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)

This page intentionally left blank

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

Copying 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 any 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 reprint-permission@ams.org .

© Copyrigh t 196 8 b y th e America n Mathematica l Society . Al l right s reserved . Reprinted wit h correction s 198 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 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).

This page intentionally left blank

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

This page intentionally left blank

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

This page intentionally left blank

This page intentionally left blank

BIBLIOGRAPHY

[ 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 .

[ 2 l M . S . Agjranovi c an d M . h Visik , a ) Parabolic boundary problems, Uspeh i Mat . Nauk 18(1963) , no . 1(109) , 206-208 . (Russian ) b) Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat . Nau k 19(1964) , no . 3(117) , 53-16 1 - Russia n Math . Survey s 19(1964), no . 3 , 53-157 . M R 3 3 #415 .

[ 3 l A * D. Aleksandrov , a ) Certain estimates concerning the Dirichlet problem, Dokl. Akad . Nau k SSS R 134(1960) , 1001-100 4 = Sovie t Math . Dokl . 1(1960) , 1151-1154. M R 26 #5290 . b) Investigation on the maximum principle. IV , V , VI , Izv . Vyss* Ucebn . Zaved. Nfetematik a I960 , no . 3(16) , 3 - 1 5 ; ibid. , I960 , no . 5(18) , 16-26 ; ibid. 1961 , no . 1(20) , 3 -20 . M R 24 #A3400d , e , £ . c) Uniqueness conditions and bounds for the solution of the Dirichlet problem, Vestnik Leningrad . Univ . Ser . Mat . Meh . Astronom . 18(1963) , no . 3 , 5 -29 ;

English transl. , Amer . Math . Soc . Transl . (2 ) 68(1968) , 8 9 - 1 1 9 . M R 29 #1434 .

D. G . Aronson-se e [ l32l .

[ 4 ] N . Aronszajn , Boundary values of functions with finite Dirichlet integral,

Conference o n Partia l Differentia l Equations , Univ . o f Kansa s Press , 1955 , pp. 7 7 - 9 3 -

[ 5 l V . M . Babic , a ) On the extension of functions, Uspeh i Mar . Nau k 8(1953) , no. 2(54) , 111-113 . (Russian ) M R 15 , 110 . V. M . Babic ' and L . N . Slobodeckii , b ) On the boundedness of the Dirichlet integral, Dokl . Akad . Nau k SSS R 106(1956) , 604-606 . (Russian ) M R 17,959 .

[61 G . I . Barenblat t an d M . I . Visik , On finite velocity of propagation in problems of non-stationary filtration of a liquid or gas, Prikl . Mat . Meh . 20(1956) ,

411-417. (Russian ) M R 18 , 256 .

[ 7 ] R . B . Barrar , Some estimates for solutions of parabolic equations, J . Math .

Anal. Appl . 3(1961) , 373-397 . M R 27 #448 .

[ 8 l S . N . Bernstein , a ) A limitation on the moduli of a sequence of derivatives of solutions of equations of parabolic type, Dokl . Akad . Nau k SSS R 18(1938) , 385-388. (Russian ) b) Collected works. HI ; Differential equations, calculus of variations and geometry (1903-1947) , Izdat . Akad . Nau k SSSR , Moscow , I960 . (Russian ) MR 23 # A 32.

631

632 BIBLIOGRAPHY

[9 ] O . V * Besov, An investigation of a family of function spaces in connection with embedding and extension theorems, Trud y Mat. Inst , Steklov . 60(1961) , 42-81; Englis h transl. , Amer . Math , Soc . Transl . (2) 40(1964) , 85-126 . MR 24 #A3501.

[10] V . M . Borok , a ) On a characteristic property of parabolic systems, Doki . AkadL Nauk SSSR 110(1956) , 903-905 - (Russian ) M R 19, 150 . b) On the numerical characteristics of systems that are correct according to /. G . Petrovskii. Izv . Vyss l Ucebn . Zaved . Matematik a 1959 , no . 1(8) , 16 -22. (Russian ) M R 24 #A737 .

[ l l ] F . E . Browder , a ) Estimates and existence theorems for elliptic boundary value problems, Proc . Nat . Acad . Sci . U . S . A . 45(1959) , 365-372 . M R 24 #A2749. b) Nonlinear parabolic boundary value problems of arbitrary order, Bull . Amer, Math . Soc 69(1963) , 858-861 . M R 27 #6049-c) Strongly non-linear parabolic boundary value problems, Amer . J . Math . 86(1964), 339-357 . M R 29 #3764 . d) Non-linear initial value problems, Ann . of Math. (2 ) 82(1965) , 51-87 . MR 34 #7923-e) Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc . Sympos . Appl . Math. , vo l 17 , Amer . Math. Soc , Providence, R . I. , 1965 , pp . 24-49- M R 33 #6092 .

[l2l A * Calderon an d A. Zygmund, On the existence of certain singular integrals, Acta Math . 88(1952), 85-139 . M R 14 , 637 .

(l3l O Ciliberto , a ) Sulle equazioni non lineari di tipo parabolico in due vari-abili, Ricerch e Mat . 3(1954) , 129-165 . M R 16, 1028 . b) Formule de maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili, Ricerch e Mat . 3(1954) , 40—75 . MR 16 , 139 . c) Sulle equazioni quasi-lineari di tipo parabolico in due variabili, Ricerch e Mat. 5 (1956), 97-125 . M R 18, 46 . d) Nuovi contributi alia teoria dei problemi al contorno per le equazioni paraboliche non lineari in due variabili, Ricerch e Mat . 5(1956) , 206-225 . MR 18, 742.

[14] E * M. Cistjakova, An application of the method of finite differences to the solution of mixed boundary value problems for certain hyperbolic and para-bolic systems, Candidate' s dissertation , Leningrad.Gos . Ped . Inst. , Lenin -grad, 1956 . (Russian )

[15] R . Courant , K . Friedrich s an d H. Lewy, a ) Vber die partiellen Differential-gleichungen der mathematischen Physik, Math . Ann . 100(1928) , 32—74 ; English transl. , Ne w York University Couran t Inst. Mat h Sciences Researc h Dept. N . Y. 0.-7689 .

BIBLIOGRAPHY 633

R. Couran t an d D . Hilbert , b ) Methoden der mathematischen Physik. Vol . I , Springer—Verlag, Berlin , 1931 ; reprint , Interscienc e , Ne w York , 1943 ; Rus -sian txansL , 3r d e d , GITTL , Moscow , 1951 ; Englis h transl. , Interscience ,

New York , 1953 - M R 5 , 97 ; M R 13 , 800 ; M R 16 , 426 .

[ l6l Czo u Jiu-lin ' (Zho u Yu-lin , Cho u Yu-lin ) ( s e e als o Kalasnikov) , Boundary value problems for non-linear parabolic equations, Dokl . Akad . Nau k SSS R

117(1957), 195-198 ; Mat . Sb . 47(89 ) (1959) , 431-484 ; Englis h transl. , Amer.

Math. Soc . Transl . '?. ) 41(1964) , 155-213 - M R 20 #3374 ; M R 2 2 #1751 .

[171 E . D e Giorgi , Sulla differenziabilita e Vanaliticita delle estremali degli inte*

grali multipli regolari, Mem , Accad . Sci . Torino . Cl . Sci . F i s . Mat . Nat. (3 ) 3(1957), 25 -43 - M R 20 #172 .

[ l8 l G . Doetsch , Theorie und Anwendung der Laplace-Transformation, Dover , Ne w York, 1943 . M R 5 , 119 .

[19] J . Douglas , Jr. , A uniqueness theorem for the solution of a Stefan problem,

Proc. Amer . Math . Soc . 8(1957) , 402-408 . M R 19 , 1060 .

[20] A . Douglis . Se e Agmon .

[2l l F . G . Dressel , The fundamental solution of the parabolic equation. I , Duk e

Math. J . 7(1940) , 186-203 . M R 2, 204 .

[22l Ju . A . Dubinskil , a ) Weak convergence in nonlinear elliptic and parabolic

equations, Mat . Sb . 67(109 ) (1965) , 609-642 ; Englis h transl. , Amer . Math . Soc. Transl . (2 ) 67(1968) , 226-258 . M R 3 2 #7958 . b) On nonlinear parabolic equations having nondivergence form, Dokl . Akad . Nauk SSS R 163(1965) , 805-80 8 = Sovie t Math . Dokl . 6 ( 1965) , 1009-1013 .

MR 3 5 #3270 .

[23] A . S . Dynin , n-dimensional elliptic boundary value problems with a single unknown function, Dokl . Akad . Nau k SSS R 141(1961) , 285-287 = Sovie t Math . Dokl. 2(1961) , 1431-1432 . M R 26 #6564 .

[24l E - B . Dynkin , Markov processes, Fizmatgiz , Moscow , 1963 ; Englis h transl. , Vols. I , II , Academi c Press , Ne w Yor k an d Springer-Verlag , Berlin , 1963 . MR 3 3 #1886 ; M R 3 3 #1887 .

[25l S . D . Eldel'ma n ( s e e als o Lipk o an d Matuctik) , a ) Parabolic systems, "Nauka", Moscow , 1964 ; Englis h transl. , Noordhoff , Groninge n an d North -Holland, Amsterda m (t o appear) . M R 29 #4998 . b) On fundamental solutions of parabolic systems. I, Mat . Sb . 38(80 ) (1956) , 51-92; Englis h transl. , Amer . Math . Soc . Transl . (2 ) 41(1964) , 1-48 ; II , Mat. Sb . 53(95) , (1961) , 73 -136 ; Englis h transl. , Amer . Math . Soc . Transl .

(2) 41(1964) , 4 9 - 1 2 0 . M R 17 , 857 ; M R 2 4 #A925 .

[26] S . Faedo , Un nuovo metodo per Vanalisi esistenziale e quantativa dei prob-lemi di propagazione, Ann . Scuol a Norm . Sup . P i s a (3 ) 1(1947) , 1—41 . MR 11 , 363 .

634 BIBLIOGRAPHY

[27] W . Feller , Zur Theorie der stockastischen Prozesse, Math . Ann . 113(1936) , 113-160.

[28l G . M . Fihtengol'c , A course in differential and integral calculus, Vol . I , 6t h ed., "Nauka" , Moscow , 1966 ; Germa n transl. , VE B DeutscherVerlag , Berlin , 1964; 2n d ed. , Hochschulbuche r fu r Math. , Ban d 61 , 1966 .

[29l A . F . Fi l ippov(se e als o Rjabe n kit) , On conditions for the existence of a solution of a quasi-linear parabolic equation, Dokl . Akad . Nau k SSS R 14 1

(1961), 568-57 0 « Sovie t Math . Dokl . 2(1961) , 1517-1519 - M R 24 #A3428 .

[30] G . E . Forsyth e an d P . C . Rosenbloom , Numerical analysis and partial dif-ferential equations, Wiley, Ne w York , 1958 .

[3ll A . Friedman , a ) Interior estimates for parabolic systems of partial differen-tial equations, J . Math . Mech . 7(1958) , 393*417 . M R 21 #7362 . b) Partial differential equations of parabolic type, Prentice-Hall , Englewoo d Cliffs, N . J . , 1964 . M R 3 1 #6062 .

[32l K . Friedrichs . Se e Ourant .

[33l E . Gagliardo , a ) Problema al contorno per equazioni differentiali lineari di tipo parabolico in n variabili, Ricerch e Mat . 5(1956) , 169-205 * M R 18 , 742 . b) Teoremi di esistenza e di unicith per problemi al contorno relativi ad equa-zioni paraboliche lineari e quasi lineari in n variabili, Ricerch e Mat . 5(1956), 239-257. M R 18 , 742 . c) Caratterizzioni delle tracce sulla frontier a relative ad alcune classi di funzioni in n variabili, Rend . Sem . Ma c Univ . Padov a 27(1957) , 284—305 -MR 2 1 #1525 . d) Proprieta di alcune classi di funzioni in piu variabili, Ricerch e Mat . 7 (1958), 102-137 . M R 2 1 #1526 .

[34] I . M . Gel'fan d an d G * E . Silov , a ) Generalized functions. Vol . Ill : Some questions in the theory of differential equations, Fizmatgiz , Moscow , 1958 ; English transl. , Academi c Press , Ne w York , 1967 . M R 2 1 #5l42b . b) The Fourier transforms of rapidly increasing functions and questions of uniqueness of the solution of the Cauchy problem, Vspehi Mat . Nau k 8(1953) , no. 6(58) , 3-54 ; Englis h transl. , Amer . Math . Soc . Transl . (2 ) 5(1957) , 2 2 1 -274. M R 15 , 867 ; M R 18 , 736 .

[35l M . Gevrey, Sur les equations aux derivees par tie lies du type parabolique, J . Math. Pure s Appl . (6 ) 9 (1913) , 305-471 ; (6 ) 10(1914) , 105-148 .

[361 N . M * Gjunter (N . M . Gunther) , La theorie du potentiel et ses applications aux problemes fondamentaux de la physique mathematique, Gauthier-Villars,Paris , 1934; Russia n transl. , GITTL , Moscow , 1953 - M R 16 , 357 .

[37] S » K. Godunov , On non-uniqueness for parabolic systems, Dokl . Akad . Nfau k SSSR 145(1962) , 498-50 0 - Sovie t Math . Dokl . 3(1962) , 1020-1023 - M R 2 8 #2361.

BIBLIOGRAPHY 635

[38l K . K . Goiovkio , a ) On embedding theorems, Dokl . Akad . Nau k SSS R 134(1960) , 19-22 « Soviet Math. Dokl . 1(1960) , 998-1001 . M R 22 #12374 . b) Two classes of inequalities for sufficiently smooth functions of n vari-ables, Dokl . Akad . Nauk SSSR 138(1961) , 22-2 5 -SovietMath . Dokl . 2(1961) , 510-513- M R 22 #12189. K. K . Goiovki o an d V. A . Solonoikov, c ) Estimates for integral operators in translation invariant norms. 1 , Trud y Mat . last. Steklov . 70(1964) , 47-58 ; English transl. , Aroer . Math . Soc. Traosl . (2 ) 61(1967) , 97-112 . M R 29 #470 .

[39l J . W . Green, An expansion method for parabolic partial differential equations, J. Researc h Nat* Bur . Standards 51(1953) , 127-132 . M R 15, 322 .

[40l A . V . Grekov , The Dirichlet problem for certain quasi-linear parabolic equa-tions, Dokl . Akad . Nauk SSS R 134(1960) , 255-25 8 ~ Soviet Math . Dokl. 1 (1960), 1043-1046 . M R 22 #1748 .

[4l1 Fr . Guglielmino , a ) Sulla regolarizzazione delle soluzioni deboli dei prob-lemi al contomo per operatori parabolici, Ricerch e Mat . 12(1963) , 44-66 . MR 28 #35 7 a* b) Ulteriori contributi alia regolarizzazione delle soluzioni deboli dei prob-lemi parabolici, Ricerch e Mat . 12 ( 1963), 140-150 . M R 28 #357b . c) Sulle equazione paraboliche del secondo ordine di tipo non-variazionale , Ann. Mat . Pure Appl . (4 ) 65(1964) , 127-151 . M R 32 #4395 .

N. M . Gunthe r - se e [361. [42] G . H. Hardy, J . E . Littlewoo d an d G. Polya , Inequalities, Cambridg e Univ .

Press, Ne w York, 1934 ; 2n d ed., 1952 ; Russia n transl. , IL , Moscow , 1948 . MR 13,727; M R 18, 722 .

[43l D - Hilbert. Se e Courant . [44] E * Hille, a ) Functional analysis and semigroups, Anier . Math . Soc. Colloq .

Publ.,vol. 31 , Aroer . Math. Soc, Providence , R . I. , 1948 . M R 9, 594 . E. Hill e an d R. S . Phillips , b ) Functional analysis and semi-groups, Amer . Mam. Soc. Colloq . Publ. , vo L 31 , rev . ed. , Amer . Math . Soc, Providence , R. L , 1957 . M R 19, 664 .

[45l E . Holmgren , Uber Systeme von linearen partiellen Differentiateichungen, Ofvers. Kongl . Vetens . -Akad . Fork . 58(1901) , 91-103 -

[46] E . Hopf , a ) Elementare Bemerkungen uber die Ldsungen partieller Differ-entialgleic hungen z we iter Ordnung vom elliptischen Typus 9 S . -B . Preuss . Akad. Wiss . Bedi n 19(1927) , 147-152 -b) Uber die Anfangswertaufgabe fttr die hydrodynamischen Crundgleichungen, Math. Nacfar. 4(1951) , 213-231 . M R 14 , 327 .

[47] A . M. Il'in, a ) On the fundamental solution of a parabolic equation, Dokl . Akad. Nauk SSSR 147(1962) , 768-771 = Soviet Math . Dokl . 3(1962) , 1697 -1700. M R 29 #1453 -

636 BIBLIOGRAPHY

A* M. II in, A . S . Kalasniko v and O. A * Oleinik, b ) Linear equations of second order of parabolic type, Uspeh i Mat . Nauk 17(1962) , no . 3(105) , 3-14 6 = Russian Math . Surveys 17(1962) , no . 3 , 1-143 . M R 25 #2328 .

[48] V . A . Il'in , On the solvability of mixed boundary value problems for hyper-bolic and parabolic equations, Uspeh i Mat . Nauk 1 5 (1960), no. 2(92) , 97 -154 * Russia n Math . Surveys 15(1960) , no . 2 , 85-142 . M R 22 #721 .

[49l V . P, . Il'in , a ) Some inequalities in function spaces and their application to the investigation of the convergence of variational methods, Trud y Mat. Inst. Steklov. 53(1959) , 64-127 . (Russian ) M R 22 #9738 . b) Properties of certain classes of differentiate functions of many variables given in an n-dimensional domain, Trud y Mat . Inst. Steklov . 66(1962) , 227 -363- ( Russian) M R 27 #3750 . V. P . Il'i n an d V. A . Solonnikov , c ) Some properties of differentiable func-tions of many variables, Trud y Mat Inst . Steklov . 66(1962) , 205-226 . (Russian) M R 27 #2768 .

[50] E . L . Ince , Ordinary differential equations, Longmans , Gree n & Co., London , 1926; reprint , Dover , Ne w York, 1944 . M R 6, 65 .

[5ll A . V . Ivanov , a ) On some properties of generalized solutions of linear equations of elliptic and parabolic type, Vestni k Leningrad . Univ . Sex* Mat . Meh. Astronom. 1 9 (1964), no . 2 , 5-15 . (Russian ) M R 30 #1292 . b) On a class of functions, Trud y Mat. Inst . Steklov . 73(1964), 159-171 . (Russian) M R 33 #6154 . c) A priori estimates for solutions of linear second-order equations of ellip-tic and parabolic types, Dokl . Akad . Nau k SSSR 161(1965) , 1270-127 3 * Soviet Math . Dokl . 6(1965) , 583-587 . M R 32 #1443-d) On certain properties of generalized solutions of parabolic equations of second order. I , Izv . Akad . Nau k SSSR Ser . Mat . (t o appear). (Russian ) e) An investigation of the properties of generalized solutions of quasi-linear parabolic equations of second order, Candidate' s dissertation , Lenin* gradGos. Univ. , 196 6 (seealso : Dok L Akad . Nauk SSS R 173(1967) , 752 -754 = Soviet Math . Dokl . 8(1967) , 463-466 - M R 35#4598). A. V . Ivanov , O . A . Ladyzenskaja , N . N . Ural'ceva an d A. L . Treskunov , f) Certain properties of generalized solutions of parabolic equations of second order, Dokl . Akad . Nau k SSS R 168(1966) , 17-2 0 « Soviet Math. Dokl. 7(1966) , 579-583 ; Trud y Mat. Inst. Steklov . 92(1966) , 57-9 2 ~PK>C .

Steklov Inst . Math . 92(196$ , 63-104 M R 35 #577 . [52l F . John , On integration of parabolic equations by difference methods. I ,

Linear and quasi-linear equations for the infinite interval, Comm . Pure AppL Math . 5(1952), 155-211 . M R 13, 947.

[53l J . Kadle c and R. Vybomy , Strong maximum principle for weakly nonlinear parabolic equations, Comment. Math. Univ. Catolinae 6(1965), 19-20 . MR 31 #1467.

BIBLIOGRAPHY 637

[54] A . S. Kalasniko v ( see als o A * M. II'in), 0 . A . Oleini k an d Czbu Jm-lin' ; The Conchy problem and boundary value problems for equations of non-stationary filtration type, Izv . Akad . Nauk SSSR Ser. Ma t 22(1958) , 667 -704. (Russian ) M R 20 #6271.

[55] S . L. K^menomostskaja , a ) On the Stefan problem, Naucn . Dok L Vyss l Skoly . 1(1958), 60-62 . (Russian ) b) On th* Stefan problem, Ma c Sb . 53(95) (1961) , 489-514 . (Russian ) MR 25 #5292 .

[56] L . I . Kamynin , a ) On the existence of a solution of boundary-value problems for a parabolic equation with discontinuous coefficients, Izv . Akad . Nauk SSSR Ser . Mat 28(1964) , 721-744 . (Russian ) M R 29 02534 (se e also : Dok L Akad. Nauk SSSR 139(1961) , 1048-105 1 * Soviet Math. Dokl . 2(1961) , 1043 -1046. M R 24 #A1519) . L. I . Kamyni n an d V. N. Masleunikova , b ) Boundary estimates of the solu-tion of the third boundary value problem for a parabolic equation, DokL Akad. Nauk SSSR 153(1963) , 526-52 9 -Sovie t Math . Dokl- 4(1953) , 1711-1714 . MR 29 #1447 .

, c ) Boundary estimates of the solution of a problem with directional derivative for a parabolic equation in a noncylindrical domain, Dokl. Akad . Nauk SSSR 160(1965) , 527-52 9 « Soviet Math. DokL 6(1965) , 128-131 . MR 31 #2516 .

[57] T . Kato, a ) Integration of the equation of evolution in a Banach space, J . Math. Soc. Japa n 5(1953) , 208-304 . M R 15, 437 . b) On the linear differential equations in Banach spaces, Comm . Pure Appl . Math. 9(1956), 479*486 . M R 19 , 279 . c) Nonlinear evolution equations in Banach spaces, Proc . Sympos. Appl . Math., vol . 17 , Amer . Math. Soc , Providence , R . L , 1965 , pp- 50—67. MR 32 #1573.

[58] I . I . Kolodner , Free boundary problem for the heat equation with applications to problems of change of phase. I , General method of solution, Comm . Pure Appl. Math . 9(1956), 1-3 L M R J9, 285 .

[59l A . S . Kompaneec . Se e Zel'dovic . [60] M.A . Krasnosel'skii, S . G . Krei n and P . E . Sobolevskti , a ) On differential

equations with unbounded operators in a Banach space, Dok L Akad . Nauk SSSR 111(1956) , 19-22 . (Russian ) M R 19, 550 . b) On differential equations with unbounded operator coefficients in a Hil-bert space, Dok L Akad . Nauk SSSR 112(1957) , 990-993-(Russian) MR W, 747.

[6l] S . G . Krein and P . E . Sobolevski x (se e als o Krasnosel'skii) , Differential equations with abstract elliptic operators in a Hilbert space, Dok L Akad. Nauk SSSR 118(1958) , 233-236 . (Russian ) M R 20 #6043-

638 BIBLIOGRAPHY

[62] S . N . Kruzkov , a ) On the a priori estimation of solutions of linear parabolic equations and of solutions of boundary value problems for a certain class of quasi-linear parabolic equations, Dokl . Akad . Nau k SSSR 138(1961) , 1005 -1008 = Soviet Math. Dokl. 2(1961) , 764-767 . M R 24 #A2147. b) A priori estimates and certain properties of the solutions of elliptic and parabolic equations of second order, Dokl . Aka d Nau k SSSR 150(1963) , 748 -751 - Sovie t Math. Dokl- 4 ( 1963), 757-760. M R 27 #1687 . c) A priori estimates and certain properties of the solution of elliptic and parabolic equations, Mat . Sb. 65(107) (1964) , 522-570 ; Englis h cransl.,Ajner . Math. Soc . Trans L (2 ) 68(1968) , 169-220 * M R 30 #1317 . d) The Cauchy problem in the large for nonlinear equations and certain quasi-linear systems of first order with several variables, Dokl . Akad . Nau k SSSR 155(1964), 743-746 = * Sovie t Math . Dokl. 5(1964) , 493-496 . M R 29 #1436 . S. N . Kruzko v an d O. A . Olefnik, e ) Quasi-linear parabolic equations of second order tvUh many independent variables, Uspeh i Mat. Nauk 16(1961), no. 5 (101), 115-155 - Russia n Math. Surveys 16(1961), no. 5, 105-146 . M R 25 #5289.

[63] M . Krzyzanski, Sur I'unicite des solutions des second et troisieme problimes de Fourier relatifs a Vequation lineaire normale du type parabolique, Ann . Polon. Math . 7(1960), 201-208 . M R 22 #1749 .

M. Kurihara - se e [ I33l-[641 A . Kzivickii . Se e Ladyzenskaja . [651 O . A . Ladyzenskaj a (se e alsolvano v an d Visik), a ) On the uniqueness of

the solution of Cauchy9s problem for a linear parabolic equation, Mat . Sb. 27(69) (1950) , 175-184 . (Russian ) M R 12,709 . b) On the closure of an elliptic operator, Dokl . Akad . Nau k SSSR 79(1951), 723-725. (Russian ) M R 14, 280. c) The mixed problem for a hyperbolic equation, GITTL , Moscow, 1953 * (Russian) M R 17, 160 . d) On the solvability of the basic boundary value problems for equations of parabolic and hyperbolic type, Dokl . Akad . Nau k SSS R 97 (1954), 395-398 . (Russian) M R 17, 495 . e) On the solution of nonstationary operator equations of various types, DokL Akad . Nauk SSSR 102(1955) , 207-210 . (Russian ) M R 17, 16 L I) On non-stationary operator equations and their applications to linear problems of mathematical physics, Mat . Sb . 45(87) ( 1958), 123—158 . (Russian) M R 22 #12290 . g) On the solution of the general diffraction problem, Dokl . Akad . Nauk SSSR 96(1954) , 433-436 . (Russian ) M R 16 , 1117 . h) The solution in the large of the first boundary-value problem for quasi-linear parabolic equations, Dokl . Akad . Nau k SSSR 107(1956) , 636-639 ; Tnidy Moskov . Mat . Obsc . 7 ( 1958), 149-177 . (Russian ) M R 22 #4880.

BIBLIOGRAPHY 639

i) A simple proof of the solvability of the fundamental boundary problems and of a problem on eigenvalues for linear elliptic equations, Vestni k Lenin -grad. Univ. Ser . Mac Meh . Astron. 10(1955) , no . 11 , 23-29* (Russian ) M R 17, 855 . j) The method of finite differences in the theory of partial differential equa-

tions, Uspeh i Mac Nauk 12(1957), no. 5 (77) , 123-148 ; Englis h transL , Amer . Math, Soc Trans L (2 ) 20(1962) , 77-104. M R 20 #3395; M R 25 #342. k) On integral estimates, convergence, approximate methods and solution in functionals for elliptic operators, Vestni k Leningrad . Univ . 13(1958) , no . 7 , 60-69. (Russian ) M R 20 #5353* 1) On the solvability "in the large** of the boundary-value problems for linear and quasi-linear parabolic equations and the Navier~Stokes equations,

Proc. Fourt h All-Unio n Math . Congress 1961 , vol . 1 , Izdac Akad . Nauk SSSR, Leningrad, 1963 , pp. 134-157 . (Russian ) M R 29 #386 . O. A * Ladyzenskaja an d A. Kzivickil , m ) The method of nets for non-stationary Navier-Stokes equations, Trud y Mac Ins c Steklov . 92(1966) , 9 3 -99 * = Proc Steklo v Ins t Math . 92U966X 105-112 . M R 34 #4651. 0 . A . Ladyzenskaj a an d N. N. Ural'ceva n ) A boundary value problem for linear and quasi-linear parabolic equations, 1 , Dokl. Akad . Nau k SSSR 13 9 (1961), 544-54 7 = Soviet Math. Dokl. 2 (1961), 969-972 . M R 25 #5288 ; 1, II, Izv . Akad . Nauk SSS R Ser. Mat . 26(1962) , 5-52 , 753-780 ; Englis h tiansL, Amer . Math. So o Transl . (2)47(1965), 217-267 , 268-299 . MR 31 #6063.

, o ) A boundary value problem for linear and quasi-linear equations and systems of parabolic type. HI , Izv . Akad . Nau k SSSR Ser. Mat . 27(1963), 161-240; Englis h transL , Amer . Math. Soc Trans L (2 ) 56(1966) , 103-192 . MR 31 #6063 .

, p ) On the H&lder continuity of solutions and their derivatives for linear and quasi-linear equations of elliptic and parabolic types, Dokl . Akad. Nauk SSSR 155(1964) , 1258-126 1 * Sovie t Math . Dokl . 5(1964) , 565 -568. M R 28 #5256; Trud y Mac Inst . Steklov . 73(1964) , 172-220 ; En^ish transL, Amer . Math. Soc. Transl . (2) 61(1967) , 207-269 . M R 30 #4063 .

, q ) Linear and quasilinear equations of elliptic type, f1 Nauka", Moscow, 1964 ; Englis h transL , Academi c Press , Ne w York , 1968 . MR 35 #1955. O. A . Ladyzenskaja , V . J a. Rivkin d an d N. N . Ural'ceva, r ) On the classical solvability of diffraction problems for equations of elliptic and parabolic types, Dokl . Akad . Nauk SSS R 158(1964) , 513-51 5 = Soviet Math. Dokl. 5(1964) , 1249-1252 . M R 29#4973; Trud y Mac Ins c Steklov . 9 2 (1966), 116-146 -Proc . Steklo v Insc Math . 92(1966) , 132-166 . E. M . Landis , Some questions in the qualitative theory of elliptic and para-bolic equations, Uspeh i Ma c Nau k 14(1959) , no . 1(85) , 21-85 ; Englis h transL, Amer . Math. Soc . Trans L (2 ) 20(1962) , 173- 2 38. M R 22 #3870 .

640 BIBLIOGRAPH Y

[67] P . D . La x an d A. N . Milgram , Parabolic equations* Contributions to the theory of partial differential equations, Ann . of Math. Studies , no. 33 , Prince -ton Univ. Press , Princeton , N . J. , 1954 . M R 16, 709 .

[68] J . Lera y and J. L . Lions , a ) Quelques resultats de Visik sur les problemes elliptiques non line aires par les methodes de Mint y-Brow der, Bull . Soc . Math. France 93(1965) , 97-107 . M R 33 #2939. J. Lera y and J. Schauder , b ) Topologie et equations fonetionnelies , Ann . Sci. Ecol e Norm . Sup. (3) 51(1934) , 45-78 ; Russia n transL , Uspeh i Mat . Nauk 1(1946) , no . 3- 4 (13-14) , 71-95 - M R 9, 606 .

[69] E « E. Levi , Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Ore . Mat . Palermo 24(1907) , 275-317 .

[70] H . Lewy . Se e Courant . [71] L . Lichtenstein , Die neue Entwicklung in der Theorie der par tie 11 en Dif~

ferentialgleichungen zweiter Ordnung, Enzyklopadie de r math. Wissenschaften , BandII,Teubner, Leipzig , 1924 , Chapter 8, pp . 1277-1334 .

[72] V . B . Lidskif , a ) The Fourier series expansion in terms of the principal functions of a non-self ad joint elliptic operator, Mat . Sb . 57(99 ) (1962) , 137 -150. (Russian ) M R 26 #460 . b) On the summability of series in the principal vectors of nonselfadjoint operators, Trud y Moskov . Ma t Obsc . 11(1962) , 3-35 ; Englis h transl.,Amer . Math. Soc. Trans L (2 ) 40(1964) , 193-228 . M R 26 #1760*

[73l J . L . Lions (se e als o [134]) . (se e als o Leray) , Equations differentielles operationelles et problemes aux limites, Di e Grundlehren de r math. Wissen * shaften, Bd . I l l , Springer-Verlag , Berlin , 1961 . M R 27 #3935 .

[74] B . Ja « Lipko an d S. D . Eidel'man , On boundary value problems for para-bolic systems in regions of general type, Dokl . Akad . Nauk SSS R 150(1963), 58-61 = Soviet Math. DokL 4(1963) , 614-618 . M R 29 #1455 .

[75] J . E . Litdewood . Se e Hardy . [76] V . E . Ljance , On a boundary value problem for parabolic systems of dif-

ferential equations with a strongly elliptic right-hand side, Mat . Sb. 3 5 (77) (1954) , 357-368 . Ma c Sb . 39(81 ) (1956) , 525 . (Russian ) MR 16, 709 .

[77] J a. B . Lopattaskii' , On a method for reducing boundary problems for sys~ terns of differential equations of elliptic type to regular integral equations, Ukrain. Mat . Z . 5(1953) , 123-151 . (Russian ) M R 17 , 494 .

[78] V . N . Masleonikova. Se e Kamynin* [79] M . I. Matiicuk an d S. D . Eide l man. On parabolic systems with coefficients

satisfying a Dint condition, Dokl . Akad . Nau k SSSR 165(1965) , 482-48 5 * Soviet Math. Dokl - 6(1965) , 1461-1464 . M R 32 #7962.

BIBLIOGRAPHY 641

[80] N . N. Meiman , a ) On the theory of partial differential equations, Dokl . Akad . Nauk SSS R 97(1954), 593-596 - (Russian ) M R 16, 254 . b) On the equation of heat conduction, Dokl . Akad . Nauk SSS R 99(1954), 209-212. ( Russian) M R 16, 485 .

[81] V . P . Mihailov , a ) On the Dirichlet problem and the first mixed problem for a parabolic equation, Dok L Akad * Nauk SSSR 140(1961) , 303-30 6 » Sovie t Math. Dokl. 2(1961) , 1204-1207 . M R 26 #487 . b) On the Dirichlet problem for a parabolic equation. I , Mat . Sb. 61(103 ) (1963), 40-64 ; H , Mat . Sb. 62(104) (1963) , 140-159 - (Russian ) M R 27 #1710; M R 28 #1410 .

[82] S - G . Mihlin, Fourier integrals and multiple singular integrals, Vestni k Leningrad. Univ . Ser . Mat . Meh. Asttonom . 1 2 ( 1957), no . 7 , 143-155 . (Russian) M R 19 , 546 .

[83] A . N. Milgram . Se e Lax . [84] G . Minty , a ) Monotone (nonlinear) operators in Hilbert space, Duk e Math.

J. 2 9 (1962) , 341-346 . M R 29 #6319 . b) Two theorems on nonlinear functional equations in Hilbert space. Bull . Amer. Math . Soc. 69 (1963), 691-692 . M R 32 #8188 . c) On a "monotonicity" method for the solution of nonlinear equations in Banach spaces, Proc . Nat . Acad . Sci . U.S.A . 5 0 (1963), 1038-104 L M R 28 #5358 .

[85] S . Mizohata , Hypoellipticite des equations paraboliques, Bull . Soc . Math. France 85 (1957), 15-50 . M R 20 #3381 .

[86] G . M . Mjunto, a ) Zum dynamischen Wdrmeleitungsproblem, Math . Z. 38(1934) , 323-338. b) Integral equations, Moscow , 1934 . (Russian )

[87] J . Moser , a ) A new proof of de Giorgi ys theorem concerning the regularity problem for elliptic differential equations, Comm . Pure Appi . Math. 13(1960) , 457-468. M R 30 #332 . b) A Harnack inequality for parabolic differential equations, Comm . Pure AppL Math. 1 7 ( 1964) , 101-134 . M R 28 #2357 .

[88] J . Nash , Continuity of solutions of parabolic and elliptic equations, Amer . J* Math. 80 (1958) , 931-954 . M R 20 #6592 .

[89] S . M . Nikol'skii, The properties of certain classes of functions of many vari-ables on differentiable manifolds, Mat . Sb. 33(75 ) (1953) , 261-326 . (Rus -sian) M R 16, 453 .

[90] L . Nirenber g (se e als o Agmon) , a ) A strong maximum principle for para* bolic equations, Comm . Pure App L Matfi . 6(1953) . 167-177 . M R 14, 1089 ; MR 16, 1336 . b) On elliptic partial differential equations, Ann * Scuola Norm. Sup. Pis a

642 BIBLIOGRAPHY

(3) 1 3 (1959), 115-162 , M R 22 #823. [9l1 O . A . Oleioi k (se e als o A * Il'in, KalaXnikov , Knizko v an d Ventcel') , a ) On

a method for solving the general diffraction problem. (Russian ) b) Boundary*value problems for linear elliptic and parabolic equations with discontinuous coefficients, Izv . Akad . Nauk SSS R S «. Mat * 25 (1961), 3-20 ; English transl. , Amer . Math. Soc* Transl . (2 ) 42(1964) , 175-194 . MR 23 #AI136. c) On quasi-linear parabolic equations with many independent variables, Dokl. Akad . Nauk SSSR 13 8 (1961), 43-4 6 * Soviet Math . DokU 2 (1961), 529-532* M R 24 #A924* d) On a method for solving the general Stefan problem, Dokl * Akad * Nau k SSSR 13 5 (1960), 1054-105 7 « Soviet Math * Dokl. 1 ( I960) , 1350-1354 . MR 23 #A2644*

[92] M « Pagni, Su un problema al contorno tipico per Vequazione del colore, Ann . Scuola Norm . Sup. Pisa (3 ) 11(1957) , 73-115* M R 25 #332*

[93] I . G . Petrovskii' , a ) Lectures on partial differential equations, GITTL , Mos-cow, 1950 ; 3r d augmented ed*, Fizmatgiz , Moscow , 1961 ; Englis h transl , Interscience, Ne w York, 1964 . M R 16, 478 ; M R 25 #2308 . b) Zur ersten Randwertaufgabe der WHoneleitungsgleichung, Compositi o Math. 1(1935) , 38M19 . c) On the Cauchy problem for systems of linear partial differential equations in a domain of nonanalytic functions, Bjull . Moskov . Gos* Univ. Ser * A. , 1 (1938), no . 7 , 1-72 * (Russian ) d) A new proof of the existence of a solution of the Dirichlet problem by the method of finite differences, Uspeh i Mat * Nauk 8 (1941), 161—170 . (Russian ) MR 3, 123 .

R* S. Phillips . Se e [441* [94l W . Pogorzelski (se e als o [l35l) , a ) Etude de la solution fondamentale de

Vequation parabolique, Ricerch e Mat . 5 (1956), 25-57 . M R 18 , 47 . b) Problemes aux limites pour Vequation parabolique normale, Ann * Poion . Math. 4( 1957), 61-92. M R 21 #3656a . c) An investigation of the integrals of a parabolic equation and of boundary value problems in an unbounded domain, Mat . Sb. 47(89) (1959) , 397-430 . (Russian) M R 22 #2791. d) Etude d'une fonction de Green et du probleme aux limites pour Vequa-tion parabolique normale, Ann. Polon . Math . 4(1958), 288-307 . M R 21 #5078.

[95l G * Polya. Se e Hardy. [96] G . Prodi , Teoremi di esistenze per equazioni cdle derivate parziali non

lineari di tipo parobolico. I , II , 1s t Lombard o Sci. Lett * Rend. C L Sci . Ma c Nat. (3 ) 17(86 ) (1953) , 3-26 , 27-47 . M R 16, 259*

BIBLIOGRAPHY 643

[97] R . D . Richtmyer , Difference methods for initial value problems, Inte r science Tracts i n Pur e and AppL Math* , no . 4 , Interscience , Ne w York , 1957 ; Rus -sian transl. , U , Moscow , I960 . M R 20 #438 ; M R 22 #3864 .

[98l V . J a. Rivkind . Se e Ladyzenskaja . [99] V . S - Rjaben'k u an d A . F . Filippov , On the stability of difference equations,

GITTL, Moscow, 1956 ; Germa n transL, Mathemati k fur Natur. und Technik , Band 3 , VE B Deutscber Verlag , Berlin , I960 . M R 19, 865 ; M R 23 #A437.

[lOOl P . C . Rosenbloom . Se e Forsydie . [lOl] £ • Rothe , W'armeleitungsgleickung mit nichtkons tanten Koeffizienten, Math .

Ana. 104(1931) , 340-362 . [102] L . h Rubinstein , a ) On a certain nonlinear problems provided by an equa-

tion of Fourier, Doctoral dissertation , Moskov . Gos . Univ. , Moscow , 1957 . (Russian) b) A two-phase Stefan problem on an interval with one~phase initial state of a heat-conducting medium, Latvija s Valst s Univ . Ziaata . Rakst i 58(1964) , vyp. 2 , 111-148 . (Russian ) M R 31 #6066 .

[103] E . S . Sabinina , a ) On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Dokl . Akad . Nau k SSS R 136(1961) , 1034-1037 » Soviet Math . Dokl. 2 ( 1961), 166-169 . M R 28 #1416 . b) On a class of nonlinear degenerating parabolic equations, Dokl . Akad . Nauk SSSR 143(1962) , 794-797 * Soviet Math. Dokl . 3 ( 1962) , 495-498 . MR 24 #A2762.

[104] A . A . Samarskii , a ) An efficient difference method for solving a multi-dimensional parabolic equation in an arbitrary domain, Z . Vycis L Mat . i Mat. Fiz . 2(1962) , 787-81 1 «U.S.S.R . Comput . Math , an d Math. Phys.196 3 (1964), no . 5 , 894-926; 3(1963) , 416 . M R 32 #609 . A. A . Samarskii'and A . N . Tihonov , b ) On finite difference schemes for equations with discontinuous coefficients, Dokl . Akad . Nau k SSS R 10 8 (1956), 393-396 . (Russian ) M R 18, 938.

[105J V.K . Saul'ev , The integration of equations of parabolic type by the method of nets, Fizmatgiz , Moscow , I960 ; Englis h transl. , Inter n at. Serie s i n Pure and AppL Math., vol . 54 , Pergamon Press , Ne w York, 1964 . M R 23 #A428; M R 33 #6153.

[106] J . Schauder . Se e Leray . [1071 L . Schwartz , Theorie des distributions. Tome s I , II , Actualite s Sci . Indust. ,

nos. 1091 , 1122 , Hermann, Paris , 1950 , 1951 . M R 12, 31 ; M R 12, 833 . [1081 G . Sestini , a ) Esistenza di una soluzione in problemi analoghi a quello di

Stefan, Rivist a Mat . Univ . Parm a 3(1952) , 3-23 > M R 14, 381 . b) Esistenza ed unicita nel problema di Stefan relativo a campi dotati di simmetria, Rivist a Mat . Univ* Parma 3(1952) , 103-113 . M R 14, 476 .

644 BIBLIOGRAPHY

[109] T . Shirota , a ) The initial value problem for linear differential equations with variable coefficients. I , H , III , Proc . Japan Acad . 33(1957) , 31-36 , 103-104, 457-461 . M R 21 #3665 ; M R 21 #3666 ; M R 21 #3667 . b) On Cauchy problem for linear partial differential equations with variable coefficients, Osak a Matfa. J. 9(1957) , 43-59 - M R 21 #3668 .

[110] G . £ . Silo v (se e als o Gel'fand) , On conditions of correctness of the Cauchy problem for systems of partial differential equations with constant coef-ficients, Uspefa i Mat . Nauk 10(1955) , no . 4(66)89-100 . (Russian ) MR 17, 495 .

[ i l l ] L . N. Slobodeckif (se e als o Babic) , a ) Generalized solutions of parabolic and elliptic systems, Izv . Akad . Nau k SSS R Ser . Mat . 21(1957) , 809-834 . (Russian) M R 21 #5084 . b) The estimation of solutions of elliptic and parabolic systems, Dokl . Akad Nauk SSS R 120(1958) , 468-471 . (Russian ) M R 21 #5060 . c) On the fundamental solution and the Cauchy problem for parabolic sys-tems, Mat . Sb. 46(88 ) (1958) , 229-258 ; Englis h transl. , Amer . Macb. Soc. Transl. (2 ) 4 1 (1964), 121-154 - M R 22 #830 . d) Estimates in Lp of solutions of elliptic systems, Dokl . Akad . Nauk SSSR 123(1958) , 616-619 . (Russian ) M R 21 #5061.

[112] V . h Smiraov , A course in higher mathematics. Vol . IV , OGIZ , Moscow, 1941; 2n d ed., GITTL , Moscow , 1951 ; Englis h transl. , Pergamo n Press , Oxford an d Addison-Wesley, Reading , Mass. , 1964 . M R 6, 42 ; MR 14, 145 ; MR 31 #1333* A course in higher mathematics. Vol * V , OGIZ , Moscow, 1947 ; rev . ed- , Fizmatgiz, Moscow , 1959; Englis h transl. , Pergamon . Press, Oxfor d and Addison-Wesley, Reading , Mass. , 1964 . M R 9, 574 ; M R 25 #5141 ; M R 29 #5964 .

[1131 S . L . Sobolev , a ) Applications of functional analysis in mathematical physics, Izdat . Leningrad . Gos . Univ. , Leningrad , 1950 ; Englis h transl. , Transl. Math . Monographs, vol . 7 , Amer . Math. Soc, Providence , R . I. , 1963. MR 14, 565 ; M R 29 #2624 . b) The equations of mathematical physics, OGIZ , Moscow, 1947 ; 2n d ed. , GITTL, Moscow, 1950 ; 4t h ed. , "Nauka" , Moscow , 1966 ; Englis h transl. , Partial differential equations of mathematical physics, Pergamo n Press , New Yor k an d Addison-Wesley, Reading , Mass. , 1964 . M R 13, 42 ; M R 31 #2478.

[1141 P . E . Sobolevski i (se e als o Krasnosel'ski i andKrein) , a ) On differential equations of parabolic type in a Hilbert space and on the approximate solu-tion of them by the Bubnov-Galerkin method* Candidate' s dissertation , Leningrad. Gos . Univ. , Leningrad , 1958 . (Russian ) b) Equations of parabolic type in a Banach space, Trud y Moskov. Mat. Obsc. 10(1961) , 297-350 ; Englis h transl. , Amer . Math . Soc Transl . (2 )

BIBLIOGRAPHY 645

49(1965), 1-62 . M R 25 #5297. c) Local and nonlocal existence theorems for nonlinear second-order para-bolic equations, Dokl . Akad . Nauk SSS R 136(1961) , 292-29 5 = Soviet Math. DokL 2(1961) , 63-<S6 . M R 25 #5291 .

[115] M. Z . Solomjak , a ) An application of the theory of semigroups to differen-tial equations in a Banach space, Dok L Akad . Nauk SSS R 122(1958) , 766 -769. (Russian ) M R 21 #3775. b) The analytic character of semigroups generated by elliptic operators in Lp, Dok L Aka d Nau k SSSR 127(1959) , 37-39 . (Russian ) M R 22 #12398. c) On a condition of Jo. £ . Lop at ins hit for the solvability of boundary value problems, Vesthi k Leningrad . Univ . 20(1965) , no . 1 , 143-144 . (Russian )

[116] V . A . Solonnikov (se e als o Golovki n an d V. P . II'in) , a ) On a priori esti-mates for certain boundary value problems, Dok L Akad . Nauk SSSR 13 8 (1961), 781-784* Soviet Math. DokL 2(1961) , 723-727 . M R 28 #2352. b) A priori estimate for equations of second order of parabolic type, Trud y Mat. Inst . Steklov . 70(1964) , 133-212 ; Englis h transL , Amer . Math. Soc . TransL (2) 65 (1967), 51-137 . M R 28 #5 267. c) On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov . 83(1965 ) = Proc . Steklov. Inst . Math . 83(1965), M R 35 #1965 . d) On general boundary value problems for systems of differential equa-tions of elliptic and parabolic types, Outline s Join t Sympos . Partia l Dif -ferential Equation s (Novosibirs k 1963) , Acad . Sci . USS R Siberian Branch , Moscow, 1963 , pp. 246-252 . M R 34 #4675. e) On boundary value problems for linear parabolic systems of differential equations of general form, Doctoral dissertation , Leningrad . Gos . Univ., , Leningrad, 1964 . (Russian )

[117] V . V . Stepanov, A course in differential equations, Fizmatgiz , Moscow , 1958; Germa n transL, 2n d rev. ed. , Hochschulbuche r fur.Mathematik , Ban d 20, VE B Deutscher Verlag , Berlin , 1963 - M R 31 #1418 .

[1181 S . Tacklind , Sur les classes quasianalytiques des solutions des equations aux derivees partielles du type parabolique, Nor d Acta Regia l Soc . Set . Uppsaliensis 4(1936) , no . 10 .

[ l l9 l A . N. Tihono v (se e als o Samarskii) , a ) A uniqueness theorem for the heat equation, Mat . Sb. 2(1935) , 199-216 . (Russian ) b) On the heat equation in several variables, Vesmi k Moskov . Gos . Univ . Ser, A . 1(1937) , no . 9 , 1-49 . (Russian ) c) On the third boundary value problem for an equation of parabolic type 9

Izv. Akad . Nauk SSSR Ser. Geograf . Geofiz . 14(1950) , 193-198 . (Rus -sian) M R 12, 29 .

646 BIBLIOGRAPHY

[1201 E * C Titchmarsh , An introduction to the theory of Fourier integrals, Claren -don Press, Oxford , 1937 ; 2n d ed., 1948 ; Russian transl., QTTL , Moscow, 1948 .

[ l2l l A . L . Treskunov . Se e Ivanov. N. S . Tr u dinger - se e [136] . [122] N . N. Uralcev a (se e als o Ivano v and Ladyzenskaja), a ) On the regularity

of solutions of multidimensional elliptic equations and variational problems, Dokl. Akad . NaukSSSR 130(1960) , 1206-120 9 = Soviet Math. DokL 1(1960) , 161-164. M R 23 #A4036. b) Boundary value problems for quasi-linear elliptic equations and systems with principal part in divergence form, Dok L Akad . Nau k SSSR 147(1962) , 313-316 « Sovie t Math . Dokl. 3(1962) , 1615-1618 , M R 26 #453-

[1231 T . D . Ventcel' , a ) The first boundary problem and the problem of Cauchy for quasi-line or parabolic equations with several space variables, Mat . Sb. 41(83) (1957) , 499-520 . (Russian ) M R 19, 657 . b) On certain quasi-linear parabolic systems, Dokl . Akad * Nauk SSSR 11 7 (1957), 21-24 . (Russian ) M R 21 #205 . c) On certain quasi-linear parabolic systems with increasing coefficients, DokL Akad . Nauk SSS R 14 0 (1961), 284-28 6 » Soviet Math . DokL 2(1961) , 1185-1187. (Russian ) d) Quasi-linear parabolic systems with increasing coefficients, Vestni k Moskov. Univ . Ser . I Mat. Men. 1963 , no . 6 , 34-44 . (Russian ) MR 28 #4252. T. D . Ventcel ' and O. A. Oleiaik , e ) Conchy's problem and the first boundary problem for a quasilinear equation of parabolic type, Dokl . Akad . Nauk SSSR 97 (1954), 605-608. (Russian ) M R 16, 259 -

, f ) The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type, Mat . Sb- 41(83 ) (1957) , 105-128 . (Russian) M R 19, 149*

[124] M . I. Visi k (se e als o Agranovic an d Barenblatt), a ) On strongly elliptic systems of differential equations, Dok L Akad . Nauk SSSR 74(1950), 8 8 1 -884; Mat . Sb. 29(71) (1951) , 615-676 . (Russian ) MR 13, 134; M R 14, 17 4 b) Mixed boundary-value problems for equations containing a first deriva-tive with respect to time and an approximate method for solving them, Dokl. Akad . Nauk SSS R 99(1954) , 189-192 . (Russian ) M R 17, 859-c) The Cauchy problem for equations with operator coefficients; mixed boundary value problem for systems of differential equations and approxi-mation methods for solving them, Mat . Sb. 39(81 ) (1956) , 51-148 ; Englis h transl., Amer . Math. Soc . Transl . (2 ) 24(1963) , 173-278 . M R 18, 215 . d) On boundary-value problems for quasi-linear parabolic systems of equations and on the Cauchy problem for hyperbolic equations, Dokl. Akad . Nauk SSSR 140(1961) , 998-1001 = Soviet Math . DokL 2(1961) , 1292-1295 -MR 28 #360 .

BIBLIOGRAPHY 647

e) On the solvability of boundary problems for quasilinear parabolic equa-tions of higher order, Mat . Sb. 59(101 ) (1962) , 289-325 ; Englis h transl. , Amen Math . Soc. Transl . (2) 65(1967) , 1-40 . M R 28 #361. M. I. Visi k an d O. A . Ladyzenskaja , f ) Boundary value problems for partial differential equations and certain classes of operator equations, Uspeh i Mat . Nauk 11(1956) , no . 6(72) , 41-97 ; Englis h transl. , Amer . Math. Soc. Transl . (2) 10(1958) , 223-281 . M R 20 #1091 ; M R 20 #1092.

[l25l L . R . Volevic' , a ) On general systems of differential equations, Dokl . Akad . Nauk SSS R 132(1960) , 20-2 3 = Soviet Math. Dokl. 1(1960) , 458-461 . MR 24 #A899 . b) Local properties of solutions of quasi-elliptic systems. Mat . Sb. 59(101 ) (1962), supplement , 3-52 . (Russian ) M R 27 #446 . c) On a problem of linear programming occurring in differential equations, Uspehi Mat . Nauk 18(1963) , DO . 3(111) , 155-162. (Russian ) M R 28 #5248 .

[12 61 N. D . Vvedenskaja , a ) An example of nonuniqueness of generalized solu-tions of quasi-linear systems of equations, Dokl . Akad . Nau k SSSR 136(1961), 532-533 = Soviet Madi. Dokl. 2(1961) , 89~9< 1 M R 23 #A1154. b) Some examples of parabolic systems of equations, Uspeh i Mat . Nauk 1 6 (1961), no . 5 , 218-219 . (Russian )

[127] R . Vybom y (se e als o Kadlec) , On the properties of the solutions of certain boundary value problems for equations of parabolic type t Ddk i Akad . Nau k SSSR 117(1957) , 563-565 . (Russian ) M R 23 #A3918.

[128] K . Yosida , a ) On the differentiability and the representation of one-para-meter semi-group of linear operators, J . Math . Soc. Japa n 1(1948) , 15—21 . MR 10 , 462 . b) On the integration of diffusion equations in Riemannian spaces, Ptoc . Amer. Math. Soc. 3(1952) , 864-879 . M R 14 , 56 a

[1291 T . J a. Zagorskii , The mixed problem for a system of partial differential

equations of parabolic type, L'vov , 1961 . (Russian )

[1301 Ju . V . Zel'dovi c an d A. S . Kompaneec , On the theory of propagation of heat with the heat conductivity depending upon the temperature. Collectio n in honor of th e seventieth birthda y o f Academician A . F . Ioffe , Izdat Akad . Nauk SSSR , Mosco w , 1950, pp . 61-71 . M R 16, 1029 .

[ l3l l A . Zygmun d (se e als o Calde ion), On a theorem of Uarcinkiewicz concern-ing interpolation of operations, J . Math . Pures Appl . (9 ) 35(1956) , 223 -248. M R 18, 321 .

References adde d b y the author s for th e Englis h translatio n [1321 D . G . Aronson , a ) On the Green's function for second order parabolic dif-

ferential equations with discontinuous coefficients, Bul L Amer . Math . Soc . 69(1963), 841-647 . M R 27 #5049 . b) Bounds for the fundamental solution of a parabolic equation (preprint) .

648 BIBLIOGRAPHY

c) On the initial value problem for parabolic systems of differential equa-tions, Bull . Amer . Math . Soc. 65(1959) , 310-318 . M R 22 #3888. D. G . Arooso n an d I. Serrin , d ) Local behavior of solutions of quasi-tin ear parabolic equations , Arch . Rationa l Mech . Anal . 25(1967) , 81—122.

[133] M . Kurihara, On UarnacVs inequality for parabolic partial differential equa-tions. I , II, Funkciala j Ekvacio j (Serio Japana ) 17(1965) , 3-57 ; 1 8 (1966), 3 — 32. (Japanese )

[1341 J . L . Lions , Sur certaines equations paraboliques non lineaires, Bul L Soc .

Math. Franc e 93(1965) , 155-175 - M R 33 #2966 . [1351 W . Porgorzelski, Proprietes des solutions du systeme parabolique d*equa-

tions aux derivees partielles, Bul L Acad . Polon . Sci . Ser . Sci . Math . Astr . Phys. 6(1958) , 369-373 - M R 21 #4301.

[136] N . S . Trudinge r 7Quasilinear elliptic partial differential equations in n variables, Doctora l Dissertation , Stanfor d Univ. , 1966 .

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

theory of differential equations , 1992 100 V . L. Popov, Groups, generators , syzygies, and orbit s in invariant theory , 1992 99 Nori o Shimakura, Partial differentia l operator s of elliptic type, 1992 98 V . A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications , 1992

(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

systems with impacts, 1991 88 A . G. Khovanskii, Fewnomials, 1991 87 Aleksand r Robertovich Kemer, Ideals of identities of associative algebras , 1991 86 V , M. Kadets and M. I. Kadets, Rearrangements o f series in Banach spaces, 1991 85 Miki o Ise and Masaru Takeuchi, Lie groups I , II , 1991 84 Da o Trong Thi and A. T. Fomenko, Minimal surfaces , stratifie d multivarifolds , an d the Plateau

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,

1990 79 N , I. Akhiezer, Elements o f the theory of elliptic functions, 199 0 78 A . V. Skorokhod, Asymptotic methods of the theory of stochastic differential equations , 1989 77 V . M. Filippov, Variational principle s for nonpotentia l operators , 198 9 76 Philli p A. Griffiths, Introductio n t o algebraic curves, 1989 75 B . S. Kashin and A. A. Saakyan, Orthogonal series , 1989 74 V . L Yudovich, The linearization metho d i n hydrodynamical stabilit y theory , 198 9 73 Yu . G. Reshetayak, Space mappings with bounded distortion , 198 9 72 A . V. Pogorelev, Bendings of surfaces an d stability of shells, 1988 71 A . S. Markus, Introduction t o the spectral theory of polynomial operato r pencils , 1988 70 N . I. Akhiezer, Lectures on integra l transforms , 198 8 69 V . N. Salii, Lattices with unique complements, 198 8 68 A . G. Postnikov, Introduction t o analytic number theory , 1988 67 A . G. Dragalin, Mathematical intuitionism : Introductio n t o proof theory , 1988

(See the AMS catalog for earlie r titles)