Some Applications of Weighted Sobolev Spaces

272 Pages·1987·18.135 MB·English

by  Anna-Margarete Sändig, Alois Kufner

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Kufner/Sandig
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Some Applications
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of Weighted
Sobolev Spaces
TEUBNER-TEXTE
zur Mathematik
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1 CiUtSWhiK-ThiAT
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" -TEXT
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Prof. RNDr. Alois Kufner, DrSc.
Born 1934 in Plzen. Director of the Mathematical Institute of the
Czechoslovak Academy of Sciences, Prague. Professor of Mathematics and
head of the Department of Mathematics of the Technical University,
Plzen.
Fields of research: Function spaces, partial differential equations.
Doz. Dr. Anna-Margarete SSndig
Born 1944 in Schwerin. Studied Mathematics in Rostock (1963 - 1968) and
Moscow (1968 - 1970). Received Dr. rer. nat. in 1973 and Dr. sc. nat.
in 1981. Associate Professor (Dozent) at the Wilhelm-Pieck-University
Rostock•
Fields of research: Elliptic differential equations - analytical and
numerical methods. v
Kufner, Alois
Some applications of
Anna-Margarete SSndig
(Teubner-Texte zur Ma
NE: Anna-Margarete SS
ISBN 3-322-00426-0
ISSN 0138-502X
® BSE -B. G. Teubner
1. Auflage
VLN 294-375/72/87 I
Lektor: Dr. rer. nat.
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02800
TEUBNER-TEXTE zur Mathematik • Band 100
Herausgeber/Editors: Beratende Herausgeber/Advisory Editors:
Herbert Kurke, Berlin Ruben Ambartzumian, Jerevan
Joseph Mecke, Jena David E. Edmunds, Brighton
Riidiger Thiele, Halle Alois Kufner, Prag
Hans Triebel, Jena Burkhard Monien, Paderborn
Gerd Wechsung, Jena Rolf J. Nessel, Aachen
Claudio Procesi, Rom
Kenji Ueno, Kyoto
Alois Kutner - Anna-Margarete Sandig
Some Applications of Weighted
Sobolev Spaces
This book is a free continuation of the book about weighted Sobo
lev spaces which appeared as Volume 31 of the series TEUBNER-TEXTE
zur Mathematik. It deals with some applications of these spaces
to the solution of boundary value problems. - Part one deals
with elliptic boundary value problems in domains whose boundaries
have conical corner points and edges; the weighted spaces make
it possible to describe in more detail the qualitative properties
of the solution including its regularity. One chapter is devoted
to the finite element method. - Part two deals mainly with
existence theorems for two types of boundary value problems:
elliptic problems with "bai behaving" right hand sides, and
equations which are degenerate-elliptic or whose coefficients
admit some singularities. It is shown how the weighted spaces
can be used to overcome these difficulties, ^lso nonlinear
problems are shortly dealt with.
1
Dieses Buch ist eine freie Fortsetzung des als Band 31 der
Reihe TEUBNER-TEXTE zur Mathematik erschienenen Buches tfber
gewichtete Sobolev-RSume. Es werden Anwendungen dieser RSume
zur LQsung von Randwertaufgaben behandelt. - Teil 1 ist ellipti-
schen Randwertproblemen auf Gebieten gewidmet, deren Rand koni-
sche Eckpunkte oder Kanten aufweist. Gewichtete ^Sume ermflgli
chen eine ausfUhrliche Beschreibung der qualitativen Eigenschaf-
ten der LSsungen bis zu Regularit^tsaussagen. Ein Kapitel ist der
Methode der finiten Elemente gewjdmet. - Teil 2 befaht sic*
hauptsachlich mit Existenzaussagen f'5r zwei "Typen von Randwert
problemen: ftfr elliptische Randwertpr obi erne, deren rechte Seiten
gewisse "schlechte" Eigenschaften haben kSnnen, und f*Jr Glei-
chungen, die ausarten oder deren Koeffizienten gewisse Singula-
ritSten aufweisen. Es wird gezeigt, wie man die entstehenden
Schwierigkeiten mit Kilfe gewichteter RSume iiberwinden kann. Es
werden auch kurz nichtlineare Probleme behandelt.
Ce volume represente une suite libre au livre sur les espaces
de Sobolev avec poids, paru comme volume 31 de la serie TEUBNET?-
TEXTE zur Mathematik. On considere ici les applications de ces
espaces a la resolution des problemes aux limites. - La premiere
partie est consacree aux problemes aux limites elliptiques
sur des domaines dont les frontieres contiennent des points angu-
laires coniques ou des aretes; les espaces avec poids permettent
de decrire en detail les proprietes qualitatives des solutions,
y compris leur regularity. Un chapitre est consacre a la methode
des elements finis. - La deuxieme partie s'occupe en principe des
theoremes d*existence pour deux types de problemes aux limites:
pour les problemes aux limites elliptiques dont les seconds
membres peuvent avoir certaines "mauvaises" proprietes et pour
les equations soit elliptiques-degenerees, soit celles dont les
coefficients presentent certaines singularites. On nontre comment
on peut surmonter les difficultes qui y surgissent a l'aide des
espaces avec poids. On traite aussi brievement des problemes
non-lineaires.
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2
CONTENTS
Preface 6
0. Preliminaries 8
Part one
Elliptic boundary value problems in non smooth domains 17
Chapter I
Elliptic boundary value problems in domains with conical points 18
Section 1 - Introducing examples 18
§ 1 - The Dirichlet problem for the Laplace operator 18
§ 2 - A mixed boundary value problem for the Laplace operator 25
§ 3 - The Dirichlet problem for the biharmonic operator 30
§ 4 - A Navier-Stokes equation 34
Section 2 - A special boundary value problem in an infinite cone K 35
§ 5 - Formulation of some boundary value problems 35
§ 6 - Solvability of the special problem in V*+2m»P(K,3) 38
§ 7 - Regularity and the expansion of the solution of the
special problem 42
§ 8 - A general boundary value problem in K 48
Section 3 - The boundary value problem in a bounded domain 51
§ 9 - Solvability in V£+2m»P(fi,e) and regularity 51
§ 10 - The expansion of the solution near a conical point 55
§ 11 - The case £ < 0 60
Section 4 - Calculation of the coefficients in the expansion 61
§ 12 - The coefficient formula for the special problem in an
infinite cone 62
§ 13 - The coefficient formula in a bounded domain 65
§ 14 - Examples 68
Chapter II
Finite element methods 71
Section 5 - Standard finite element methods in domain with conical
points 71
§ 15 - Weak solutions. Existence and uniqueness 72
§ 16 - Finite element spaces 74
§ 17 - Error estimates in W™' (ft) 77
§ 18 - Error estimates in Lp(ft) , 2 ^ p ^ » 8^
3
Section 6 - A Modified Finite Element Method in domains with conical
points 89
§ 19 - An iterative method 89
§ 20 - Dual Singular Function Method 94
Chapter III
Elliptic boundary value problems in domains with edges 97
Section 7 - A special boundary value problem in a dihedral angle 97
§ 21 - An introducing example - 97
§ 22 - Formulation of some boundary value problems 99
§ 23 - Solvability of the special problem in V£+2m,p(D,3) 103
§ 24 - Regularity of the special problem in a dihedral angle 106
§ 25 - General boundary value problem in D 107
Section 8 - Boundary value problem in a bounded domain 110
§ 26 - Solvability in V*+2m»P(G,K(.)) and regularity 110
§ 27 - The case £ < 0 113
§ 28 - Example 114
Section 9 Expansions near the edge 115
§ 29 - Definition of some function spaces 115
§ 30 - Expansions in a dihedral angle with and without tangential
smoothness conditions 118
-
§ 31
Expansions in a bounded domain 126
§ 32 - Example 128
Section 10 - Calculation of the coefficients 130
§ 33 - The coefficient formula in a dihedral angle 130
§ 34 - The coefficient formula in a bounded domain 139
Part two
Elliptic boundary value problems with "non regular" right hand sides
and coefficients 141
Chapter IV
Elliptic problems with "bad" right hand sides 141
Section 11 - The Dirichlet problem in spaces with power type weights 141
§ 35 - Bounds for the admissible powers 141
Section 12 - The Neumann problem 145
§ 36 - Formulation of the problem 145
§ 37 - Existence theorems 148
§ 38 - The case N - m < 2k 162
4
Section 13 - A modified concept of the weak solution 176
§ 39 Formulation of the problem 176
§ 40 - The Dirichlet problem 181
§ 41 - Power type weights. Other boundary value problems 194
Chapter V
Elliptic problems with "bad" coefficients 204
Section 14 - Singular and degenerate equations - a simple case 204
§ 42 - An example. Formulation of the problem 204
§ 43 - Existence theorem 208
§ 44 - Weakening conditions A. 1 - A.4 211
Section 15 - Singular and degenerate equations - a more complicated
case 222
§ 45 - Conditions on the coefficients 222
§ 46 - Existence theorem. Some generalizations. Examples 228
Section 16 - Strong singularities and strong degeneration 234
§ 47 - Modified spaces. Existence theorem 234
§ 48 - Examples. Remarks 238
Chapter VI
Nonlinear differential equations 243
Section 17 - Problems with "bad coefficients" 243
§ 49 - Formulation of the problem. Some auxiliary results 243
§ 50 - The main existence theorem 248
Section 18 - Elliptic boundary value problems 254
§ 51 Formulation and some existence results 254
References 261
Index 266
P R E F A CE
This book is in fact a free continuation of the book of the first author
Weighted Sobolev Spaces, which appeared in 1980 as Volume 31 of the series
TEUBNER-TEXTE zur Mathematik and, as the second edition, in Wiley & Sons Pu
blishing House in the year 1985 (in the sequel , this book is refered to as [I]).
In the above mentioned book some fundamental properties of Sobolev spaces
with weights were established. In a motivating introduction, several possibi
lities of application of these spaces were indicated : solution of boundary
value problems for partial differential equations with nonstandard domains
(i.e., domains with a more complicated geometrical structure) or nonstandard
differential operators (coefficients of the equation or of its right hand side
or of the boundary values make it impossible to use "current "methods). The
book [I] touched only briefly the possibilities of exploiting the weighted
spaces, and therefore, the present publication is an attempt to acquaint an
interested reader in a little wider framework with the possibilities which
the weighted spaces offer when applied to the solution of boundary value
problems.
This book was written by two authors and consists of two parts. Both parts
are self-contained and can be studied independently. Let us briefly mention
their contents.
Part One, whose author is A.-M. SANDIG, concerns the first of the
above mentioned domains of practicability. Here elliptic boundary value pro
blems for domains with conical corners and with edges are studied. In this case
the weight functions make it possible to describe in more detail the qualita
tive properties of the solution, first of all as concerns its regularity. This
field, in which a pioneering work was done by V. A. KONDRAT'EV in the sixties,
has attracted the interest of quite a number of authors, concerning analytical
as well as numerical methods. The application of weighted spaces assumes here
a very immediate character also in numerical methods, which is demonstrated
by a modification of the popular finite element method. The account presented
in this book is an attempt to give a survey of analytical results of V. G.
MAZ'JA and B. A. PLAMENEVSKII and of their application in the finite element
methods. It was especially the last field to which the author herself has con
tributed by her own results. The restriction to two types of "singular boun-
6
daries" - that is, corners and edges - is caused by the technical diffi
culties with which the investigation meets; in a book of the given extent and
destination it was not possible to present many further existing results.
Part Two is devoted to rather more theoretical applications , namely
to existence theorems for elliptic differential equations ( in this aspect it
is tied up with [I], where these problems were studied for the Dirichlet pro
blem) , and further for problems of the type of degenerate equations and equa
tions with singular coefficients. The aim is to show that even here the weigh
ted spaces can provide a useful tool enlarging the scope of boundary value
problems solvable by functional-analytical methods. The author, A. KUFNER,
included in it primarily the results he has lately obtained together with his
colleagues.
The authors do hope that the book will arouse the reader's interest in
weighted spaces and convince him (at least a little) of the usefulness of these
mathematical objects. They welcome any comments which could help them to im
prove further work in the field , and they use the opportunity to extend their
thanks to all who in any way took part in the preparation of this book. Among
them, at least four names should be mentioned explicitly : Dr. Jifi JARNlK who
improved the authors' English, Dr. Jifi RAKOSNlK who drew the figures , Mrs.
Ruzena PACHTOVA who carefully typed the manuscript, and Dr. Renate MULLER from
the TEUBNER Publishing House who by her patient support has eventually succee
ded in making the authors complete the text.
Prague/Rostock 1984 - 1987 A.-M. S.
A. K.
7
0. P r e l i m i n a r i es
0.1. THE DOMAIN OF DEFINITION. In what follows we shall work with functions
u = u(x) defined on an (in general arbitrary) measurable set
fiCRN •
In most cases ft will be a domain, i.e. an open and connected set and we will
suppose that the boundary 8ft of ft will satisfy certain regularity conditions.
Mainly we will work with domains of the class
what means that the boundary can be locally described by a Lipschitz-continuous
function of N - 1 variables (for details, see [I], Chapter 4, or A. KUFNER,
0. JOHN, S. FUCIK [1], Sections 5.5.6 and 6.2.2). Such a boundary can contain
conical -points or edges.
Let us give two typical examples of domains considered in Part one of
this book.
«N
0.2. EXAMPLES. (i) Let ft be a domain in IR with one or more conical
points 0. (i = 1, .,s) on 3ft . Here, 0 6 8ft is a conical point if there
exists a neighbourhood U of 0 such that U O ft is diffeomorphic to a cone
with vertex at 0 . For a more detailed explanation see Subsection 5.1 (ii).
(ii) Let ft be a domain with an M . This means that M is a
smooth (N-2)-dimensional manifold on
8ft which divides 8ft into tw odis
joint parts T and T . E.g., an
infinite roof can serve for 8ft , M
being the ridge of the roof, or the
figure from the Fig. 0.
For u = u(x) with x e ft and
for a = (a ,...,a ) a multiindex,
we will denote by
Fig. 0
V...+aN ax aN
the derivative 8 u/8X]L ...8xN of u in the sense of distributions.
0.3. CLASSICAL S0B0LEV SPACES. (i) For 1 < p ^ « , let