolation type -- 10.2. The limits as Ø^→Ø under Hyp. 2.2 (proof of Lemma 7.20) -- 10.2.1. Statement of the convergence results -- 10.2.2. Ideas of the proof of Lemmas 10.7 and 10.8 -- Chapter 11. Step 4: partial Carleman estimate (proof of Lemmas 7.6 and 7.21) -- 11.1. The partial Carleman estimate under Hyp. 2.4 (proof of Lemma 7.6) -- 11.1.1. The consequence of the estimate of the scalar product given in Lemma 7.5 -- 11.1.2. Some adapted Hardy-type inequalities -- 11.1.3. Consequence: proof of Lemma 7.6 -- 11.2. The partial Carleman estimate under Hyp. 2.2 (proof of Lemma 7.21) -- 11.2.1. The consequence of the estimate of the scalar product given in Lemma 7.20 -- 11.2.2. Proof of Lemma 7.21 -- 11.2.3. Uniform estimates when \a→1⁻ (under Hyp. 2.12). Chapter 12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas 7.9-7.11) -- 12.1. Estimate of the zero order term: proof of Lemma 7.8 -- 12.2. Estimate of the first order spatial derivatives: proof of Lemma 7.9 -- 12.2.1. The non uniform estimate -- 12.2.2. The uniform estimate -- 12.3. Estimate of the second order spatial derivatives: proof of Lemma 7.10 -- 12.4. Estimate of the first order time derivative: proof of Lemma 7.11 -- Chapter 13. Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14 and 7.15) -- 13.1. The global Carleman estimate for : proof of Lemma 7.12 -- 13.2. The first global Carleman estimate for : proof of Lemma 7.14 -- 13.2.1. Zero order term estimates. -- 13.2.2. First order spatial derivatives estimates. -- 13.2.3. First order time derivative estimate. -- 13.2.4. Second order spatial derivatives estimates. -- 13.2.5. Conclusion: proof of Lemma 7.14. -- 13.3. The second global Carleman estimate for : proof of Lemma 7.15 -- Chapter 14. Proof of observability and controllability results -- 14.1. Proof of Theorem 3.13 -- 14.2. Equivalence between null controllability and observability -- Chapter 15. Application to some inverse source problems: proof of Theorems 2.9 and 2.11 -- 15.1. Proof of Theorem 2.9 -- 15.2. Proof of Theorem 2.11 -- Part 2 . Strongly degenerate operators with Neumann boundary conditions -- Chapter 16. Controllability and inverse source problems: notation and main results -- 16.1. Notation and assumptions -- 16.1.1. Geometric assumptions and properties of the domain -- 16.1.2. Assumptions on degeneracy -- 16.2. Statement of the controllability problem and main results -- 16.2.1. A null controllability result for \a∈[1,2). -- 16.2.2. Counterexample for \a∈[2,+∞). -- 16.2.3. Explosion of the controllability cost as \a→2⁻ in space dimension 1. 16.3. Statement of the inverse source problems and main results -- 16.3.1. First inverse source problem -- 16.3.2. Second inverse source problem -- Chapter 17. Global Carleman estimates for strongly degenerate operators -- 17.1. Functional spaces and well-posedness -- 17.1.1. Function spaces -- 17.1.2. Normal trace theory and integration-by-parts formula -- 17.1.3. Regularity results -- 17.1.4. Well-posedness -- 17.2. Observability: inequality and cost -- 17.3. Global Carleman estimates for strongly degenerate operators -- 17.4. Fundamental tools -- 17.5. Some extensions -- 17.5.1. Global Carleman estimate for a more general degenerate parabolic equation -- 17.5.2. Weakened geometric assumptions -- Chapter 18. Hardy-type inequalities: proof of Lemma 17.10 and applications -- 18.1. Some Hardy-type inequalities in dimension 1 -- 18.1.1. The classical Hardy inequality when \a∈(1,2) -- 18.1.2. A first extension of the classical Hardy inequality -- 18.2. Proof of Lemma 17.10 -- 18.2.1. Proof of Lemma 17.10, part (i), under Hypothesis 17.9. -- 18.2.2. Proof of Lemma 17.10, part (ii), under Hypothesis 17.9. -- 18.2.3. Proof of Lemma 17.10 under Hypothesis 16.1. -- 18.3. Some Hardy-type inequalities adapted to our problem -- 18.3.1. The natural extension of Lemma 17.10. -- 18.3.2. Consequence of Lemma 18.3: another Hardy-type inequalities. -- 18.3.3. Consequences of Lemma 18.4. -- Chapter 19. Global Carleman estimates in the strongly degenerate case: proof of Theorem 17.7 -- 19.1. Outlines of the proof of Theorem 17.7 -- 19.2. Proof of Theorem 17.7 under Hyp. 17.9 -- 19.2.1. Steps 1 and 2 (under Hyp. 17.9): computation and estimate of the scalar product on subdomains -- 19.2.2. Step 3 (under Hyp. 17.9): the limits as Ø^→Ø -- 19.2.3. Step 4 (under Hyp. 17.9): partial Carleman estimate. 19.2.4. Steps 5 and 6 (under Hyp. 17.9): from the partial to the global Carleman estimate (proof of Theorem 17.7).
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.
9781470427498
Elliptic operators.
Parabolic operators.
Carleman theorem.
Electronic books.
QA329.4 .C36 2015
515/.3534