UID:
almahu_9947363014302882
Umfang:
XVI, 388 p. 10 illus.
,
online resource.
ISBN:
9781461208952
Serie:
Universitext,
Anmerkung:
I. Notation and function spaces -- §1. Some notation -- §2. Basic facts aboutW1,p(?) andWo1,p(?) -- §3. Parabolic spaces and embeddings -- §4. Auxiliary lemmas -- §5. Bibliographical notes -- II. Weak solutions and local energy estimates -- §1. Quasilinear degenerate or singular equations -- §2. Boundary value problems -- §3. Local integral inequalities -- §4. Energy estimates near the boundary -- §5. Restricted structures: the levelskand the constant ? -- §6. Bibliographical notes -- III. Hölder continuity of solutions of degenerate parabolic equations -- §1. The regularity theorem -- §2. Preliminaries -- §3. The main proposition -- §4. The first alternative -- §5. The first alternative continued -- §6. The first alternative concluded -- §7. The second alternative -- §8. The second alternative continued -- §9. The second alternative concluded -- §10. Proof of Proposition 3.1 -- §11. Regularity up tot= 0 -- §12. Regularity up toST. Dirichlet data -- §13. Regularity atST. Variational data -- §14. Remarks on stability -- §15. Bibliographical notes -- IV. Hölder continuity of solutions of singular parabolic equations -- §1. Singular equations and the regularity theorems -- §2. The main proposition -- §3. Preliminaries -- §4. Rescaled iterations -- §5. The first alternative -- §6. Proof of Lemma 5.1. Integral inequalities -- §7. An auxiliary proposition -- §8. Proof of Proposition 7.1 when (7.6) holds -- §9. Removing the assumption (6.1) -- §10. The second alternative -- §11. The second alternative concluded -- §12. Proof of the main proposition -- §13. Boundary regularity -- §14. Miscellaneous remarks -- §15. Bibliographical notes -- V. Boundedness of weak solutions -- §1. Introduction -- §2. Quasilinear parabolic equations -- §3. Sup-bounds -- §4. Homogeneous structures. 2 -- §5. Homogeneous structures. The singular case 1 〈p〈 2 -- §6. Energy estimates -- §7. Local iterative inequalities -- §8. Local iterative inequalities $$ \left( {p 〉 max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\}} \right) $$ -- §9. Global iterative inequalities -- §10. Homogeneous structures and $$ 1 〈 p \leqslant max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\} $$ -- §11. Proof of Theorems 3.1 and 3.2 -- §12. Proof of Theorem 4.1 -- §13. Proof of Theorem 4.2 -- §14. Proof of Theorem 4.3 -- §15. Proof of Theorem 4.5 -- §16. Proof of Theorems 5.1 and 5.2 -- §17. Natural growth conditions -- §18. Bibliographical notes -- VI. Harnack estimates: the casep〉2 -- §1. Introduction -- §2. The intrinsic Harnack inequality -- §3. Local comparison functions -- §4. Proof of Theorem 2.1 -- §5. Proof of Theorem 2.2 -- §6. Global versus local estimates -- §7. Global Harnack estimates -- §8. Compactly supported initial data -- §9. Proof of Proposition 8.1 -- §10. Proof of Proposition 8.1 continued -- §11. Proof of Proposition 8.1 concluded -- §12. The Cauchy problem with compactly supported initial data -- §13. Bibliographical notes -- VII. Harnack estimates and extinction profile for singular equations -- §1. The Harnack inequality -- §2. Extinction in finite time (bounded domains) -- §3. Extinction in finite time (in RN) -- §4. An integral Harnack inequality for all 1 2) -- §4. Hölder continuity ofDu (the case 1 〈p〈 2) -- §5. Some algebraic Lemmas -- §6. Linear parabolic systems with constant coefficients -- §7. The perturbation lemma -- §8. Proof of Proposition 1.1-(i) -- §9. Proof of Proposition 1.1-(ii) -- §10. Proof of Proposition 1.1-(iii) -- §11. Proof of Proposition 1.1 concluded -- §12. Proof of Proposition 1.2-(i) -- §13. Proof of Proposition 1.2 concluded -- §14. General structures -- §15. Bibliographical notes -- X. Parabolicp-systems: boundary regularity -- §1. Introduction -- §2. Flattening the boundary -- §3. An iteration lemma -- §4. Comparing w and y (the casep〉 2) -- §5. Estimating the local average of |Dw| (the casep〉 2) -- §6. Estimating the local averages of w (the casep〉 2) -- §7. Comparing w and y (the case max $$ \left\{ {1;\tfrac{{2N}} {{N + 2}}} \right\} 〈 p 〈 2 $$) -- §8. Estimating the local average of |Dw| -- §9. Bibliographical notes -- XI. Non-negative solutions in ?T. The casep〉2 -- §1. Introduction -- §2. Behaviour of non-negative solutions as |x| ? ? and as t ? 0 -- §3. Proof of (2.4) -- §4. Initial traces -- §5. Estimating |Du|p?1 in ?T -- §6. Uniqueness for data inLloc1(RN) -- §7. Solving the Cauchy problem -- §8. Bibliographical notes -- XII. Non-negative solutions in ?T. The case 1 The uniqueness theorem -- §6. An auxiliary proposition -- §7. Proof of the uniqueness theorem -- §8. Solving the Cauchy problem -- §9. Compactness in the space variables -- §10. Compactness in thetvariable -- §11. More on the time—compactness -- §12. The limiting process -- §13. Bounded solutions. A counterexample -- §14. Bibliographical notes.
In:
Springer eBooks
Weitere Ausg.:
Printed edition: ISBN 9780387940205
Sprache:
Englisch
DOI:
10.1007/978-1-4612-0895-2
URL:
http://dx.doi.org/10.1007/978-1-4612-0895-2
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