退化拋物方程

內(nèi)容概要

數(shù)學(xué)真正意義上研究退化和奇異拋物偏微分方程是近些年才開始的，起源于60年代中葉DeGiorgi，Moser，Ladyzenskajia和Ural’tzeva這些人的工作。本書是近些年來該領(lǐng)域的進(jìn)展的綜述。其基本思想來自上個世紀(jì)90年代作者在波恩大學(xué)的Lipschitz講義。目次：函數(shù)空間；弱解和局部能量估計；退化拋物方程的Holder連續(xù)性；奇異拋物方程解的Holder連續(xù)性；弱解有界性；Harnack 估計：p]2；Harnack 估計和；退化和奇異拋物系統(tǒng)；拋物p系統(tǒng)：Du的Holder連續(xù)性；拋物p系統(tǒng)：邊界奇異；ΣT中的非負(fù)解：p]2； ΣT中的非負(fù)解：1[p[2。    本書適用于數(shù)學(xué)專業(yè)的研究生和科研人員。

書籍目錄

Preface1. Elliptic equations： Harnack estimates and Holder continuity ...2. Parabolic equations： Hamack estimates and holder continuity ..3. Parabolic equations and systems4. Main resultsI. Notation and function spaces1. Some notation2. Basic facts about3. Parabolic spaces and embeddings4. Auxiliary lemmas5. Bibliographical notesII. Weak solutions and local energy estimates1. Quasilinear degenerate or singular equations2. Boundary value problems3. Local integral inequalities4. Energy estimates near the boundary5. Restricted structures： the levels k and the constant 76. Bibliographical notesIII. Holder continuity of solutions of degenerate parabolic equations1. The regularity theorem2. Preliminaries3. The main proposition4. The first alternative5. The first alternative continued6. The first alternative concluded7. The second alternative8. The second alternative continued9. The second alternative concluded10. Proof of Proposition 3.111. Regularity up to t = 012. Regularity up to ST. Dirichlet data13. Regularity at ST. Variational data14. Remarks on stability15. Bibliographical notesIV. Holder continuity of solutions of singular parabolic equations1. Singular equations and the regularity theorems2. The main proposition3. Preliminaries4. Rescaled iterations5. The first alternative6. Proof of Lemma 5.1. Integral inequalities7. An auxiliary proposition8. Proof of Proposition 7.1 when （7.6） holds9. Removing the assumption （6.1）10. The second alternative11. The second alternative concluded12. Proof of the main proposition13. Boundary regularity14. Miscellaneous remarks15. Bibliographical notesV. Boundedness of weak solutions1. Introduction2. Quasilinear parabolic equations3. Sup-bounds4. Homogeneous structures. The degenerate case 1 p > 25. Homogeneous structures. The singular case 1 < p < 26. Energy estimates7. Local iterative inequalities8. Local iterative inequalities9. Global iterative inequalities10. Homogeneous structures and 111. Proof of Theorems 3.1 and 3.212. Proof of Theorem 4.113. Proof of Theorem 4.2..14. Proof of Theorem 4.315. Proof of Theorem 4.516. Proof of Theorems 5.1 and 5.217. Natural growth conditions18. Bibliographical notesVI. Harnack estimates： the case p>2l. Introduction2. The intrinsic Hamack inequality3. Local comparison functions4. Proof of Theorem 2.15. Proof of Theorem 2.26. Global versus local estimates7. Global Hamack estimates8. Compactly supported initial data9. Proof of Proposition 8.110. Proof of Proposition 8.1 continued11. Proof of Proposition 8. i concluded12. The Cauchy problem with compactly supported initial data13. Bibliographical notesVII. Hamack estimates and extinction profile for singular equations1. The Harnack inequality2. Extinction in finite time （bounded domains）3. Extinction in finite time （in RN）4. An integral Hamack inequality for all5. Sup-estimates for6. Local subsolution.7. Time expansion of positivity8. Space-time configurations9. Proof of the Hamack inequality10. Proof of Theorem 1.211. Bibliographical notesVIII. Degenerate and singular parabolic systems1. Introduction2. Boundedness of weak solutions3. Weak differentiability of Du and energy estimates for IOul4. Boundedness of lOut. Qualitative estimates5. Quantitative sup-bounds of6. General structures7. Bibliographical notesIX. Parabolic p-systems： Hiolder continuity of Du1. The main theorem2. Estimating the oscillation of Du3. Hlder continuity of Du （the case p > 2 ）4. HOlder continuity of Du （the case 1 < p < 2 ）5. Some algebraic Lemmas6. Linear parabolic systems with constant coefficients7. The perturbation lemma8. Proof of Proposition l.1-（i）9. Proof of Proposition 1.l-（ii）10. Proof of Proposition 1.1-（iii）11. Proof of Proposition 1.1 concluded12. Proof of Proposition13. Proof of Proposition 1.2 concluded14. General structures15. Bibliographical notesX. Parabolic p-systems： boundary regularity1. Introduction2. Flattening the boundary3. An iteration lemma4. Comparing w and v （the case p > 2）5. Estimating the local average of IDwl （the case p > 2 ）6. Estimating the local averages of w （the case p > 2 ）7. Comparing w and v （the case max 1

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