Linear and Quasi-linear Equations of Parabolic Type

Citation preview

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Other Titles in This Series 148 Vladimi r I . Piterbarg , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 1 99 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 , 1 99 6 146 L . Ya. Adrianova , Introductio n t o linea r system s o f differentia l equations , 1 99 5 145 A , N , Andriano v an d V. G. Zhuravlev , Modula r form s an d Heck e operators , 1 99 5 144 O . V . Troshkin, Nontraditiona l method s i n mathematica l hydrodynamics , 1 99 5 143 V . A. Malyshe v an d R . A . Minlos , Linea r infinite-particl e operators , 1 99 5 142 N . V . Krylov, Introductio n t o th e theor y o f diffusio n processes , 1 99 5 141 A . A . Davydov , Qualitativ e theor y o f contro l systems , 1 99 4 140 Aizi k I . Volpert , Vital y A . Volpert , and Vladimi r A . Volpert , Travelin g wav e solution s o f paraboli c systems, 1 99 4 139 I . V . Skrypnik, Method s fo r analysi s o f nonlinea r ellipti c boundar y valu e problems , 1 99 4 138 Yu . P . Razmyslov , Identitie s o f algebra s an d thei r representations , 1 99 4 137 F . I. Karpelevic h an d A . Ya , Kreinin , Heav y traffi c limit s fo r multiphas e queues , 1 99 4 136 IMasayosh i Miyanishi , Algebrai c geometry , 1 99 4 135 Masar u Takeuchi , Moder n spherica l functions , 1 99 4 134 V . V. Prasolov* Problem s an d theorem s i n linea r algebra, 1 99 4 133 P . I. Naumki n an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n th e theor y o f waves , 1 99 4 132 Hajim e L/rakawa , Calculu s o f variation s an d harmoni c maps , 1 99 3 131 V . V. Sharko, Function s o n manifolds : Algebrai c an d topologica l aspects , 1 99 3 130 V . V. Vershinin, Cobordism s an d spectra l sequences , 1 99 3 129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions , 1 99 3 128 V . P. Orevkov, Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 1 99 3 127 F . L . Zak , Tangent s an d secant s o f algebrai c varieties , 1 99 3 126 M . L . Agranovskil , Invarian t functio n space s o n homogeneou s manifold s o f Li e group s an d applications, 1 99 3 125 Masayosh i Nagata , Theor y o f commutativ e fields, 1 99 3 124 Masahis a Adachi , Embedding s an d immersions , 1 99 3 123 M , A . Akivi s an d B . A. Rosenfeid , Eli e Cartan (1 869-1 951 ) , 1 99 3 122 Zhan g Guan-Hou, Theor y o f entir e an d meromorphi c functions : Deficien t an d asymptoti c value s and singula r directions , 1 99 3 121 LB . Fesenk o an d S. V . Vostokov, Loca l fields an d thei r extensions: A constructiv e approach , 1 99 3 120 Takeyuk i Hid a an d {vlasuyuk i Hitsuda , Gaussia n processes , 1 99 3 119 M . V . Karasev an d V . P. Maslov, Nonlinea r Poisso n brackets . Geometr y an d quantization , 1 99 3 118 Kenkich i lwasawa , Algebrai c functions , 1 99 3 117 Bori s Zilber , Uncountabl y categorica l theories , 1 99 3 116 G . M - Fel'dman , Arithmeti c o f probabilit y distributions , an d characterization problem s o n abelia n groups, 1 99 3 115 Nikola i V . Ivanov, Subgroup s o f Teichmiille r modula r groups , 1 99 2 114 Seiz o ltd . Diffusio n equations , 1 99 2 113 Michai l Zhitomirskii , Typica l singularitie s o f differentia l 1 -form s an d Pfaffia n equations , 1 99 2 112 5 . A . Lomov , Introductio n t o th e genera ! theor y o f singula r perturbations , 1 99 2 111 Simo n Gindikin , Tub e domain s an d th e Cauch y problem , 1 99 2 110 B . V. Shabat, Introductio n t o comple x analysi s Par t II . Function s o f severa l variables , 1 99 2 109 Isa o Miyadera , Nonlinea r semigroups , 1 99 2 108 Take o Yokonuma , Tenso r space s an d exterio r algebra , 1 99 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, 1 99 2 (Continued in the back of this publication) Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

This page intentionally left blank

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

10.1090/mmono/023

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

Linear and Quasi-linear Equations of Parabolic Type

American Mathematical Society

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

O . A . Jla^bDKeHCKa n B . A . COJIOHHHKO B H. H . ypajibijeB a

JIHHEMHblE H KBA3MJIMHEHHbI E YPABHEHM H nAPABOJIMHECKOrO THn A H3^;aTejibCTBO « H a y K a » TjiaBHaii PeAaKirHf l H3HKO-MaTeMaTHMecKoS J I i r r e p a T y p b i MocKBa 1 96 7 T r a n s l a t e d fro m t h e Russia n b y S . S m i t h 2000 Mathematics Subject

Classification.

Primar

y 35-XX .

L i b r a r y o f Congres s C a r d N u m b e r 68-1 944 0 I n t e r n a t i o n a l S t a n d a r d Boo k N u m b e r 978-0-821 8-1 573- 1 I n t e r n a t i o n a l S t a n d a r d Seria l N u m b e r 0065-928 0

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting 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 e in 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 n reviews, 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 an y materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission 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 02904-229 4 USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © Copyrigh t 1 96 8 b y th e America n Mathematica l Society . Al l right s reserved . Reprinted wit h correction s 1 98 8 Printed i n th e Unite d State s o f America . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . @ 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 s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 71 6

61 51 41 31 21 1

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

PREFACE Equations o f paraboli c typ e ar e encountere d i n man y branches o f mathematic s and mathematical physics , an d the form s i n whic h the y ar e investigate d var y 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 linea r an d quasi-linear , mak e u p the subjec t o f investigation o f th e presen t 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 t o 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 mutua l membershi p i n th e mos t commonl y 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 depend s only o n th e smoothnes s o f th e function s definin g the proble m (i . e. th e function s considere d t o b e know n i n th e problem , namel y th e coefficients an d th e fre e term s o f th e equations , th e function s assignin g th e in * itial an d boundar y condition s an d th e boundar y o f th e domai n i n whic h 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 deterioratio n (a s woul d equall y 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 domai n of defini tion is determine d onl y b y th e smoothnes s of th e coefficients an d fre e term s o f the equatio n an d doe s no t depen d 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 i n th e coefficient s singu larities o f hig h order) . Ther e exist s a limit t o admissibl e deteriorations , beyon d which suc h propertie s o f th e problem s a s uniquenes s ar e lost . A s i n th e analysi s v

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

PREFACE

VI

carried ou t b y us fo r ellipti c equation s i n th e boo k [ 5 q] w e begi n b y determin ing thi s limit , fo r whic h w e construc t appropriat e examples . With these example s (and example s fro m [6 5 I, n, o]) w e hav e manage d t o outline wit h sufficien t accu racy th e limit s o f a possibl e 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 fre e terms , whic h i s later presente d i n Chapte r III . As a characteristic o f suc h "bad " known function s 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 respec t t o x an d of orde r l /2 wit h respec t 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 th e 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 th e smoothness o f th e solution s an d th e solvabilit y "i n th e large M o f th e boundar y value problem s an d of th e Cauch y proble m i s determine d no t only b y th e smooth ness o f th e know n function s a- • (x, t 9 u , p) , a(x, t, u, p) makin g u p the equatio n but als o b y thei r behavio r 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 nonfulfilmen t o f whic h implie s 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 th e 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 proble m fo r 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 terminolog y use d i n th e book , a descriptio n 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 giv e a brief historica l survey . I n Chapter I I we have assemble d proposition s that ar e use d throughou t th e boo k an d describe th e properties , no t o f the solutions of an y differentia l equations , bu t o f arbitrar y function s belongin g t o variou s func tion space s o r classes . I t is perhap s bette r t o treat thi s chapter a s a reference o n DFor function s u(x, t) o f a s p a c e L

(T r_

r

(.Qr) th e nor m

. 1

i s finite .

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

PREFACE

va

its differen t assertions . Th e mai n tex t begin s wit h Chapte r II L I t and Chapter IV are devote d t o linear equations . I n Chapters V and VI we investigat e quasi-linea r equations. Finally , i n Chapte r VII we examin e linea r an d quasi-linear system s o f second orde r with commo n principal part s an d give a surve y o f th e result s o n general boundary-valu e problem s fo r linear paraboli c systems , th e mos t general o f those considere d u p to th e present time . Th e main content s o f eac h chapte r ca n be understoo d independentl y 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 wor k 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 o f 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. Smirno v for havin g looked ovei th e manuscrip t o f th e entir e boo k an d having mad e a number of impor tant critica l remark s an d suggestions- The y wer e take n int o accoun t durin g th e final revision . The author s expres s thei r heartfelt thank s t o thei r colleague s an d student s A* Treskunov, A . Oskolkov , M . Faddeev , I . Krol' , V . Matvee v and technician L. M. DikuSina fo r thei r help i n th e preparatio n 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 u s throughou t th e writin g o f th e boo k an d obtained durin g thi s tim e som e interestin g result s o n linea r equation s (se e Bibli ography).

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

This page intentionally left blank

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Prefatory Note t o th e Translatio n The activ e cooperatio n o f th e Russia n author s has mad e it pos sible t o bring the presen t translatio n up-to-dat e an d to improve i t i n several respects . Sligh t addition s an d corrections hav e bee n mad e throughout, an d som e o f 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 th e edito rial staf f wis h t o thank th e Russian author s fo r their long-continue d and cordial assistance .

IX

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

This page intentionally left blank

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

TABLE O F CONTENT S Page Preface v Prefactory not e t o th e Translatio n i

x

Chapter I . Introductor y materia l §1. Basi c notatio n an d 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 solutio n 2 5 §4. Basi c result s an d their possibl e developmen t „.. .42 Chapter II. Auxiliar y proposition s •

5

7

§!• Som e elementar y inequalitie s 5 §2. The space s W lq (Q ) an d //'(J2). Embeddin g theorem s 6 §3* Differen t functio n space s dependin g o n x an d t. Embeddin g theorems 7 §4. O n averagings an d cuts o f element s o f L 9(Q), L q r (QT) an d

^•° •

•8

§5- Som e othe r auxiliary proposition s 3 §6. O n estimates o f max|u| . Th e clas s W>(Q 1 T, y, r , k 9 K) 0 §7- The class 8 2 ((? T , Mf y, r, S, K)

.„

§8« Th e functio n classe s B 2 « ? T U r'> * ' •) an< 1 §9. Th e functio n classe s 8^ 1 2

J

£

2«?r

8 0 4

2 9 2 HO

U F'» • 1 ••) 2

Chapter III . Linea r equation s wit h discontinuou 1 s coefficient s 3

2 8 3

§1. Statemen t o f th e problem . Generalize 1 d solution s 3 4 §2. Th e energ y inequalit y 3 9 §3- Uniquenes s theorem s 4 5 §4. Solvabilit y an d stability o f th e firs t boundar y valu e proble m i n the classes 1 VY A{Qr) an d W\'*{Q T) 5 3 §5. O n the solvabilit y o f othe r boundar y valu e problems . Th e Cauch y problem 6 7 §6. O n estimates i n th e spac e JF 2 ,l ((? r ) an d thei r consequence 1 s 7 2 §7. A n estimate o f maxg r |u|. Th e maximu 1 m principle 8 1 1 §8. Loca l estimates o f ma x \u\ 9 1 §9- Estimate s o f som e norm s o f Orlic z fo r generalized solution 1 s 9 4 xi

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

xii § 1 0 . A n estimat e o f Holder' s constant . Harnack' s inequalit y 20

4

§11* A n estimat e o f1 m a x ^ , ) ^ ! an d < (" X /^ X) 2

0

§ 1 2 . 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 e1 dat a o f th e proble m 2

9

§13- O n diffractio n problem s 22

4

§ 1 4 . 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

§ 1 6 . Rothe' s metho d an d th e metho d o f finit e difference s 24 1 § 1 7 . 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 §1

9

. 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 - Estimate s o f th e hea t potential s i n th e norm s o f W* § 4 - Domains . Som e auxiliar y proposition s •

•

3 m

m

* 28

8

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

7 3 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

§ 1 0 , Loca l estimate s o f th e solution s o f problem s (5-3 ) an d (5.4 ) 35 1 § 1 1 . A fundamenta l solutio n o f th e paraboli c equatio n o f secon d orde r ... . 35 6 § 1 2 . Som e auxiliar y inequalitie s fo r th e functio n Q 36

4

§ 1 3 . Estimate s o f th e fundamenta l solutio n 37


§ 1 4 . Solutio n o f th e Cauch y proble m 38

9

§15- Th e single-laye r potentia l 39

5

§ 1 6 . Solutio n o f th e firs t boundar y valu e proble m 40

6

§17. O n th 1 e estimate s o f S . N . Bernstei n 4

4

Chapter V . Quasi-linea r equation s wit h principa l par t i n divergenc e for 1 m4 §1

. Bounde d generalize d solutions . Holde 1 r continuit y 4

7 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

0

§ 4 . A n estimat e o f raax|u

x|

43

i n th e whol e domai n 43

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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

§ 1 0 . 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^ § 1 . A proo f o f th e smoothnes s o f generalize d solution s o f c l a s s 3 t 1 and a n estimat e o f (px/ffi 5

5 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 § 3 . A n estimat e o f |u|fe § 4 . O n estimate s o f | u

| 57

a

x

4

>. 57

9

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

6 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

§ 1 0 . 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 1 s6

5

Bibliography

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

631

This page intentionally left blank

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

10.1090/mmono/023/01

EDITOR'S NOTE For reason s o f economics , mos t displaye d formula s i n this translation have been inserted fro m th e origina l Russian . Thi s means that certain letters unfortunately have a different appearanc e i n formula s fro m thei r counterparts in the text. Th e principal instances are summarized in the followin g table: isplayed formulas

Text

V

*

A

a

a e

%

e

&

2.

o4i

%

K

K

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

This page intentionally left blank

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

10.1090/mmono/023/02

CHAPTER I

INTRODUCTORY MATERIA L

This boo k i s devote d t o th e basi c linea r an d quasi-linear second-orde r partia l differential equation s o f paraboli c type . Fo r the m th e solvabilit y o f th e basi c boundary valu e problem s an d o f th e Cauch y proble m i n variou s functio n space s i s studied an d investigation s ar e carrie d ou t concernin g th e dependence s o f th e smoothness propertie s o f th e solution s o f thes e equation s o n th e know n function s making u p th e equation s an d o n th e propertie s o f th e othe r know n function s i n th e problems. W e begin wit h a descriptio n o f certai n example s tha t permi t on e t o out line wit h sufficien t accurac y th e contour s o f a possibl e theor y fo r thes e questions , and wit h a n enumeratio n o f th e basi c result s o f th e presen t book . Thes e section s ( § § 3 ana


q ' v r ( .4

)

» i s take n ove r al l nonnegativ e integer s r an d s s a t i s -

fying th e conditio n 2? + s = / . Spaces W (Q) an d Wy *' 2(Sy») wit

h nonintegra l / wil l b e use d i n Chapter s

IV an d VII . Th e forme r spac e is define d in § 2 o f Chapte r II , an d th e latte r i n § 3 o f Chapte r II .

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

I. INTRODUCTORY MATERIAL In addition t o W* * \Q>p), we wil l encounte r tw o space s wit h different ratio s of th e uppe r indices : W*'Q(Qjf) is th e Hilber t spac e wit h scala r produc t

Or and W^^Qj-) i s th e Hilber t spac e wit h scala r produc t

VAQf) i s th e Banac h spac e consistin g o f al l element s o f W\'°(Q T) havin g a finit e nor m

|«| Qr =vral o max r ||«(*. 0il a. 0+JI«xll 1 2, Qf. ( .5

)

where her e an d belo w

idxdt. V^'°{QT) i s th e Banac h spac e consistin g o f al l element s o f V 2(Qj.) tha t are continuous i n t i n th e nor m of L,(Q) f wit h nor m ' • ^ ' " o < " r , , ' ( j f - * > t a + H « * l k v d-6

)

The continuit y i n t o f a function u(x, t) i n the nor m of LAO) mean s tha t ||ttU, t + A*) - u(% , 0|U Q -* 0 fo r A t —• 0. Th e spac e Fj»°((?|. ) i s obtaine d b y completing th e se t Vh^iQj) i n th e nor m of VAQ T). V\M(QT) i s th e subse t o f thos e element s u U, * ) o f V*'°(Q T) fo r whic h

JJ

A' 1 [«(X, * + * ) — «(*. t)]*dx dt _H>0.

0Q

A zero ove r I ^ 0 « ? r ) , f j ' 1 ^ ) , F 2 ( 0 r ) , V\'°{Q T)9 V\M{Q T) mean s that onl y thos e element s o f thes e space s ar e take n whic h vanis h o n Sj » We now define space s consistin g o f function s tha t ar e continuou s i n th e sense o f Holder .

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

§ l. BASI C NOTATION AND TERMINOLOGY 7 We will sa y tha t a function u(x) define d i n O satisfie s a Holder conditio n i n x wit h exponen t a , a € (0 , l ) , an d Holde r constan t (*)$ i a th e domai n Q i f sup p-tt osc {a; Q i n H l {jQT)

> 0 the y ar e equivalent , an d therefor e thei r

wil l no t b e note d i n th e s e q u e l .

All o f th e functio n s p a c e s jus t describe d ar e complete . H (O ) i s th e se t o f function s belongin g t o H ( 0 ' ) fo r an y close d subdomai n Q ' CQ . fil»l^2(QT) is subdomain Q

9

CQ

th e s e t o f function s belongin g t o H l'l^2Q') fo

r an y close d

T*

C ( 0 ) (C(fl) ) i s th e set o f al l function s tha t ar e continuou s i n Q (i n ft)* C{Qj) an

d C(Q T) ar

e define d analogously . C (fl ) (C (fl) ) i s th e se t o f al l

continuous function s i n O ( 0 ) havin g continuou s derivative s i n Q (Q ) u p t o order I inclusively . C2ll(QT) (C

2fl

(QT)) is

th e s e t o f al l continuou s function s i n Q

having continuou s derivative s u C^^iOj*) is

%,

v,

%%>

T

(i n Q

u % i n Q^ (i n Qj>).

th e se t o f al l continuou s function s i n Qj satisfyin

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

g

T)

§ 1 . BASI C NOTATION AN D TERMINOLOGY

9

i Lipschit z conditio n i n x an d a Holde r conditio n i n t wit h exponen t % 0 (1 2 ) (I - 1 , 2 ) i s th e se t o f al l continuou s function s i n 1 2 having continuou s derivatives i n 1 2 up to orde r Z - 1 , wit h th e derivative s o f orde r I - 1 havin g a first differentia l a t eac h poin t o f 0 an d th e derivative s o f orde r / bein g bounde d in 1 2 . 02*l(Qj.) (0 2,l(QT)) i s th e se t o f al l continuou s function s i n Qy (i n Qj) having a t eac h poin t o f Qy (o f Qj) derivative s u % an d u wit h th e u % bein g continuous i n x an d having a firs t differentia l wit h respect t o x a t eac h poin t o f Q

) and th e function s u x, u t> u xx bein g bounde d i n Qj* (i n Qj\ 3« O n domains, thei r boundaries , an d functions prescribe d o n these bound * aries. Throughou t th e boo k w e limi t ou r considerations t o domains havin g Apiece wise-smooth boundarie s wit h nonzer o interio r angles' * or , mor e briefly , Apiece wise-smooth boundaries. " B y thi s restrictio n w e wil l understan d a domain whose closur e ca n b e represente d i n th e for m 1 2 = 12, (J • • * \J 12^y, 12. fl 12- = 0, where eac h o f th e 1 2 ^ can b e homeomorphically mappe d ont o the uni t ball o r cub e by means o f function s satisfyin g a Lipschit z conditio n i n 1 2 . an d suc h tha t th e Jacobians |d( z )/d(x) | o f thes e transformation s ar e bounde d from below b y a positive constant . We will sa y tha t th e boundar y 5 o f a domai n 1 2 (o r a par t S - o f it ) satisfie s condition (A ) i f ther e exis t tw o positive number s a Q an d $ Q suc h tha t the inequality mes Qp < ( 1 — 0 O) mes K 0 holds fo r an y bal l K wit h cente r o n S (o n S , respectively ) o f radiu s p < a^ and any o f th e connecte d component s 1 2 ' o f th e intersectio n 1 2 o f th e bal l K wit h 1 2 . Let * = (a r j» • • • f x®) be any poin t o f th e boundar y S o f a domain 0 * W e will cal l (y« 9 • • •» y n) a local Cartesia n coordinat e syste m wit h origi n a t x i f y an d x ar e connecte d b y the equation s y i - a HfSxk ~ x %)9* ' ~ *»"" " » n > w n e r e t n aih for m a n orthogona l numerica l matri x an d the y n axi s ha s th e directio n o f th e outward (wit h respec t t o 1 2 ) normal t o S a t x . We will sa y tha t a surface S belong s t o class H , / > 1 (o r C , o r 0 , / > 1 ) i f ther e exist s a number p > 0 suc h tha t the intersectio n o f S wit h a bal l K o f radiu s p wit h cente r a t a n arbitrary poin t x ° € 5 i s a connected surface ,

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

e

10

L INTRODUCTOR Y MATERIA L

the equatio n o f whic h i n th e loca l coordinat e syste m (y. , • • • , y n) with origi n a t *° ha s th e for m y n = cuCy** *' * > >7i-i^ ' w here (s) i s give n o n a surfac e 5 o f clas s H * > 1 + > 1 ( C I o r O'l) , W e will sa y tha t 1 . Bu t for / < 1 al l o f wha t ha s bee n sai d concernin g func tions o n S tha t belon g t o H (S) (C (5 ) o r 0 (5) ) i s no t true . O f two possibl e but differen t definition s i t i s mor e convenien t fo r u s t o take th e on e tha t preserves the equivalenc e o f th e norms |0(s)| £ an d |