Degenerate Parabolic Equations [Paperback]

Evolved from the author's lectures at the University of Bonn's Institut f?r angewandte Mathematik, this book reviews recent progress toward understanding of the local structure of solutions of degenerate and singular parabolic partial differential equations.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. H?lder 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. H?lder 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 strl3!