other titles in this series - american mathematical society · 137 f. i. karpelevich and a. ya,...

Other Titles in This Series

148 Vladimi r I . Piterbarg , Asymptoti c method s i n the theor y of Gaussia n processe s an d fields, 199 6

147 S . G . Gindiki n an d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s wit h quasihomogeneous principa l part , 199 6

146 L . Ya. Adrianova , Introductio n t o linear system s o f differentia l equations , 199 5

145 A , N , Andriano v an d V. G. Zhuravlev , Modula r form s an d Heck e operators , 199 5

144 O . V . Troshkin, Nontraditiona l method s i n mathematica l hydrodynamics , 199 5

143 V . A. Malyshev an d R . A. Minlos , Linea r infinite-particle operators , 199 5

142 N . V . Krylov, Introductio n t o th e theory o f diffusion processes , 199 5

141 A . A . Davydov , Qualitativ e theor y o f control systems , 199 4

140 Aizi k I . Volpert , Vital y A . Volpert , and Vladimi r A . Volpert , Traveling wav e solutions o f paraboli c

systems, 199 4

139 I . V . Skrypnik, Method s fo r analysi s o f nonlinea r ellipti c boundar y valu e problems , 199 4

138 Yu . P . Razmyslov , Identitie s o f algebra s an d thei r representations , 199 4

137 F . I . Karpelevic h an d A . Ya , Kreinin , Heav y traffi c limit s for multiphase queues , 199 4

136 IMasayosh i Miyanishi , Algebrai c geometry , 199 4

135 Masar u Takeuchi , Moder n spherica l functions , 199 4

134 V . V. Prasolov* Problem s an d theorem s i n linea r algebra, 199 4

133 P . I. Naumkin an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n the theor y o f waves , 199 4

132 Hajim e L/rakawa , Calculus o f variation s an d harmoni c maps , 199 3

131 V . V. Sharko, Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3

130 V . V. Vershinin, Cobordisms an d spectra l sequences , 199 3

129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions , 199 3

128 V . P. Orevkov, Complexit y o f proof s an d thei r transformations i n axiomatic theories , 199 3

127 F . L. Zak , Tangent s an d secant s o f algebrai c varieties , 199 3

126 M . L . Agranovskil , Invarian t functio n space s o n homogeneou s manifold s o f Li e groups an d

applications, 199 3

125 Masayosh i Nagata , Theor y o f commutativ e fields, 199 3

124 Masahis a Adachi , Embedding s an d immersions , 199 3

123 M , A . Akivi s and B . A. Rosenfeid , Eli e Cartan (1869-1951) , 199 3

122 Zhan g Guan-Hou, Theor y o f entire an d meromorphi c functions : Deficien t an d asymptoti c value s

and singula r directions , 199 3

121 LB . Fesenk o and S. V . Vostokov, Loca l fields an d thei r extensions: A constructive approach , 199 3

120 Takeyuk i Hid a an d {vlasuyuki Hitsuda , Gaussia n processes , 199 3

119 M . V . Karasev an d V. P. Maslov, Nonlinea r Poisso n brackets . Geometry an d quantization , 199 3

118 Kenkich i lwasawa , Algebrai c functions , 199 3

117 Bori s Zilber , Uncountabl y categorica l theories , 199 3

116 G . M - Fel'dman , Arithmeti c o f probabilit y distributions , an d characterization problem s o n abelia n

groups, 199 3

115 Nikola i V . Ivanov, Subgroup s o f Teichmiille r modula r groups , 199 2

114 Seiz o ltd . Diffusio n equations , 199 2

113 Michai l Zhitomirskii , Typica l singularitie s o f differentia l 1-form s an d Pfaffia n equations , 199 2

112 5 . A . Lomov , Introductio n t o th e genera ! theor y o f singula r perturbations , 199 2

111 Simo n Gindikin , Tube domain s an d th e Cauch y problem , 199 2

110 B . V. Shabat, Introductio n t o comple x analysi s Par t II . Function s o f severa l variables , 199 2

109 Isa o Miyadera , Nonlinea r semigroups , 199 2

108 Take o Yokonuma , Tensor space s and exterio r algebra , 199 2

107 B . M . Makarov , M . G . Goluzina , A . A . Lodkin , an d A. N . Podkorytov , Selecte d problem s i n rea l analysis, 199 2

(Continued in the back of this publication)

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O.A. Ladyzenskaja V.A. Solonnikov N. N. Ural'ceva

Linear and Quasi-linear Equations of Parabolic Type

American Mathematical Society

10.1090/mmono/023

O . A . Jla^bDKeHCKa n

B . A . COJIOHHHKO B

H . H . y p a j i b i j e B a

JIHHEMHblE H KBA3MJIMHEHHbI E YPABHEHM H

nAPABOJIMHECKOrO THn A

H3^;aTejibCTBO «HayKa » TjiaBHaii PeAaKirHf l

<!>H3HKO-MaTeMaTHMecKoS JI i r repaTypb i MocKBa 196 7

Transla ted fro m th e Russia n b y S . Smi t h

2000 Mathematics Subject Classification. P r i m a r y 35 -XX .

Library o f Congres s Car d N u m b e r 68-1944 0 In te rna t iona l S t anda r d Boo k N u m b e r 978-0-8218-1573- 1

In te rna t iona l S t anda r d Seria l Numbe r 0065-928 0

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10 9 8 7 6 1 6 1 5 1 4 1 3 1 2 1 1

PREFACE

Equations o f paraboli c typ e ar e encountere d i n many branches o f mathematic s and mathematical physics , an d the form s i n whic h the y ar e investigated vary wide -ly. The equation s encountere d mos t frequentl y (an d in adjoinin g field s o f stud y almost exclusively ) ar e those o f secon d order . Suc h equation s (and certain classes of system s o f secon d order) , bot h linear and quasi-linear , mak e up the subjec t o f investigation o f th e present book . Ou r study o f thes e equation s i s concerne d main -ly wit h th e solvabilit y o f thei r boundar y valu e problem s an d with a n analysis o f the connection s betwee n th e smoothnes s o f th e solution s an d the smoothnes s o f the known functions enterin g int o th e problem.

A basic conditio n tha t i s assume d t o b e fulfille d fo r all equation s considere d is th e conditio n o f unifor m parabolicity . Fo r suc h equation s w e have manage d t o give sufficientl y complet e answer s t o centra l question s o n the solvabilit y o f th e above-indicated problem s an d to establish a serie s o f exac t dependence s o f th e properties o f th e solution s o n the propertie s o f th e know n function s i n terms o f * their mutual membershi p i n th e mos t commonly occurrin g functio n spaces .

For linear equation s th e solvabilit y o f th e basi c boundar y valu e problem s and of th e Cauch y proble m depends only o n th e smoothnes s o f th e function s definin g the problem (i . e. th e function s considere d t o be know n in th e problem, namel y th e coefficients an d the fre e term s o f th e equations , th e function s assignin g th e in* itial an d boundary condition s an d th e boundar y o f th e domai n i n which th e solu -tion exists) . The smoothe r thes e know n functions , th e bette r behave d will b e th e solution. Conversely , i f on e worsen s th e propertie s o f th e know n function s i n th e problem, the n th e differentia l propertie s o f th e solution s als o becom e worse , where the deterioration (a s woul d equally b e tru e wit h a n improvement) ha s a local char -acter (fo r example , th e smoothnes s o f th e solution s insid e thei r domain of defini -tion is determine d onl y b y th e smoothness of th e coefficients an d free term s o f the equatio n an d does no t depend o n th e smoothnes s o f th e boundar y or of th e in -itial an d boundary functions) . Bu r one canno t arbitraril y worse n th e properties o f the function s definin g th e proble m (for example , admi t in th e coefficient s singu -larities o f hig h order) . Ther e exist s a limit t o admissible deteriorations , beyon d which suc h propertie s o f th e problems a s uniquenes s ar e lost . A s i n the analysi s

v

VI PREFACE

carried out by us fo r ellipti c equation s i n the book [<> 5 q] w e begin b y determin -ing thi s limit , fo r whic h w e construc t appropriat e examples . With these example s (and examples fro m [6 5 I, n, o]) w e hav e manage d to outline wit h sufficien t accu -racy th e limit s o f a possible theor y o f boundar y valu e problem s fo r equation s wit h discontinuous and , i n general, unbounde d coefficient s an d free terms , whic h i s later presented i n Chapte r III .

As a characteristic o f suc h "bad " known functions w e hav e selecte d thei r membership in th e space s L r iQf\^ Th e solution s her e fal l int o a certain func * tion space , th e element s o f whic h hav e derivative s o f firs t orde r with respect t o x an d of orde r l /2 wit h respect t o t. W e then observe tha t th e propertie s o f thes e solutions improv e a s th e differentia l propertie s o f the function s definin g th e equa-tion or problem improve .

A qualitatively differen t situatio n hold s fo r nonlinear equations . Fo r them the smoothness o f th e solution s an d the solvabilit y "i n th e large M o f th e boundary value problem s an d of th e Cauch y proble m is determine d not only b y th e smooth-ness o f th e know n functions a- • (x, t 9 u , p) , a(x, t, u, p) makin g up the equatio n but also b y their behavior a s u an d p increase withou t limit . I n § 3 of Chapte r I we cit e a number of example s elucidatin g certai n restriction s o n thi s behavior , the nonfulfilment o f whic h implies a nonsolvability o f thes e problem s "i n th e large." An d in subsequen t chapter s (Chapter s V , VI , VII ) i t i s prove d tha t thes e restrictions, togethe r wit h a certain no t large smoothness , ar e o n the whol e als o sufficient fo r th e uniqu e solvabilit y o f th e basi c boundar y valu e problem s an d of the Cauch y problem for quasi-linea r equations .

The genera l pla n o f th e boo k i s a s follows. I n Chapter I we presen t th e basi c notation an d terminology used i n th e book , a description o f th e main results proved in it , an d a number of example s indicatin g th e exactnes s o f thes e results ; finally , we give a brief historica l survey . I n Chapter II we have assemble d proposition s that ar e use d throughou t the boo k an d describe th e properties , no t o f the solutions of an y differentia l equations , bu t of arbitrar y function s belongin g t o various func -tion space s o r classes . I t is perhap s bette r to treat thi s chapter a s a reference o n

DFor function s u(x, t) o f a spac e L r (.Qr) th e nor m

(T r_ . 1

i s finite .

PREFACE va

its differen t assertions . Th e main text begin s wit h Chapter II L I t and Chapter IV are devote d t o linear equations . I n Chapters V and VI we investigate quasi-linea r equations. Finally , i n Chapter VII we examin e linea r an d quasi-linear system s o f second orde r with commo n principal part s and give a survey of th e results o n gen-eral boundary-value problem s fo r linear parabolic systems , th e mos t general of those considere d u p to the present time . Th e main contents o f eac h chapte r ca n be understoo d independently o f th e others .

The content s o f al l th e chapters , excep t Chapte r I V and parts o f Chapter s II and VII, ar e based on th e work of 0 . A . Ladyfcenskaj a an d N. N . Ural'ceva . These chapters wer e writte n by them* Chapte r IV and §§8—1 0 of Chapte r VII were writ -ten b y V. A . Solonnikov , wh o is responsibl e fo r many of th e result s i n thi s par t of the book *

Hie author s ar e extremel y grarefu l r o Academician V . I . Smirnov for having looked ovei th e manuscrip t o f th e entir e boo k and having made a number of impor-tant critica l remark s an d suggestions- The y were taken into accoun t durin g the final revision .

The authors expres s thei r heartfelt thank s t o their colleagues an d students A* Treskunov, A . Oskolkov , M . Faddeev , I . Krol' , V . Matvee v and technician L. M. DikuSina fo r thei r help in the preparation o f th e book . A particularly larg e amoun t of quit e exper t assistanc e wa s rendere d by A. Treskunov , a graduate studen t a t Leningrad University , wh o worked with us throughou t the writing o f th e book and obtained during thi s tim e som e interestin g result s o n linea r equations (se e Bibli -ography).

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Prefatory Note t o th e Translatio n

The activ e cooperatio n o f th e Russian author s has mad e it pos-sible t o bring the presen t translation up-to-dat e an d to improve i t in several respects . Sligh t addition s an d corrections hav e bee n mad e throughout, an d som e of th e materia l ha s bee n entirel y rewritten , most notabl y Chapte r I I § 2 o n embeddin g theorems , Chapte r IV §4 o n certai n supplementar y theorems , an d Chapte r V § 6 o n solv -ability o f th e firs t boundar y problem. Th e translato r an d the edito -rial staf f wis h t o thank the Russian author s fo r their long-continue d and cordial assistance .

IX

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TABLE O F CONTENTS

Page

Preface v

Prefactory not e t o th e Translatio n i x

Chapter I . Introductor y materia l 1

§1. Basi c notatio n and terminology „ 2 §2. Classica l statement s o f th e problems . Th e maximu m principle 1 1 §3. O n admissible extension s o f th e concep t o f a solution 2 5 §4. Basi c result s an d their possible developmen t „.. . 4 2

Chapter II . Auxiliar y proposition s • 5 7

§!• Som e elementary inequalitie s 5 8 §2. The space s W l

q (Q) an d //'(J2). Embeddin g theorems 6 0 §3* Differen t functio n space s dependin g on x an d t. Embeddin g

theorems 7 4 §4. O n averagings an d cuts o f element s o f L 9(Q), L q r (QT) an d

^•°<er> • • 8 2 §5- Som e othe r auxiliary proposition s 3 9 §6. O n estimates o f max|u| . Th e clas s W>(Q T, y, r , k 9 K) 10 2 §7- The class 82((?T, Mf y, r, S, K) .„ HO §8« Th e functio n classe s B 2 «? T U r'> * ' •) an< J £ 2 « ? r U F'» • • • ) 12 2 §9. Th e functio n classe s 8^ 1 12 8

Chapter III . Linea r equations wit h discontinuou s coefficient s 13 3

§1. Statemen t o f th e problem . Generalize d solution s 13 4 §2. Th e energ y inequalit y 13 9 §3- Uniquenes s theorem s 14 5 §4. Solvabilit y an d stability o f th e firs t boundar y value proble m i n the

classes VY A{Qr) an d W\'*{Q T) 15 3 §5. O n the solvabilit y o f other boundar y value problems . Th e Cauch y

problem 16 7 §6. O n estimates i n the space JF 2

,l((?r) an d their consequence s 17 2 §7. A n estimate o f maxg r|u|. Th e maximum principle 18 1

§8. Loca l estimates o f ma x \u\ 19 1 §9- Estimate s o f som e norm s o f Orlic z fo r generalized solution s 19 4

x i

xii

§10 . A n estimat e o f Holder' s constant . Harnack' s inequalit y 20 4

§11* A n estimat e o f max^ , )^ ! an d < ("X/^X) 21 0

§12. O n th e dependenc e o f th e smoothnes s o f generalize d solution s

on th e smoothnes s o f th e dat a o f th e proble m 21 9

§13- O n diffractio n problem s 22 4

§14. Functiona l method s fo r th e solutio n o f boundar y valu e problem s 23 3

§15. Th e metho d o f continuit y i n a paramete r 23 9

§16. Rothe' s metho d an d th e metho d o f finit e difference s 24 1

§17. O n Fourier' s metho d 25 2

§18. O n th e Laplac e transfor m metho d * 25 5

Chapter IV . Linea r equation s wit h smoot h coefficient s 25 9

§ 1 . Th e hea t equatio n an d hea t potential s 26 1

§ 2 . Estimate s o f th e hea t potential s i n Holde r norm s 27 3

§ 3 - Estimate s o f th e hea t potential s i n th e norm s o f W* m*m 28 8

§ 4 - Domains . Som e auxiliar y proposition s • • 29 4

§ 5 - Formulatio n o f basi c result s o n th e solvabilit y o f th e Cauch y

problem an d boundar y valu e problem s fo r equation s wit h variabl e coefficients i n Holde r functio n c la s se s • 31 7

§ 6 . Mode l problem s i n a hal f spac e 32 3 § 7 . O n th e solvabilit y o f proble m (5.4' ) , 32 8

§ 8 . O n th e solvabilit y o f proble m (5.4 ) 33 8

§ 9 . Th e firs t boundar y valu e proble m i n c l a s s e s W* ,l(QT) 34 1

§10 , Loca l estimate s o f th e solution s o f problem s (5-3 ) an d (5.4 ) 35 1

§11 . A fundamenta l solutio n o f th e paraboli c equatio n o f secon d orde r ... . 35 6

§12 . Som e auxiliar y inequalitie s fo r th e functio n Q 36 4

§13 . Estimate s o f th e fundamenta l solutio n 37< >

§14 . Solutio n o f th e Cauch y proble m 38 9

§15- Th e single-laye r potentia l 39 5

§16 . Solutio n o f th e firs t boundar y valu e proble m 40 6

§17. O n th e estimate s o f S . N . Bernstei n 41 4

Chapter V . Quasi-linea r equation s wit h principa l par t i n divergenc e for m 41 7

§ 1 . Bounde d generalize d solutions . Holde r continuit y 41 8

§ 2 . O n th e boundednes s o f generalize d solution s 42 3

§ 3 . Estimate s o f max^ , \u %\ an d (u^ty 43 0

§ 4 . A n estimat e o f raax|u x| i n th e whol e domai n 43 8

xiii § 5 - Estimat e o f y ^ / ( 0 ? an d highe r derivative s i n a n arbitrar y sub -

domain o f th e domai n Q T ... » 44 4

§ 6 . Th e solvabilit y o f th e firs t boundar y valu e proble m 44 9

§ 7 , Othe r boundar y valu e problem s 47 5

§ 8 , Th e Cauch y proble m 49 2

§ 9 - O n th e Stefa n proble m 49 6

§10 . Anothe r metho d o f estimatin g th e Holde r constan t fo r solution s ....... . 50 3

Chapter VI . Quasi-linea r equation s o f genera l for m 5^ 5

§ 1 . A proo f o f th e smoothnes s o f generalize d solution s o f c las s 3 t

and a n estimat e o f (px/ffi 51 6

§ 2 . A n estimat e o f (u^) W 52 4

§ 3 * Th e estimatio n o f maxlu^ l 53 3

§ 4 . Existenc e theorem s 55 ^

§ 5 . Equation s wit h on e spac e variabl e 56 O

Chapter VII . System s o f linea r an d quasi-linea r equation s 57 1

§ 1 . Generalize d solution s o f linea r system s 57 1

§ 2 . O n th e boundednes s o f max ^ |u | 57 4

§ 3 . A n estimat e o f |u|fe a>. 57 9

§ 4 . O n estimate s o f | u x | ^ an d o f othe r highe r norm s o f th e solution s 58 3

§ 5 - Quasi-linea r paraboli c systems . Estimate s o f th e norm s

M Q I * * * * > ! » i n term s o f max ^ |u , u x\ 58 5

§ 6 . A n estimat e o f max ^ (u^ l 58 8

§ 7 . A n existenc e theore m fo r quasi-linea r system s 59 6

§ 8 . Linea r paraboli c system s o f genera l for m 59 7

§ 9 - Statemen t o f th e boundar y valu e problem s an d th e Cauch y prob -lem fo r paraboli c system s 60 4

§10 . Basi c result s o n th e solvabilit y o f th e Cauch y proble m an d o f the genera l boundar y valu e problem s fo r paraboli c system s 61 5

Bibliography 631

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[ l ] S . Agmon , A . Doughs an d L . Nirenberg , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , Comm . Pur e Appl . Math , 12(1959) , 623-727 ; Russia n trans!. , IL, Moscow , 1962 . M R 23 #A2610 .

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Other Titles in This Series , Jr ue , u u , (Continued from the front of this publication)

106 G.-« C Wen, Conformal mapping s and boundary valu e problems, 1992 105 D . R. Yafaev, Mathematical scatterin g theory: Genera l theory , 1992 104 R . L. Dobrushin, R« Koteck^, and S. Shiosman, Wulff construction: A global shape from loca l

interaction, 1992 103 A . K. Tsikh, Multidimensional residue s and thei r applications , 1992 102 A . M, Il'in, Matching of asymptotic expansions of solutions of boundary valu e problems, 1992 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Dong Zhen-xi, Qualitative

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(revised edition, 1994 ) 97 Itir o Tamura, Topology of foliations: A n introduction, 199 2 96 A . I. Markushevich, Introduction t o the classical theory o f Abelian functions , 199 2 95 Guangchan g Dong, Nonlinear partia l differentia l equation s of second order, 1991 94 Yu . S. IK'yashenko, Finiteness theorems for limi t cycles, 1991 93 A . T. Fomenko and A. A. Tuzhilin, Elements of the geometry and topolog y of minimal surfaces in

three-dimensional space , 1991 92 E . M« Nikishln and V. N. Sorokin, Rational approximations an d orthogonality , 199 1 91 Mamor u Mimura and Hirosi Toda, Topology of Li e groups, I and II , 1991 90 S . L. Sobolev, Some applications of functional analysi s in mathematical physics, third edition, 1991 89 Valeri l V. Kozlov and Dmitril V. Treshchev, Billiards: A genetic introduction t o the dynamics of

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problem, 1991 83 N . I. Pbrtenko, Generalized diffusio n processes , 1990 82 Yasutak a Sibuya, Linear differentia l equation s in the complex domain: Problem s of analytic

continuation, 199 0 81 LM . Geifan d and S. G. Gindikin, Editors, Mathematical problem s of tomography, 199 0 80 Junjir o Noguchi and Takushiro Ochiai, Geometric function theor y i n several complex variables,

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(See the AMS catalog for earlie r titles)