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AuthorDiBenedetto, Emmanuele. author
TitleDegenerate Parabolic Equations [electronic resource] / by Emmanuele DiBenedetto
ImprintNew York, NY : Springer New York : Imprint: Springer, 1993
Connect tohttp://dx.doi.org/10.1007/978-1-4612-0895-2
Descript XVI, 388 p. 10 illus. online resource

CONTENT

I. Notation and function spaces -- ยง1. Some notation -- ยง2. Basic facts aboutW1,p(?) andWo1,p(?) -- ยง3. Parabolic spaces and embeddings -- ยง4. Auxiliary lemmas -- ยง5. Bibliographical notes -- II. Weak solutions and local energy estimates -- ยง1. Quasilinear degenerate or singular equations -- ยง2. Boundary value problems -- ยง3. Local integral inequalities -- ยง4. Energy estimates near the boundary -- ยง5. Restricted structures: the levelskand the constant ? -- ยง6. Bibliographical notes -- III. Hรถlder continuity of solutions of degenerate parabolic equations -- ยง1. The regularity theorem -- ยง2. Preliminaries -- ยง3. The main proposition -- ยง4. The first alternative -- ยง5. The first alternative continued -- ยง6. The first alternative concluded -- ยง7. The second alternative -- ยง8. The second alternative continued -- ยง9. The second alternative concluded -- ยง10. Proof of Proposition 3.1 -- ยง11. Regularity up tot= 0 -- ยง12. Regularity up toST. Dirichlet data -- ยง13. Regularity atST. Variational data -- ยง14. Remarks on stability -- ยง15. Bibliographical notes -- IV. Hรถlder continuity of solutions of singular parabolic equations -- ยง1. Singular equations and the regularity theorems -- ยง2. The main proposition -- ยง3. Preliminaries -- ยง4. Rescaled iterations -- ยง5. The first alternative -- ยง6. Proof of Lemma 5.1. Integral inequalities -- ยง7. An auxiliary proposition -- ยง8. Proof of Proposition 7.1 when (7.6) holds -- ยง9. Removing the assumption (6.1) -- ยง10. The second alternative -- ยง11. The second alternative concluded -- ยง12. Proof of the main proposition -- ยง13. Boundary regularity -- ยง14. Miscellaneous remarks -- ยง15. Bibliographical notes -- V. Boundedness of weak solutions -- ยง1. Introduction -- ยง2. Quasilinear parabolic equations -- ยง3. Sup-bounds -- ยง4. Homogeneous structures. 2 -- ยง5. Homogeneous structures. The singular case 1 <p< 2 -- ยง6. Energy estimates -- ยง7. Local iterative inequalities -- ยง8. Local iterative inequalities $$ \left( {p > max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\}} \right) $$ -- ยง9. Global iterative inequalities -- ยง10. Homogeneous structures and $$ 1 < p \leqslant max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\} $$ -- ยง11. Proof of Theorems 3.1 and 3.2 -- ยง12. Proof of Theorem 4.1 -- ยง13. Proof of Theorem 4.2 -- ยง14. Proof of Theorem 4.3 -- ยง15. Proof of Theorem 4.5 -- ยง16. Proof of Theorems 5.1 and 5.2 -- ยง17. Natural growth conditions -- ยง18. Bibliographical notes -- VI. Harnack estimates: the casep>2 -- ยง1. Introduction -- ยง2. The intrinsic Harnack inequality -- ยง3. Local comparison functions -- ยง4. Proof of Theorem 2.1 -- ยง5. Proof of Theorem 2.2 -- ยง6. Global versus local estimates -- ยง7. Global Harnack estimates -- ยง8. Compactly supported initial data -- ยง9. Proof of Proposition 8.1 -- ยง10. Proof of Proposition 8.1 continued -- ยง11. Proof of Proposition 8.1 concluded -- ยง12. The Cauchy problem with compactly supported initial data -- ยง13. Bibliographical notes -- VII. Harnack estimates and extinction profile for singular equations -- ยง1. The Harnack inequality -- ยง2. Extinction in finite time (bounded domains) -- ยง3. Extinction in finite time (in RN) -- ยง4. An integral Harnack inequality for all 1 2) -- ยง4. Hรถlder continuity ofDu (the case 1 <p< 2) -- ยง5. Some algebraic Lemmas -- ยง6. Linear parabolic systems with constant coefficients -- ยง7. The perturbation lemma -- ยง8. Proof of Proposition 1.1-(i) -- ยง9. Proof of Proposition 1.1-(ii) -- ยง10. Proof of Proposition 1.1-(iii) -- ยง11. Proof of Proposition 1.1 concluded -- ยง12. Proof of Proposition 1.2-(i) -- ยง13. Proof of Proposition 1.2 concluded -- ยง14. General structures -- ยง15. Bibliographical notes -- X. Parabolicp-systems: boundary regularity -- ยง1. Introduction -- ยง2. Flattening the boundary -- ยง3. An iteration lemma -- ยง4. Comparing w and y (the casep> 2) -- ยง5. Estimating the local average of w| (the casep> 2) -- ยง6. Estimating the local averages of w (the casep> 2) -- ยง7. Comparing w and y (the case max $$ \left\{ {1;\tfrac{{2N}} {{N + 2}}} \right\} < p < 2 $$) -- ยง8. Estimating the local average of w| -- ยง9. Bibliographical notes -- XI. Non-negative solutions in ?T. The casep>2 -- ยง1. Introduction -- ยง2. Behaviour of non-negative solutions as | ? ? and as t ? 0 -- ยง3. Proof of (2.4) -- ยง4. Initial traces -- ยง5. Estimating u ?1 in ?T -- ยง6. Uniqueness for data inLloc1(RN) -- ยง7. Solving the Cauchy problem -- ยง8. Bibliographical notes -- XII. Non-negative solutions in ?T. The case 1 The uniqueness theorem -- ยง6. An auxiliary proposition -- ยง7. Proof of the uniqueness theorem -- ยง8. Solving the Cauchy problem -- ยง9. Compactness in the space variables -- ยง10. Compactness in thetvariable -- ยง11. More on the timeโ{128}{148}compactness -- ยง12. The limiting process -- ยง13. Bounded solutions. A counterexample -- ยง14. Bibliographical notes


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