Nonlinear Partial Differential Equations Of Second Order
NonlinearPartial Differentia l Equation sof Second Orde r
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10.1090/mmono/095Translations o fMATHEMATICALMONOGRAPHSVolume 9 5NonlinearPartial Differentia l Equation sof Second Orde rGuangchang Don gAmerican Mathematica l Societ yProvidence, Rhod e Islan d
**»#«tt BHa«##8m m R S %tTranslated fro m t h e Chines e b y K a i Sen g C h o u [Raisin g Tso ]2000 Mathematics SubjectClassification.Primary 35Kxx ; Secondar y 3 5 J x x , 35Lxx .ABSTRACT. Thi s i s a treatis e o n nonlinea r partia l differentia l equation s o f secon d order . Wit h th eexception o f the firs t chapter , al l t he remainin g chapter s ar e based o n th e publishe d o r unpublishe dwork o f th e author .A prior i estimatio n i s the mai n them e o f this book . Emphasi s i s placed o n ellipti c an d paraboli cequations. Nevertheless , som e hyperboli c equation s ar e als o discussed . Eac h chapte r o f thi s boo khas it s ow n physica l background .This boo k ca n b e use d a s a tex t fo r graduat e student s i n mathematics , o r a s a referenc e fo rresearchers, teachers , o r universit y seniors .Library o f Congres s Cataloging-in-Publicatio n D a t aTung, Kuang-ch'ang .[Fei hsie n hsin g er h chie h p'ie n we i fe n fan g ch'eng . English ]Nonlinear partia l differentia l equation s o f secon d order/Guangchan g Dong ; [translate d fro mthe Chines e b y Ka i Sen g Cho u (Raisin g Tso)] .p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-9282 ; v . 95 )Translation of : Fe i hsie n hsin g er h chie h p'ie n we i fe n fan g ch'eng .Original Chines e titl e o n vers o o f t.p. : Fe i hsie n hsin g er h chie h p'ie n we i fe n fan g ch'eng .Includes bibliographica l references .ISBN 0-821 8-4554- 3 (alk . paper )1. Differentia l equations . Partial . 2 . Differentia l equations . Nonlinear . I . Chou , Ka i Seng .II. Title . III . Title : Fe i hsie n hsin g er h chie h p'ie n we i fe n fan g ch'eng . IV . Series .QA377.T86131 99 1 9 -27853515'.353—dc20 CIPAMS softcover ISB N 978-0-8218-4685-8Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie sacting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us ein teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i nreviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n this publicatio nis permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc hpermission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society ,201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b ye-mail t o reprint-permission@ams.org . 1 99 1 by th e America n Mathematica l Society . Al l right s reserved .Reprinted b y th e America n Mathematica l Society , 2008 .Translation authorize d b y th e Qinghu a Universit y Pres s t o b e publishe dexclusively b y th e America n Mathematica l Society .The America n Mathematica l Societ y retain s al l right sexcept thos e grante d t o th e Unite d State s Government .Printed i n th e Unite d State s o f America .@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline sestablished t o ensur e permanenc e an d durability .This publicatio n wa s typese t usin g A/ S-T ,the America n Mathematica l Society' s T j K macr o system .Visit th e AM S hom e pag e a t http://www.ams.org /10 9 8 7 6 5 4 3 2113 1 2 1 1 1 0 09 0 8
ContentsIntroductionNotes for the English Translation 3CHAPTER I. The First Boundary Valu e Problem for Second-Orde rQuasilinear Parabolic Equations with Principal Par t inDivergence Form 7§ 1. Uniform an d Holder estimates fo r the solution 71§2. A uniform bound for D xu1§3. A Holder estimate for D xu§4. Existence an d uniqueness of th e solution fo r the first boundaryvalue problem 2073CHAPTER II . A Periodic Boundary Value Problem fo r a NonlinearTelegraph Equation 3 1§ 1. Solvability fo r higher-dimensional telegrap h equations in thenonresonance cas e 32§2. A discussion on the resonance cas e 36§3. Regularity of a generalized solutio n 39CHAPTER III. The Initial Value Problem for a Nonlinear Schrodinge rEquation 4§1. Background materials 4§2. The initial value problem for the linear Schrodinger equation 4§3. The initial value problem for a nonlinear Schrodinger equation 43358CHAPTER IV . Multi-Dimensional Subsoni c Flow s Around anObstacle 59§1. Introduction 59§2. Background material fo r the linear problem 6 1§3. Solution t o the auxiliary problem 64§4. Resolution o f the problem o f a flow passing an obstacle andelementary propertie s of the solution 73§5. Further properties of the solution 74
vi CONTENTSCHAPTER V . The Initial-Boundary Valu e Problem for Degenerat eQuasilinear Parabolic Equations 77§ 1. Formulation o f the problem and a Holder estimate for thesolution 78§2. Solvability fo r the first boundary value problem 981§3. Uniqueness o f the solution 0 11§4. The initial value problem 07CHAPTER VI. The Speed of Propagation of th e Solution o f aDegenerate Quasilinea r Parabolic1Equation 0§1. An estimate on the domain1of dependenc e 0§2. A lower11estimate for the solution997CHAPTER VII . Aleksandrov an d Bony Maximum Principle s fo r1Parabolic Equations 2 1§1. Introduction 2 1§2. Some properties1of conve x function s 25§3. Convex envelopes 35§4. Several Aleksandrov1maximu m principles 391§5. Bony maximum principles 45CHAPTER VIII . The Density Theore m and Its Applications1 49§1. The statement1o f th e density theorem 49§2. Several lemmas and the proof of the density1theorem 5 1§3. The Harnack inequality fo r parabolic equations wit h1measurable coefficient s 58§4. A Holder estimate fo r the solution of a quasilinear parabolicequation 59§5. A Holder estimate fo r the solution for a quasilinear paraboli csystem 62CHAPTER IX. Full y Nonlinear Parabolic1Equations 69§ 1. A uniform boun d for a solution u an d an interior estimate fo rDxu 70§2. An interior estimate fo r the second1derivatives 74§3. An interior Holder estimate fo r the second derivatives1 8 1§4. A near boundary1Holde r estimate fo r u 84§5. Uniform an d Holder estimates for D xu nea r the boundary1 89§6. Near boundary unifor m an d Holder estimates fo r the secon dderivatives 98§7. Uniqueness an d existence o f a solution fo r the first boundaryvalue problem under the natural structure conditions 200
CONTENTSvnCHAPTER X. Full y Nonlinear Parabolic Equations (Continued ) 20§ 1. The density theorem for quasilinear parabolic equation s withnatural structure condition o f the second kind 20§2. A Holder estimate fo r the solution and unique solvabilit y fo rthe first boundary value problem 22§3. Certain apriori estimates for the solution s of a fully nonlinea rparabolic equation with natural structur e condition o f thesecond kind and the unique solvability o f th e first boundaryvalue problem 2377Symbols 245References24778
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SymbolsQ, usuall y a bounded (connected ) ope n setQ th e closure o f Q3Q th e boundary o f QCC AccBmean s A c BQ' Q' cc Qrf(x, 4) th e distanc e betwee n x an d th e se t A, i.e. , d(.x, t ) infyeAxyQp npQCl an d rf(Q p, dCl) pdx dx d(x,dCl)dxy dxy niin{rf JC, }G usuall y the cylinder 1 x (0 , T]d*Q th e paraboli c boundar y o f Q, i.e. , d*Q Q x { r 0 } U dQ,x[0,T]Q' Q'QQan d d(Q'9d*Q) 0Qp QpPQQ an d d{Q p9d'Q) dp the distance between the point J P (x , 0 an d d*Qdp,p2 dK{p), B aBp(x ) akR(x ) aplp2 minidPrdP2}ball with radius pball with radius p , centere d at x , i.e. , {* : J C — JC p}square with length 2R , centere d at x (JC ,. , JC ) , i.e.,{A:: \x t-x \ R,\ i n}KR(x ,t ) kR(0)x(0,T)\A\ th e measure of th e set A , o r \A\ meas/ 1a.e. almost everywher e0 empty setdu/dN Ni s th e norma l a t th e boundary , du/dN i n genera l i sthe norma l derivativ e o f u alon g th e boundary ; usuall y w etake N t o be the inner one.con th e area of th e unit spher e in R nKm th e volume o f the unit ball in R n245
SYMBOLS246pl/pll«ll„ \\u\\give np (J a\u\ dx) , Qk aC* u u{x l9.,xn) is continuousl y differentiabl e u p t oA:th order and its kth derivatives satisfy a Holder condition withexponent a ( 0 a 1 )C2'1 u u(x {, . . , xn, t) twic e continuously differentiable inx{, . . , xn an d continuously differentiable i n tC 2 A,i a/2 M( 1 / , J w ) an d u t ar e Holde r continuou s i n xwith exponen t A and Holder continuou s i n t wit h exponen tA/2GlML,fl wC a (Q) ( 0 a 1 ) , w a fi i s the Holder constant\u\a[Q ueC a a/2(Q)9 Q Q x ( 0, 7 1 andW (fl) The functio n spac e consisting o f al l function s i n Q whos eweak derivatives up to fcth order belong to L P(Q)Wkp(Q) Th e closur e (unde r th e W (Q)-norm ) o f function s i nWk(Q) whic h vanis h nea r dQ. I n othe r words , i t i s theclosure of C (Q ) unde r (Q)-nor mk k/2W ' (Q) Q & x (0 , T], th e function spac e consistin g o f al l func tions i n Q whos e wea k derivative s i n x u p t o kth orde rand weak derivatives in t u p to k/2 orde r (k i s even) belon gto L P(Q)1a"" a max{a , 0}a" oT - min{a , 0}P
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248REFERENCES20. B . Ou, On "Subsonic flowsinspace and its higher dimensional generalization." Preprint.(Chinese)21. ,Existence, uniqueness and continuous dependence for subsonic flows with "source"in space. Preprint. (Chinese )22. Carl o Miranda , Equazioni alle derivate parziali di tipo ellittico, Ergebniss e de r Mathematik un d ihre r Grenzgebiete , Hef t 2 , Springer-Verlag , Berlin , 1 955 ; English transl , Springer Verlag, Berlin, 1 970 .23. J . Bear , Dynamics of fluids in porous media, American Elsevier , New York, 1 972 .24. B . H. Gilding, A nonlinear degenerate parabolic equation, Ann. Scuol a Norm . Sup . Pisa,CI. Sci. (4) 4 (1977), no. 3, 393-432.25. Y . Z . Chen , Uniqueness of the generalized solution of quasilinear degenerate parabolicequations, Proceedings o f th e 1 98 2 Changchun Symposiu m o n differentia l geometr y an d differ ential equations, Science Press, 317-332.26. G . C . Dong, A higher dimensional nonlinear degenerate parabolic equation, Proceedingsof the 1 98 2 Changchun Symposiu m o n differential geometr y and differential equations , SciencePress, 373-382 .27. Y . Z. Chen, Holder estimates for solutions of uniformly degenerate quasilinear parabolicequations, Chinese Ann. Math. Ser . B, 5 (1984), no. 4, 661-678.28. G . C . Dong , The first boundary value problems for solutions of degenerate quasilinearparabolic equations. Preprint .29. Y . Z. Chen, On finite diffusing speed for uniformly degenerate quasilinear parabolic equations. Preprint.30. B . J . Bian , On the finite propagation for solutions of second order degenerate parabolicequations. Preprint. (Chinese )31. A . D . Aleksandrov , Studies on the maximum principle, Izv . Vyssh . Uchebn . Zaved .Matematika 1 961 , no. 1 , 3-20. (Russian )32. C . Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pur a Appl. (4 ) 7 4(1966), 1 5-30 .33. N . V. Krylov, Sequences of convexfunctions, and estimates of the maximum of the solutionof a parabolic equation, Sibirsk. Mat. Zh. 1 7 (1976), no. 2, 290-303; English transl . in Siberia nMath. J. 1 7(1 976) , no. 2.34. Y . Z . Chen , Aleksandrov and Bony maximum principles for parabolic equations. Preprint. (Chinese )35. A . D. Aleksandrov, Intrinsic geometry of convex surfaces, OGIZ, Moscow, 1 948 ; Germantransl., Akademie-Verlag, Berlin , 1 955 .36. S . Y . Chen g an d S . T . Yau , On the regularity of the Monge-Ampere equationdet u/dxJdXj) F(x, u), Comm . Pure Appl. Math. 30 (1977), no. 1 , 41-68.37. J . M. Bony , Principe du maximum dans les espaces de Sobolev, C. R . Acad . Sci . Paris,Ser. A-B 265 (1 967) , A333-A336.38. P.-L . Lions , A remark on Bony maximum principle, Proc. Amer. Math . Soc . 88 (1983),no. 3 , 503-508.39. N . V . Krylo v an d M. V. Safonov , A certain property of solutions of parabolic equationswith measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1 , 161-175; Englishtransl. in Math. USSR-Izv. 1 6 (1981), no. 1 , 1 51 -1 64 .40. N . S . Trudinger , Local estimates for subsolutions and supersolutions of general secondorder elliptic quasilinear equations, Invent . Math. 61 (1980), no. 1 , 67-79.41. G . C . Dong, Local estimates for subsolutions and supersolutions of general second orderparabolic quasilinear equations. Preprint .42. N . V. Krylov, On estimates for the derivatives of solutionsof nonlinear parabolic equations,Dokl. Akad . Nau k SSS R 27 4 (1 984) , no . 1 , 23-26 ; Englis h transl . i n Sovie t Math . Dokl . 2 9(1984), no. 1 , 14-17.43. Y . Z . Chen, On Krylov's a priori estimates for fully nonlinear equations, Adv. Math . 1 5(1986), 63-101 . (Chinese )44. N . V . Krylov , Boundedly nonhomogeneous elliptic and parabolic equations, Izv. Akad.Nauk SSS R Ser . Mat . 4 6 (1 982) , no. 3 , 487-523; English transl. , Math . USS R Izv . 2 0 (1983),no. 3 , 459-492.
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EpilogueOur discussion o n quasilinea r an d full y nonlinea r equation s i s fa r fro mcomplete. Man y importan t aspects , suc h a s the obliqu e derivativ e proble mfor full y nonlinea r equations , a s wel l a s th e application s o f full y nonlinea requations, ar e left out . Hopefull y the y wil l b e covere d i n a future publica tion.251
Nonlinear partial differential equations of second order/Guangchang Dong; [translated from the Chinese by Kai Seng Chou (Raising Tso)]. . Dong and S. Li, A boundary value problem for nonlinear telegraph equations, Non-linear Anal. 5 (1981), no. 7, 705-711. 8. H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators
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