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Applied Mathematical Sciences I Volume 52
Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-Linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions.
15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 19. Marsden/McCracken: The Hopf Bifurcation and its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. I1. 25. Davies: Integral Transforms and Their Applications. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory.
(continued after Index)
M. Chipot
Variational Inequalities and Flow in Porous Media
Springer-Verlag New York Berlin Heidelberg Tokyo
M. Chipot Department of Mathematics University of Nancy I B.P. 239-54506 Vandoeuvre
Cedex France
AMS Classification: 76505, 49A29
Library of Congress Cataloging in Publication Data Chipot, M. (Michel) Variational inequalities and flow in porous media. (Applied mathematical sciences ; v. 52) Bibliography: p. Includes index. 1. Fluid dynamics. 2. Porous materials. 3. Variational inequalities (Mathematics) I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 52. QA1.A647 vol. 52 [QA911] 510s [532'.051]
84-5598
With 13 Illustrations
© 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, NY, 10010, U.S.A. Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America. 9 8 7 6 5 4 3 2 1
ISBN 0-387-96002-3 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-96002-3 Springer-Verlag Berlin Heidelberg New York Tokyo
Preface
These notes are the contents of a one semester graduate course which I taught at Brown University during the academic year 1981-1982.
They are
mainly concerned with regularity theory for obstacle problems, and with the dam problem, which, in the rectangular case, is one of the most interesting applications of Variational Inequalities with an obstacle. Very little background is needed to read these notes.
The main re-
sults of functional analysis which are used here are recalled in the text. The goal of the two first chapters is to introduce the notion of Variational Inequality and give some applications from physical mathematics. The third chapter is concerned with a regularity theory for the obstacle problems.
These problems have now invaded a large domain of applied
mathematics including optimal control theory and mechanics, and a collection of regularity results available seems to be timely.
Roughly speaking,
for elliptic variational inequalities of second order we prove that the solution has as much regularity as the obstacle(s).
We combine here the
theory for one or two obstacles in a unified way, and one of our hopes is that the reader will enjoy the wide diversity of techniques used in this approach.
The fourth chapter is concerned with the dam problem.
This problem
has been intensively studied during the past decade (see the books of Baiocchi-Capelo and Kinderlehrer-Stampacchia in the references).
The
relationship with Variational Inequalities has already been quoted above. Starting with a new point of view introduced by Brezis-KinderlehrerStampacchia, and in a different setting by Alt, we develop the theory for general domains, and with relatively elementary techniques we give the main results on the subject including a first study of the free boundary (i.e., the one limiting the wet set of the porous medium considered). v
vi
I am especially indebted to H. Brezis from whom I learned most of this subject.
I wish also to express my thanks to my colleagues in the Applied Mathematics Division of Brown University and particularly to Constantine Dafermos and Jack Hale who helped to create such a friendly atmosphere in the Division.
Thomas Sideris and Jalal Shatah helped me to correct the manuscript; I thank them very much for their friendly assistance. Finally, I thank Roberta Weller and Katherine MacDougall for the excellent typing of the manuscript. Michel Chipot Providence, R.I. February 1982
Table of Contents Page
PREFACE CHAPTER 1.
v
ABSTRACT EXISTENCE AND UNIQUENESS RESULTS FOR SOLUTIONS OF VARIATIONAL INEQUALITIES 1.1. 1.2. 1.3.
CHAPTER 2.
EXAMPLES AND APPLICATIONS 2.1. 2.2. 2.3. 2.4. 2.5. 2.6.
CHAPTER 3.
CHAPTER 4.
Fixed Point Theorems Motivation Existence and Uniqueness Results Comments
1 1 1
3
9
10
Some Functional Analysis The Dirichlet Problem The Obstacle Problem Elastic Plastic Torsion Problems Nonlinear Operators Fourth Order Variational Inequalities Comments
10
A REGULARITY THEORY
22
THE OBSTACLE PROBLEMS:
14
16 17 18 20 21
3.1. 3.2.
Monotonicity Results The Penalty Method
22 24
3.3.
W2'p(O)-Regularity (2 < p
= 0 (
,
Vv E K
(1.4)
denoting the duality between
X',X), we can prove that
u
satis-
fies:
u E K,
D(0) Hence,
0 < lim 0(t)
t
'D (0)
= f (u) + > 0
Vv E K.
Thus V.I.'s arise naturally in variational problems on convex sets.
In
the next paragraph, we shall investigate more closely problems like (1.5). More precisely, let
be an element of
f
X', A
:
K + X'.
We shall study
problems of the type: u E K,
>
(1.6)
V v E K.
Before solving (1.6), let us quote a special case where Proposition 1.3 arises, and moreover, where we know that a minimum is achieved. Let
K
be a nonempty closed convex set in a Hilbert space
the inner product
(
,
and let
)
well known that there exists a unique f
on
K, such that with u E K,
Moreover, u
be an element of
f
u
1lu-f112 < ilv-f112
H
with
Then it is
K, called the projection of
in
Dull = (v,v) 1/2
H.
we have
Y v E K.
is the unique element such that (apply, for instance, Proposi-
tion 1.3) u E K,
(u,v-u) > (f,v-u)
Vv E K
(1.7)
and thus projection on a closed convex set in Hilbert spaces provide us with a very large class of V.I. holds (see corollary 1.13). map
1.3.
where
f + u
u
Note also that in this case uniqueness
In the sequel, we shall denote by
PK
the
the
is the solution of (1.7).
Existence and Uniqueness Results 1.3.1.
Let
X
The finite dimensional case. be a finite dimensional space, X'
pairing between Theorem 1.4.
Let
K
be a nonempty compact convex subset of
a continuous mapping of exists a solution u E K,
its dual, and
X',X.
u
K
into
X'.
Then for every
f E X'
X
and
A
there
of the problem:
>
Vv E K.
(1.8)
1.
4
ABSTRACT EXISTENCE AND UNIQUENESS RESULTS
Choose a Euclidean structure on
Proof:
and denote by
j
X' - X
:
<x',x> = (j(x'),x) If
PK
i.e., an inner product
X
the mapping defined by Vx E X.
(1.8)
denotes the projection of
X
on
for the above Euclidean
K
structure, then the mapping
x H PK(x - j(Ax - f)) is continuous from
into
K
and has a fixed point
K
u
which satisfies
(see 1.7))
u E K,
(u,v - u) > Cu - j(Au - f),v - u)
and so by (1.8) Remark 1.5.
u
Vv E K,
satisfies (1.6).
Note that in the case where
necessarily have a
K
is unbounded (1.6) doesn't
Indeed, choose for example
solution.
X = X' =7R = K
with the pairing <x',x> = x'x
For
A
.
: ]R -r (0,+-), the problem
Au-(v - u) > 0 is equivalent to where
K
Vv EIR
Au = 0, which doesn't have a solution.
Thus, in the case
is unbounded some assumptions must be added.
To study the case when
K
is unbounded, we make the following defini-
tion:
be a convex unbounded subset of a Banach space
Definition 1.6.
Let
K
with dual space
X'.
We shall say that
if there exists
v0 E K
,
>
X
K
such that
/I1v-v0jj - +(
Since the map
x H Ax - f
can solve (1.6) in the case assume
Then for every
f = 0, and let
BR
f = 0
Vv E K.
(1.10)
is continuous and coercive, if we then general case will follow.
denote the closed ball of center
0
Thus
and radius
Existence and Uniqueness Results
1.3.
R
(for the norm
X
in
uR E K Choose
n BR,
II
X), uR
in
II
5
> 0
R > IIvOII
with
Vv E K n BR.
as in (1.9).
v0
the solution of (1.10)
Then we have by (1.10)
> 0.
(1.11)
Moreover,
- +
R
< - + IIAvOII'IIuR - v01 IIuR - v0II(-/IIuR - v0II + IIAvOII') where
II
for all
denotes the strong dual norm in
II'
we may choose
R
coercivesness of A
R
X'.
Now if
IIuRII = R
big enough so that the above inequality and
imply
< 0 which contradicts (1.11).
Now for every
on
v
So there exists an we can choose
K
uR + e(v - u ) E K n BR
> 0 > 0
1.3.2.
A in
IIuRII < R.
uR
Vv E K
is the solution of (1.6).
The infinite dimensional case.
In this part we shall denote by dual,
such that
small enough such that
Vv E K
R
which proves that
R
and thus by (1.10)
R
Gei
e > 0
the pairing between
will be a mapping of
K
and
X'
into
a reflexive Banach space, X'
X
X, K
and
X'
II
a closed convex set of II
its X.
will denote the norm
X.
Let us give some definitions. Definition 1.8.
We shall say that
> 0 and
A
A
is monotone if
Vv,u E K
(1.12)
is strictly monotone if equality holds in (1.12) only for
Definition 1.9.
We shall say that
A
is continuous on finite dimensional
subspaces if for every finite dimensional subspaces A :
K n m -- X'
v = u.
M
of
X
the mapping
is weakly continuous, that is to say, if for all
x E X,
ABSTRACT EXISTENCE AND UNIQUENESS RESULTS
1.
6
v H
is continuous on
K fl M.
With these definitions we now can state Theorem 1.10.
Let
be a nonempty closed convex subset of
K
flexive Banach space.
Let
f E X'
and
A
K + X'
:
continuous on finite dimensional subspaces. K
(i)
is bounded,
there exists a solution
u
K
of the V.I.
>
u E K,
Moreover, if A
Then, if
or
is coercive on
(ii) A
X, a re-
be a monotone map
Vv E K.
(1.6)
is strictly monotone, the solution of (1.6) is unique.
Before giving a proof of this theorem, let us note the following useful lemma.
Minty's Lemma. space
Let
A
and
X
K
be a nonempty closed convex subset of a Banach
a monotone operator from
ous on finite dimensional subspaces of
X.
K
into
Then for
u E K, >
Vv E K
u E K, >
Vv E K.
which is continu-
X'
f E X'
a.e
For (b) it suffices to remark that by monotonicity
Proof:
> > For (a) replace
v
in (1.13) with
Vv E K.
u + t(v - u)
for
t E (0,1].
Then
we have > But
is strictly positive, hence
t
> Now using the continuity of ting
Vv E K.
t
A
Vv E K.
on finite dimensional subspaces and let-
go to zero, we obtain (1.6).
We are now able to prove Theorem 1.10. Proof of Theorem 1.10: assume
Consider case (i).
Without loss of generality we
f = 0.
Consider C(v) = {u E KI > 0}. It is easy to see that bounded.
So
C(v)
C(v)
is a weakly closed subset of
is weakly compact, and if
K
which is
1.3.
Existence and Uniqueness Results
n vEK
7
C(v) = 0
we can find
in
vl,...,vn
K
such that
C(vl) n C(v2) n ... n C(vn) _ 0.
But now consider M i
(1.14)
the subspace of X
the canonical injection of M -+ X
spanned by
and by
vl ... vn.
Denote by
the "restriction" mapping
r
defined by
Vx' E V.
r(x') = x'IM
By applying the result of Theorem 1.4 and Minty's Lemma to the operator
v -+ r-A°i(v) we have that there exists a solution of the problem
uE KOM, > 0 Of course such a
u
belongs to
tradiction to (1.14), and thus
VvE KOM. C(vl) n ... fl
C(v)
which gives a con-
fl C(vn)
The theorem in case (i),
0.
vEK now follows from another application of Minty's Lemma.
For (ii) the proof
now follows easily as in Theorem 1.7. To prove uniqueness in the case of strictly monotone operators, it suffices to note that if u1 E K,
ul,u2
are two solutions of (1.6), then we have
>
Vv E K
I
u2 E K,
Thus setting
> v = u2
Vv E K.
in the first inequality and
in the second
v = uI
and adding leads to < 0.
By strict monotonicity this gives
ul = u2
and the result.
As a very useful corollary, let us note (compare with (1.2)): Corollary 1.11.
A
Let
X
be a reflexive Banach space, X'
be a monotone, coercive operator of
on finite dimensional subspaces of f E X'
X.
X
into
Then
A
X'
its dual.
is onto i.e., for all
one can solve the equation:
Au = f. If
A
Let
which is continuous
is strictly monotone, the solution of (1.15) is unique.
(1.15)
ABSTRACT EXISTENCE AND UNIQUENESS RESULTS
1.
8
By Theorem 1.10, we deduce that there exists
Proof:
u E X,
>
Taking v= u ± w, w E X =
u
such that
Vv E X.
we have
Vw E X
and thus (1.15) and the theorem follows. In view of this corollary, the coerciveness assumption seems
Remark 1.12.
Indeed, if
more natural.
is clear that for
Axx x
IR, it
IxI + +m
ATi
=
into
to be onto one needs that
A
when
IAxI + +'o
is a monotone mapping from IR
A
+m when
IxI + +10
In the particular case of Hilbert spaces, let us note also: Corollary 1.13. and
a(u,v)
H
Let
a(u,u) > viIuiI2 Let
K
satisfying for some
u
Denote by
A
v - VO
and
f E H'.
Vv E K.
Then
K = H, u
is the unique solution of (1.15)'
such that
Vv E H.
v,v0 E K
we have a(v-v0,v-v0)
V - v0
v - v0
-
IIV V0ll ++W
and the result follows from Theorem 1.10. (1.15)' follows immediately from Corollary 1.11. Remark 1.14. Theorem.
)
(1.6)'
the linear operator of H + H'
Then by (1.16), for all
H
Vv E H.
a(u,v) =
,
v > 0
such that
a(u,v - u) >
a(u,v) =
(
(1.16)
be a nonempty closed convex subset of
Moreover, in the case
Proof:
H
Vu E H.
there exists a unique u E K,
be a Hilbert space with an inner product
a bilinear continuous form on
The second part of this corollary is known as Lax-Milgram
Comments for Chapter 1
9
Comments
The first results about Variational inequalities were given by Stampacchia [99] and Lions-Stampacchia [86], see also Fichera [58] and Hartmann-Stampacchia [68].
The presentation that we have adopted here
borrows widely from Kinderlehrer-Stampacchia [76] (see also Brezis [27] and the interesting introductions given in Baiocchi [14], Baiocchi-Capelo [15] and Kinderlehrer [71]).
Some generalizations that will not be used here are possible. is, for instance, to find
One
such that
+ (v) - (u) >
u E K,
where
u
Vv E K
is a convex, weakly lower semi-continuous function (see Moreau
$
[90], Brezis [27], Lions [83]). depend on
Another one is to allow the set
K
to
u; the problem is then called a quasi-variational inequality
(see Bensoussan-Lions [18], Lions [84], Tartar [105], Baiocchi-Capelo [15]).
See also Lions [83] and Brezis [27] for a generalization of
Corollary 1.11 and Mosco [91] for a different point of view of variational inequalities.
Chapter 2
Examples and Applications
2.1.
Some Functional Analysis Let
c2
By
be a bounded connected open subset of IRn.
LP(c2), we denote the usual spaces of "equivalence classes" of
real functions whose p-th power is integrable (1 < p < +-). denote the space of functions which are essentially bounded.
will The norms
on these spaces will be
1/p
(
Jul
p
= (J lu(x) IPdx)
(1 < p < +.o) (2.1)
Jul_ = ess sup lu(x)I. xEQ
For
k
a positive integer, we shall denote by
space defined for every
Wk'P(c) = {u E LP (c2) where in (2.2)
Dau
Wk'P(Q), the Sobolev
1 < p < +- by IDa'u E LP(0)
V lal < k)
denotes the derivative
(2.2)
Blal
in the distri-
axI1...2xn1
butional sense (jai = al + ... + an). It is well known that these spaces are Banach spaces with the norm
Ilullk,P
(2.3)
Ia en> -
Vv E K (Resp. v E
K*
Since (3.14) in case (ii),
26
THE OBSTACLE PROBLEMS:
3.
By the monotonicity of >
-
Vt > 0, we would obtain the
of:
under the assumption that K = {v E H0'(2)lv(x) < i(x) Remark 3.6.
a.e.} , 0.
It is easy to see that the above proof also holds for a non-
linear operator
A
from
H1(H)
into
H-1(0)
which satisfies the assump-
tions of Minty's Lemma and the coercivity property > vllu - vuI1,2
Vu,v E H1(92).
In the next part, we will use
Remark 3.7.
uE
to denote the solution of
a slightly more complicated problem, that is to say 0
-Au. +
(uE
+
e 2
(uE-4)
+
E HO(B).
With the same proof as above, one can easily see that Theorem 3.4 holds without any change for such a
uE.
3.3. W2'p(S2)-Regularity (2 < p < +W).
27
W2'P(S2)-Regularity (2 < p < +.o),
3.3.
In this part, we shall assume that aij
are sufficiently smooth.
any
2 < p < +.o
ax.
1
uE is in
when
r
the boundary of
SZ
and the
More precisely, we shall assume that for
f E LP(53), the solution of the Dirichlet problem f
(aij axj)
(3.17)
H0 (S3)
and that we have the estimate
W2'P(0)
I1u1I2,p < CPIfIp
(see [4), [19]) where
(3.18)
is a constant which doesn't depend on
Cp
Now let ,J be two functions in H1(f) $(x) < V(x)
a.e.
in
u
or
f.
such that (3.19)
S2
and such that < 0
0.
*)aij 3x1(uE J
Moreover, 91
is negative, so is
v, and
dx.
f
(3.27)
J
Finally, we note that
v
0
only when
ue < $ < *
and hence when
92 E
(ue-V) = 0.
From this it follows that
(!e (u£-#) +
02(u e-V')) v dx (3.28)
Is£(uE-O)Ip.
= Jo
Combining (3.25)-(3.28), we get I8E
J (f -
By applying Holder's Inequality, we get with
q = p/p-l
(-A$)+IplvIq
ICE (uE-t)Ip t If -
But recalling that
Iviq = ISl(uE
v =
ISl(uC_$)Ip 2.
l(uewe have
Ip-1
-
01
and (3.23) follows.
Now if a (ue-p)
the truncated function if
ItI
is not bounded, it is clear that
F (t) = F(t) if n > n, leads to a v = Fn(S1(ue-$))
ItI < n, F (t) = Sign n
which is in
H0((2).
The result
3.3. W2'p ((2) -Regularity (2 < p < +)
29
t in then follows after an easy passage to the lim!2(,,
Finally, to ob-
n.
v = F(E i))
tain (3.24), it is enough to consider
(or
s
v = Fn(e (uE - )) which concludes the proof.
Assuming that
Remark 3.9. as
p .+ +-
EL
((2)
f E L%1),#,* E W2'w(f2)
in (3.23), (3.24) leads to the fact that
and taking the limit £1(uE-+), s?(uE-1P)
with
l- (uE a) l , _ I f -
I
< I f + (-AP)
I
IW.
(3.29)
For the obstacle problems, we claim now that under the assumption (3.18), we have:
Theorem 3.10:
solution
uI
be in
Let
measure such that
with
H1(()
(-A$)+ E LP((2)
and if
on
f E LP(S2)
If A is a
r.
(p > 2), then the
of
u1 E KV > is in
0 < 0
Vv E K0
and there exists a constant
W2'P(SZ)
Cp
(3.30)
independent of ul,f,$
such that
IIulII2,P
Theorem 3.11. (3.21).
(3.31)
< Cp(IfIp + I(-A$)+Ip). Let
Then for
O,ip
be in
f E LP((2)
H1(0)
and satisfy (3.19), (3.20) and
the solution
u2 E KV, > is in
of the problem
Vv E K*
and there exists a constant
W2'P(c)
u2
CP
(3.32)
independent of
u2,f,$,ip
such that (3.33)
Cp(Iflp + I(-Ab)+Ip + I(-AMY) Ip)
IIu2II2,P
0
81(t) < 0,
uE
From (3.18), (3.22),
Vt > 0.
(3.23), and (3.24), we get Iluell2,p
Since
Remark 3.12.
fied for
a -
when
ue - u2
When the
0
a..
il
(by Theorem 3.4), the result follows as above.
are smooth, the above assumptions are satis-
f E LP(S2),0,* E W2'P(c2).
Remark 3.13.
(3.35)
Cp(iflp + I(-A$)+IP + 1(-A*) Ip)
n, by the Sobolev embedding theorem, we get that
u 1
and
u2
are in
with
Cl'a(S2)
a = 1 - P.
Note that in general ui f C2(0)
(see below).
For optimal results of regularity see [29].
Remark 3.14.
Estimates (3.34), (3.35) could be used in place of (3.15) to
go to the limit in (3.22).
Some Complementary Results
3.4.
In the unidimensional case, the solution of (3.30) or (3.32) is always continuous since
H1(52) a C(SC).
In higher dimension, under the assumptions
on the previous part, we can prove:
Theorem 3.15.
Assume that
such that the solution tion
u1
u
and
¢ E C(S2)
If f E H(S2) is
of (3.17) is continuous on
S2, then the solu-
of
ul E KV > is continuous in Proof:
K4 # 0.
(3.36)
SZ.
(See [84].)
smooth functions
Vv E K
0n
Assume first that such that
0n < $
f = 0
and
and choose a sequence of $n i $
in
L%2).
be the sequence defined by un E Ken, > 0 By Theorem 3.10, we have
Iun So
- ull
> 0
k
we have
uE > , uE > 4
and so
a1(u6-#) + a2(uE-*) > 0.
Now we deduce from the above inequality (see Proposition 3.2) that The inequality Aue E L (52).
uE > -k
uE
u
< k. e
We thus have 3.53 and
Moreover, we have:
Proposition 3.17. solution
follows in a similar way.
If
,ip E W2'w(l)
of (3.47) is such that
satisfy (3.19) and (3.20), then the
Aue E L%)
and satisfies the
estimate:
IAuEI., < Max[I
(-A*)
(3.54)
W2'p(1)-Regularity
3.5.
35
f = e(ue-$) + e(uwe obtain,
Applying the Remark 3.9 with
Proof:
since
and
a1 < 0
s2 > 0
Se
101(u6
(-A0)+I., I f
I., < Max[ I f -
+ (-A,) I.,]
< Max[1(-A0)+1.,,1(-A1P)
Ifl_.
Now the result follows from (3.47) since by (3.53) we have
Ifl
2.
Boundary Estimate
First let us note that since deduce from (3.46), (3.47) that
uE E W2'p(52)
(see Remark 3.18) we
ue E W4'p(1) (p > n).
This will allow
us in the sequel to apply the maximum principle on the second derivatives of
uE
in the sense of Bony [25].
and use an approximation argument.)
(One could choose at first smooth data
36
THE OBSTACLE PROBLEMS:
3.
Now
u
e
is at least in
A REGULARITY THEORY
so the following proposition makes
C2(S2)
sense:
Under the assumptions of the Proposition 3.17, if uE
Proposition 3.21.
denotes the solution of (3.47) there exists a constant depend on
or on $, such that for all
c E (0,1]
C
which doesn't
i,j = 1,...,n
we
have 2 a
u
i
E
ax ax j
Yx E r.
C(I*I2,m + IIkHI2,m)
(x)
(3.57)
We will use here ideas from H. Brezis - D.,Kinderlehrer [34] and
Proof:
R. Jensen [69]. Step 1:
We will first show that there is no loss of
x0 E r.
Let
is flat at
aij's satisfy
generality in assuming that
r
some additional properties.
(The arguments below are classical in bound-
and that the
x0
ary value problems and to follow the idea of the proof instead of technical tools the reader is invited to go directly to (3.67) and (3.68).) Since
is smooth, there exists a diffeomorphism
r
from a neighborhood centered at
of
V
onto
0 = (61162,...,6n)
an open ball of IRn
B
and
6(x0) = 0
such that
0
in in
x0
0(SZ n v) = B+ = {x = (X1,.... Xn) E BIXn > 0}
0(r n v) = a0B+ = {X = (X1,.... Xn) E BIXn = 0}.
For each function on
defined on
v
v n 52, we define a function
(and conversely) by setting
B+ U a0B+
v' (X) = v(6-1(X)) - v(x) = v'(6(x)). So we have (setting n
ax. (x) _
i
v'
k=1
k
axiaxj(x) =
in place of
(3.59)
(6 (x)) . axk (x)
i
a2u,
Be
axit
n
aek
x(x)
axj(x)
axk(6(x))'
k,t=l
uE)
a6
,
aX
n
a2u
u
(3.58)
(3.60)
i
aXk(6(x)).
+ kIl
axiax.(x). 7
2
Since 8X'
and
aXaX
are given by a similar formula it results are bounded on
from (3.56) that the derivatives 2X
i
B+
by the quantity
'a
3.5.
W
(s)-Regularity
37
Moreover, from this bound and (3.59) (3,60) we
C(110112,- + 1I'P112,a).
easily see that to have (3.57) in a neighborhood of
in
x0
it is
r fl v
2
a u ax.ax.
enough for
to satisfy an inequality of type (3.57) with 30B+.
in place of formation
on a neighborhood of we have to study
0
u'
0
Thus, via this trans-
in
satisfying (see (3.52), (3.59), (3.60))
with the summation convention: 2
aij BXiaxj + bk aXk =
a1('-$') + a2(u'-,p')
on
B+ (3.61)
I& = 0 on
a0B+
the coefficients (X) a!
being given by (see (3.60))
aij
ao. ae.
n
.
13
k,k= l (akR axk ax9)(e
Replacing
(a!. + a!.)/2
by
a!.
(3.62)
(x))
it is clear that there is no loss of
generality in assuming that in (3.61) we have (3.63)
Vi,j = 1,...,n.
aij =
(Note also that from (3.62) we can deduce that (3.1) holds for some
in place of
v'
T = (T1,...ITn)
v.)
with
aij
Now let us perform a second transformation
defined by (see [61], [69]). if
x = (X1....,xn) _ (x,xn),
Ti(x) = X. + xn$i(x)
n
i
Tn(x) = xn where
is given by
ai
fi(x) =
(3.64)
ani(x'0)/ann(X,0).
An easy computation shows that the Jacobian of T a neighborhood of maps
into
B+
0, T
B+.
at
is
0
is an isomorphism which preserves
1
and so in and
B0B+
Because of this second transformation, and with ob-
vious notations for functions the study (in some ball
AB+
v"
(see (3.58)), we are now concerned with
(a < 1) still denoted by
B+) of
u"
the
solution of
a'..
a2ulf
i3 axiaxj
+ b" LL LL = a (uif- fit) + a (urt-prr) k aXk
1
2
on
B+
(3.65) lull
= 0
a0B+
on
38
THE OBSTACLE PROBLEMS:
3.
A REGULARITY THEORY
aj = ani, but now (see (3.62), (3.64)) for all
with (see (3.62), (3.63)) i = 1,...,n-1 n
ain = ani
aT
BT.
annbi = 0
axX = ani +
ak1C' aXk
k,t=1
on
BOB+
Rewriting the first equality in (3.65) as a
aXi
(a
+ (b -
au'l )
ij aXj
aXi
k
aXk
+a
(u"-®"")
2
1
be the solution of
w
and letting
Balik) au It = a
a' .
ij aX) = (bk
ax.
B+
aXk)3Xk on
(3.66)
w=0
aB+,
on
u* = u" - w, * = V - w, j* = 1" - w
we see by setting
l(u* - *) + a2(u* -*) on
aX(sij aX.) = 1
u* = 0
that we have
B
30B+
on
Now from (3.56), (3.59), (3.60) it follows that the right-hand side of the first equation in (3.66) is bounded in quantity we have
W
C(110I12,- + II'II2 .,), and so on w E W3'p I_
I IwI
W2' if p > n
12,,,_ C(11#112,- +
1,p
(B+), for all
or on some
B+
p, by the AB+ (A < 1)
and an estimation of type
I I*112,,,)
Thus to prove (3.57) we have only to prove it on a neighborhood of
in
in place of
and with
a0B+
u,b,i.
0
in
Reverting to our notation
u, we have proved that in place of (3.52) we can assume that
u = uE
satisfies
as
(aij 2-U-
= a1(u-$) + a2(u-*)
on
B+
az)
(3.67) 1
with
u = 0
on
a.. = a.. 13
31
30B+
and
ain = ani = 0 $,i
being in
on
W2,w(B+)
(3.56) now being true on
30 B+
Vi = 1,...,n-1,
(3.68)
satisfying always (3.19), (3.20) and (3.55), B+.
W2,w
3.5.
(S2) -Regularity
39
The fact that (3.55) is preserved by the different trans-
Remark 3.22.
formation results from (3.56), (3.59), (3.60), (3.62). the fact
u E W4'p
Note also that
is preserved since by (3.46), (3.64), ToB
is of class
C4.
We deal now with
Step 2:
u = 0
First since
on
satisfying (3.67), and (3.68) holding.
u
we have:
a0B+
2
on
= 0
aa.u
aOB+
Vi,j = 1,...,n-1.
J
Using the first equation in (3.67) and (3.68) we get (see (3.19), (3.48), (3.49), (3.50)) 2
as
a
Yx E a0B+.
ann(x) Z(x) + axn WD u (x) = n ax n
(3.69)
n
(i.e., ann > v > 0) and from (3.56)
A
And thus by the ellipticity of E E (0,1]
we get for 2
()
0.
Q+ > 0)
ai (u - ) (A+u - A+$) + aZ (u - S) (A+u - A+W) (x+) < 0
and by (3.51) we get A+u(x+) < A+t(x+)
or
which clearly leads to (3.73).
and
al (u - $)(x< 0
and
since in this case
Q
A-u(x)
].
(3.78)
We have indeed from the definition of x+,x
Bu + Au(x+) = A+u(x+) > A+u(x) = Bu + Au(x ) Bu - Au(x-) = A u(x-) > A u(x+) = Bu - Au(x+). Adding these inequalities, we get [al(u-0) + a2(u-*)](x+) = Au(x+) > Au(x ) = [a
1(u-#)
+ a2(u-V')](x )
42
THE OBSTACLE PROBLEMS:
3.
A REGULARITY THEORY
But if (3.78) holds the above left hand side is strictly negative (see (3.48), (3.49)) and the right side is strictly positive.
This is a con-
The inequality in the
tradiction and hence (3.72) holds in all cases.
other side follows in the same way; note for instance that tion of a problem of the same type as (3.71) with
is a solu-
-u
in place of
Let us now return to (3.67).
With an eye to deriving the analogs of (3.75), (3.76), set 2
Bu =
where
k
ax ax + c ax , k n k
is a positive constant satisfying
c
as c
nn
Applying
(3.79)
axnn > n
al
Au
to
B
we easily obtain aa..
B(Au) = A(Bu) +
axi
B(aij)axj
xk
+
aaij
+ axn
Now, let
ax n
axj
(3.80)
a2u axkaxj)
be a smooth function such that
a
0 < a < 1
2
on
a = 1
B+
in a neighborhood of
0
(3.81) 30B+
a = 0
on
BB+ - B0B+
= 0
on
n
ax
(One of the roles of this function is to avoid the problems created by the fact that
u
is not prescribed on a part of the boundary of
Setting for
(with the summation convention in
i = 1,...,n aaiJ .
au
Si = B(aij)ax.
+ axk
and multiplying (3.80) by
B+.)
2
j)
2
aai] .
a u axnaxj + axn
a u axkaxj
a, we get after an easy computation:
a -Si) + as (CFSj - 2aij Bx3 Bu)
oB(Au) = A(ciBu) +
ax
x
(3.82)
af. = A(aBu) + f0 + ax.
i
with obvious definition for the
f.'s. 1
3.5.
W2''A-Regularity
43
Now let us temporarily assume that we have proved the following lemma: Lemma 3. in
If the
denote the functions of (3.82), there exists a
fi
W2'p(B+) (p > n)
w
which is a unique solution of the problem
af.
-Aw = f0 + x1
on
B+
(3.83)
i
w = 0
on
BB+-BOB+,
= 0
a,D-
on
90B+
n
Moreover, there exists a constant b,lp
which doesn't depend on
C
e E (0,1],
such that:
i lwl lW_ C(I 1+112,., + I III 12,1,).
(3.84)
Now from (3.82) we deduce A(aBu - w) = a8(Au).
Using (3.67) to compute
we get
B(Au)
A(aBu-w) = a1(u-d)(oBu-aB+) + a2(u-1P)(aBu-a8P) (3.85)
+ al(u-0)a as (u-$)BX (u-0) +
On the other hand, applying
axk
n
k
A
to
Au
ax (u-V+)
n
given by (3.67) leads to
A(Au) = ai(u-f)(Au-A+) + a2 (u-*)(Au-A*) (3.86)
+ a1(u-#) aij
Multiplying (3.85) by
Xi(u-$)aa
(u4) + a2(u-*)a ij
v, adding and subtracting
A(Au)
give us now the
following equations to which we will apply the maximum principle: A(V(aBu-w) + Au) = al'(u-O)(A+u-A+*) + a21(u-')(A+u-A+*)
+ al(u-$)Q+(u-0) + a2"(u-4V)Q+(u-'P) (3.87)
A(v(OBu-w) - Au) = ai(u-+)(A_u-A_+) + a21(u-4)(A-u-A-*)
+a
(u-0) + a2 (u-1P)Q _
In these equalities we have set by definition:
A+v = vaBv ± Av Q+v = va L Vaxk
By '
+ a.. av av i7 azi axj
44
THE OBSTACLE PROBLEMS:
3.
A REGULARITY THEORY
We shall note that by (3.1), (3.81) we have: av axk
8z. >_ vI0v12 > vv
aij
X.
i
av axn
7
is positive and
and thus, Q+
Let us now denote by (Resp. v(oBu - w)
Q-
is negative.
(Resp. x) a point where
x+
achieves its maximum in
- Au)
B+.
v(QBu - w) + Au We want to prove
that 2
-
aXkax (x) k n
c(i1*1
By the definition of
Vx E B.
2,m +
x+,x
(3.89)
it suffices to (see (3.55), (3.56), (3.84))
,
prove that (v(UBu-w) + Au)(x+)
(v(1Bu-w) - Au)(x-)
or
(3.90)
< c(I I01I2,m + I I1I I2,,,) Different cases are possible: x+ E BB+ - 90B+.
Case 0:
since in this case
We have then, by definition of
v(x+) = 0
and
x+
and
w(x+) = 0:
(v (QBu - w) + Au) (x+) = Au (x+) , Clearly the case
and by (3.55) we get (3.90).
x
E aB+ - a0B+
follows
in the same way. +
Case 1:
(see Case 1 of the heuristic proof).
E a B
x
0
then (recall that
We have
v(a'Bu-w) + Au E W2'p(B+), (p > n))
axn(v (QBu - w) + Au) (x+) < 0. Using (3.67), (3.81), (3.83), the above inequality can be rewritten as
[va.
But on
ax
n
Bu + ai(u-O)aX (u-0) + az(u-*)aX (u-V)](x+) < 0. we have
9OB+
n
n
u = 0
and thus, ai(u $)(x+) =
Combined with the definition of 3
a u2 Lvcr
axk axn
e
this leads to
B
2
+ vac ax k
+ E 8x (2u - $ -
ax
n
0.
n
To estimate the third derivative of
u
the derivative of (3.69) in the
direction and we get
xk
in the above inequality we take
3.5.
W2,cQM -Regularity
3 a
45
2
aann
u
a
u
_
ann axkaxn E axkaxn
2
3a
22u
nn axk
a ann 8u 8xk8xn axn -
_
axn
ann, which is greater or equal to
Dividing by
8
e aXk($ + )
v, and replacing
a3u 2 axkaxn
by its value in our inequality above leads after using (3.56), (3.70) to as
vc7(c
where
C
"Q ..
2
- al
axnn) axk axn (x+) n nn
doesn't depend on u
k
n
(x+)
< C0101 2,W +
e E (0,1].
- C(I l,I
I
But now from (3.79) we deduce that
I'PI 12,00).
Clearly the same argument holds if
and (3.90) follows easily.
x
E a0B+
and thus we can assume that we are in the: Case 2.
x+
and
are in
x
Let us assume first that
B+.
x+ E [u > #].
Then we have (see (3.48),
(3.49))
and
a1" (u - $)(x+) = 0
(3.91)
az(u - i)(x+) > 0.
Now the maximum principle in
W2'p(n) (p > n)
(see [25]) leads to
if A(v(aBu - w) + Au)(x) < 0.
x-'x But from (3.91), (3.87) and the fact that
ess l im inf [ai (u-$) (A+u-A+m) + az
Q+
is positive we get
(A+u-A+V') ] (x) < 0.
From (3.51) one easily deduces that this implies
A+u(x+) < IA+,IW
or
IA+V'I..
Using now (3.84) we get (3.90).
The case
x
E [u < V']
follows in
the same way, see Case 2 in the heuristic proof, and the case where x+ E [u < 4]
and
x
E [u > ]
is impossible.
To see this, just add the
two inequalities v(ciBu-w)(x+) + Au(x+) > v(aBu-w)(x-) + Au(x-)
V(aBu-w)(x) - Au(x) > V(aBu-w)(x+) + Au(x+) to get a contradiction.
(As after 3.78) thus (3.90) holds in all cases.
46
THE OBSTACLE PROBLEMS:
3.
The lower bound can be obtained by noting that
-u
satisfies a prob-
-m, 4.
lem of the form (3.67) with now m, changed to
By (3.89) the
and a compactness argument
x0
estimate (3.57) holds in a neighborhood of
A REGULARITY THEORY
Thus it remains only to prove Lemma 3.
shows that (3.57) is true globally. Proof of Lemma 3.
Step 4:
can be written as
f0 + afi/axi
First we remark that agi go + ax.
where the bounded in
gi's are of the same type as the
(i.e., in
fi
W2'p(B+)
and
C(Iloll2,- + 11*112,,,) -- see (3.82),
by the quantity
Lp(B+)
(3.55) but also with gn(x) = 0
on
(3.92)
a0B+.
in (3.82)) in
fn
But (see the form of
can be written like
afn/axn
Clearly, it is enough to check that this.
afn/axn
only the terms
2
2
and
(a axiax.
5
# n
j
n
To see that they can be suitably rewritten,
may not be of the above type. note that we have: a2u
a
ax
92U
a
n
axn
n
aann
1
(a 2) = ax a(ax
n
+ a
nn
(a
au ax + e4 + CC) n
n
as ax
as an (nn n
nn
2
au a _ a u axi axn(a xj) axn(B axiaxj) a
3u
+ et + EVp)
n
au as a axn (axi ax
as (3.69) allows us to conclude.
We then set for
denoting the sign of
x = (x,xn) E B, sign(xn)
(i.e., ±1),
in(x) = ani(x) =
sign(xn)ain(x,Ixn1)
9ij(x) = aji(x) = aij(x,Ixn1)
Vi = 1,...,n-1
for all pair
(i,j) # (k,n)
(k = 1,...,n-1)
gi(x) = gi(x,lxnl)
Vi = 0,...,n-1,
gn(x) = sign(xn)'gn(x,Ixnl)
xn
W
3.5.
2,w
(I)-Regularity
47
Now it is easy to check that the
iij(x)
satisfy the ellipticity
condition (3.1) and thus, there exists an unique solution
w E H1(B)
of
the problem
x.(aij 1,_
= 0
on
axj) = g0 +
B
P.
The gi's are in
W1'p(B+)
g0 + agi/axi E LP(B)
thus
on
z1 gi
and the
gi's are in
w(x,xn) = w(x,-xn)
tions are solutions of the above problem. on
(see (3.92)),
and (see [4], [65]) w E W2'p(B) a C1(B).
over, it is easy to check that
a (x) = 0
W1'p(B)
More-
since these two func-
Hence
a0B+
n
and the restriction of w
to
results simply from the theorem Remark 3.22.
satisfies (3.83).
B+
In the case where
The estimate (3.84)
of the appendix combined with (3.55).
Al
on
= 0
$ =
r
this proposition is
simpler to prove using an unpublished result of H. Brezis.
(See [51].)
We can now conclude the estimates of the second derivatives of u
e
by:
3.5.3.
Global Estimates.
Proposition 3.23.
If
$,V'
W2'-(S2)
satisfy (3.19), (3.20), and under
the assumptions of Proposition 3.17, then the solution in
W2'"(l)
e E (0,1],$,*
1 lu6112,Proof:
and there exists a constant such that
C(1 ICI 12,E + I ICI
(3.93)
First, let us estimate the derivative
in place of
Bu =
of (3.47) is
uE
which doesn't depend on
C
ague/axkaxt.
Set (with
ue)
a2u axkaxR
We have
aa.. B(Au) = A(Bu) + axi(B(aij)axj
By introducing the solution
2
as J
2
axk]
axjaxR + axR
+
w
of
axjaxk
u
48
3.
Aw
_
au (B(aij)ax
a
axi
THE OBSTACLE PROBLEMS:
2
aai.
a
u
a2u
aai
+ axk ax.ax
A REGULARITY THEORY
+ axk ax.axk)
w=0 on r the above inequality becomes A(Bu - w) = B(Au)
with, moreover, (see the Theorem AI of the Appendix and (3.55))
11w11-
C(11#112,,, + I I*I I2,o)
Now computing
(3.94)
by replacing Au by its value (see (3.52))
B(Au)
we get
A(Bu - w) = ai(u-0)(Bu-B0) + +
a2(u-Vi)(Bu-BW)
(u-$)ax (u-*) +
z(u
ax
R
k
Multiplying the two sides of this equality by
ax
k
t
v, then adding and
subtracting (3.86) we get the following two equations: A(v(Bu-w) + Au) = al(u-CQ+(u-$) + a2(u-lP)Q+(u-'P)
A(v(Bu-w) - Au) = al(u-O)(A_u-A_t) + a2(u-*)(A_u-A_Vp)
+ al(u-#)Q_(u-$) + aZ
_
where we have set A+v = vBv ± Av av
Q+v = v axk (Note that
of the sign
aij
av
av
av
axR ± aij axi axj
av Lv_
av
axi axj ± v axk axR
v(Iovl
av
2
av
± axk axt) > 0
Q+
is
±.)
The proof is now as in the preceding proposition. we denote by
and
x+
(Resp. x) a point in
v(Bu - w) - Au) achieves its maximum.
where
S2
If
x+
then by (3.55), (3.57) and the definition of
That is to say,
v(Bu - w) + Au
(or x) belongs to x+
(Resp. r,
we have
v(Bu-w) (x) + Au(x) < v(Bu-w) (x+) + Au(x+) < C(l I+I 12,,, + I hPI ! 2,,,) , VxESZ.
3,5.
2,w
W,
(Std-Regularity
49
Applying (3.55) and (3.94) we get
1B-(-) I < C( I 1$ 12,,, + I ['P112,.)' Thus we can assume
x+,x
(3.95)
E Q. but applying verbatim (with
B = 52,
a = 1) the arguments of Case 2 of the previous proposition leads again to (3.95) which completes the proof (for a variant of this technique, see
[52]). As a consequence for the obstacle problems we now have: Under the assumptions (3.46) if
(See [34] and [69]).
Theorem 3.24.
0 E W2'p(S2), 0 < 0
on
P, f E Wl'p(a), p > n
then the solution
u1
of
the problem >
u
1
EK
0
is in W2(St) u1,0,f
Vv E K0
= {v E Hl0 (S2) v(x) > 0(x)
a. e. in
and there exists a constant
C
S2)
which doesn't depend on
such that (3.96)
11u11 12,- < C(IIfIIl,p + 11+112,,,).
Under the assumptions (3.46) if
Theorem 3.25.
(See [51].)
0 < 0 < *
r, f E W1'p(S2), p > n, then the solution
on
{ >
u is in
2
a
W2'(S2)
u2,0,'P,f
0,i, E W2'w(S2),
of the problem
Vv E KV
(x) > v(x) > 0(x) a.e.
E K* _ {v E H1(52) 0
u2
and there exists a constant
C
in
S2)
which doesn't depend on
such that
I1.2112,,, < C(I Ifl Il,p + 11+112,., + I1*112,m) Proof:
w
Let
ue
be the solution of (3.16).
By introducing the solution
of
-Aw = f,
w E H1(S2)
we easily see that of
(3.97)
0,*.
ue - w
satisfies (3.47) with
Thus from the Theorem 3.23 we get that
0 - w, 'P - w
in place
uc - w E W2'-'(1)
and
by (3.93) I Iue - wl12,W < C(IIO-wl
1'-w112,.)
12,-- + I
By regularity results for the Dirichlet problem (see [4]) and the Sobolev
50
A REGULARITY THEORY
THE OBSTACLE PROBLEMS:
3.
embedding theorem (see [1])
11-1 12,-
C'1lfl Il,p
C1 1-113,p
and the above inequality leads now to
(3.98)
I IuEI12,., < C(l Ifl1l,P + 1101 12,- + I I*I Under the assumptions (3.48), (3.49) taking
s(t) < o l
Vt < o,
8 2(t)
= 0,
v+ = 110112,,
then we have that (see Proposition 3.4 and Remark 3.7) uE + u1
as
e + 0
and a standard argument gives us Theorem 3.24, (3.96) resulting from Taking
(3.98).
Vt < 0,
a1(t) < 0
Vt > 0
02(t) > 0
leads to Theorem 3.25.
W1'°(S2)-Regularity
3.6.
First let us take a look at the case of an operator with constant In this case the translation method works and we are even
coefficients.
able to get
here that
0 < *.
Let $, be two functions in Assume that
we have:
H0I(S2) n C0'a(37)
is a constant and
u
Indeed, assuming
0 < a < 1.
is a bounded Lipschitz domain of IRn
n
Theorem 3.26. with
regularity for all
CO'a(S2)
u
(0 < a < 1)
is the solution of
V..V(v-u)dx >
u E f 12
J
S1
and we have
u E CO'a(S2)
lula, < Max(IOla,I*1,,) where IvIa
=
Iv(x) - v(y) I
sup
yIa
x#y
Ix -
(x,y)E52 Proof:
(See H. Brezis - N. Sibony [36] and P. Hartman - G. Stampacchia
[68].)
For
outside
S.
v E H1(1) for
let us denote by
v E C0'a(52)
1"v(x + h) -v(x)I < Ivlalhla Now fix
h EIRn
v
the extension of
v
by
0
we have clearly
and consider for
Vx,h EIRn. C = Max(lO1a,1v,la)
(3.100)
the functions
3.6. W1'p(S2) -Regularity
51
v(x) = Max(u(x),u(x+h) - CIhIa), By (3.100) we have in
w(x) = Min (u(x),ii (x-h) + CIhIa)
IRn
$(x) < u(x) < v(x) < Max(i(x),i(x+h) - CIhIa) = (x) $(x) < Min($(x),+(x-h) + CIhIa) < w(x) < u(x) < (x) and so the restrictions to and
of
52
v(x) = u(x) + (u(x+h) - u(x) - CIhIa)+
w(x) = u(x) - (u(x-h) - u(x) + CIhIa)
are in
0.
Applying (3.99)
we get IRn
u(x) - CIhIa)+dx > fe u(u(x+h) - u(x)
fe
-
CIhIa)+dx
u(x) + CIhIa) dx > fin -1(u(x-h) - u(x) + CIhIa) dx.
Now changing
x
into
x + h
in the integrals of the
second inequality
gives us
e
Vu(x+h)-V-(u(x) - u(x+h) + CIhIa) dx
fIR
> -f
n
}i(u(x) - u(x+h) + CIhIa) dx.
]R
Thus adding with the first one, using that
fn V(u(x) -
(-f)
= f+, leads to
u(x) - CIhIa)+dx > 0
IR
tia
CIhIa)+I2
fn IV(u(x+h) - u(x) -
< 0
(Vh EIRn)
IR
So for almost all
x
we have
u(x + h) - u(x) < CIhIa
Vh EIRn.
That is to say, after possibly changing
lu(x) - u(y)I
Remark 3.27.
x
and
on a set of measure
y
CIx - yla,
gives the result, i.e., Vx,y E N.
The above method works also, of course, for inequalities of
type u E K*,
0
Vx,y EIRn.
u(x) - u(y) < Clx - yla,
Now a permutation of
u
>
Vv E 0
52
THE OBSTACLE PROBLEMS:
3.
is an operator with constant coefficients and
A
where
A REGULARITY THEORY
(Indeed, if
able space.
problem to the case
is in a suit-
f
is smooth enough one can easily reduce the
S2
f = 0.)
to give a well-known application of this re-
f = u
We now choose
Let us denote by
sult to the elastic-plastic torsion problem.
the
6
Euclidean distance to the boundary, i.e., 6(x) = Infjx - yj yEP
with
for the usual Euclidean norm in IRn.
I
I
Vx E 52
Then under the previous assumptions upon Theorem 3.28.
If
we have:
12
is a positive constant the problems
.i
u E K = {v E H(12)Iv(x) < 6(x)
in
a. e.
S2}
(3.101)
Vv E K
v(v-u)dx J52
1S2
and
u E K' = {v E
H1(c2) I I Ov(x)
in
a. e.
kpD-(P+l) = Max(IIoo+II,0,1lV
IIm)
(3.110)
VxE52. Thus, if (x,0)
and if
x E S2
meets
y
denotes the first point where the segment
we have (note that
r
6+
is radial) 4+(x)-O+(r)=0+(x)
d+(x) > d+(x)
-
6+(y) > Max(IIO4+II.,,IIOV+ IiIx-yj >
Hence if (3.110) holds we have in < b+ < d+
and
6
a
kplxl-(P+2)Iv(P+2)
- M]
(3.112)
55
W1'"(Sd)-Regularity
3.6.
where
M
is some constant which bounds from above on
the quantity
SZ
n E
i,j=1
[6ijaij(x)!!a(x) + _x.]I. 1
Assuming M
is greater than
2v, we can clearly choose
k,p
such that
(see (3.110))
v(p+2) - M = 0,
kpD-(P+1)
= Max(IIO++II.0,1 vg
From (3.111), (3.112) we then deduce in
+ s +
I,.).
(3.113)
(see (3.9), (3.10))
0
s -A6+
-A6
S2
I
- (6 -b)
SE
(6
+
!2
-A6+ > 0 = -Aue + !-,(u e-
e
s2
-A6- < 0 = -Aue +
Hence using (2.15), (2.16) and test functions
(ue
(ue-1P)
(ue - 6 )
one deduces from these inequalities that
< u(x) < 6+ (x)
6 -( X )
Vx E S2
and thus IDuE(x0)1 < lV6+(x0)I = kpR (P+1). By (3.113) this finishes the proof.
Let us now deduce global estimates for Proposition 3.32. if
u
E
:
Under the assumptions of the previous proposition and
is the solution of (3.105) there exists a constant
ue
pends only on
52,v,R
and the norms
Ilaijlll,.o
C
which de-
(in particular not on
E)
such that: (3.114)
Iv.110, I Io*I I.,)
I IVuel 1p
First let us prove that we have
Proof:
loo+I
where
C
(3.115)
I.,, I IV i I
I IVu6112
T} k
S22 = {x E 52I aX < -T}. k Moreover, in au
521,522
vl = axk - T,
respectively, let us set: au
v2 = - xk - T.
We thus obtain two positive functions in fy (see (3.117) and note that
Si > 0)
HO(521)
and
H1(c2)
which satis-
3.6.
Wl'W (0)-Regularity
57
as
-Avl = -Au - T)
-A(8x
k A(
-Av2 =
k
ax(a i
)
au aa axkl
auk - T) = -A(- axk)
ax.(
But now we can apply the Corollary A2
in
S2
k
(3.119)
aX)
of the Appendix.
in
522.
More precisely,
if n = 2, using (iii) of the Corollary A2 from (3.115), (3.119) it re-
vi E L(c)
sults that
for all
(i = 1,2) and that
q
Iviiq,s2i < C Max(IIV II,,,IIo*II.,) denotes the
(I
Iq,S2
depend on
Lq-norm in
By dividing
E.)
provides us with au E Lq(1)
52
and
SZi
into
S21,S22
some constant which doesn't
C
and
this
{x E SZI Iau I < T}
and an estimate
k
auax
k
for all
-< C Max(I Iaol I, , I IV*1 L,),
q
q > 2.
choosing
q > n
But now since this estimate holds clearly for all
Thus with the same decomposition of au axk
k,
we can apply the point (i) of Corollary A2 to (3.119). 0
as above we now get
< C Max(IIo0IIm,1IVOI.)
which is (3.114). In the case
n = 1, by (3.115) we are in case (i) of Corollary A2 and
one can conclude as above (of course, in this case one can provide a shorter proof).
vi E L
If now
n > 2, (3.115) and (ii) of Corollary A2 show that
and an estimate
(SZ.)
IviI2*,s2 < C Max(IIooII,,,IIopjL).
i
But the decomposition of
S1
into
5211522
and
{x E lI I aX I < T}
k-
combined
with the above inequality provide us now with: Du axk
Thus if viously.
2* < C Max(IIv$II.0.IIvl)II.) 2* > n If
we can apply (i) of Corollary A2 to conclude as pre-
2* < n, and since the above inequality holds for all
get by applying the point (ii) of the Corollary A2 with
I Uk p
* < C.Max(I IooIIm,I IVpII.,)
p < 2*:
k, we
58
p < 2*.
for all n
THE OBSTACLE PROBLEMS:
3.
for some
p*
A REGULARITY THEORY
Repeating the procedure, it is clear that we can exceed and the result is now an application of (i) of Corollary
A2 which concludes the proof. Because of (3.106) and the estimate (see (3.115))
Remark 3.33.
IVuII2 < C Max(IIoo+II2'IIo I12) one might expect that an estimate of type
JIVueI Itl < C Max(I Ioo+I I.o, I IVI-I I, )
(3.120)
holds (this would lead to a similar estimate for the solution of the homogeneous obstacle problems).
But this is not the case in general (see Re-
mark 3.36).
Note that with the same proof as in (3.53) we have
u£ I .o < Max (I c+ I
.,,
I 'P
I.,) .
As a consequence of the above estimates, we are now able to prove for the obstacle problems:
Let
Theorem 3.34.
(52)
Vv E KIP
and there exists a constant
C
which doesn't depend on
f, ",u2 such that
IVu2I I® < C(I flp + I ldol l + ! IVPI IW). Proof:
For
81.82
(3.122)
satisfying (3.9), (3.10) and 4, satisfying (3.19),
(3.20) let us consider the solution
u6
of:
WJ,W(Q)-Regularity
3.6.
-AuE +
59
e (°E - O) +
uE E H (S2)
SE (ue - V+)
=f
.
Introducing the solution
of
w
w E H1
0(52)
uE - w
we see that
satisfies (3.105) i.e.,:
A(uE - w) =ES ((uE - w) - ($
w) ) +
s((uE -w) -
w) )
E
uE - w E H1(S2) and by Proposition 3.32 we get
< C Max(IIV(e - w)II.,,IIV(* -
I lv(uE - w)II
But by (3.18) and the Sobolev embedding theorems we have
1IOwII. < Ciiui'2,p `- CIfip
and thus we get easily
IlvuCIL < C(Iflp + iIveli + where when
doesn't depend on
C
(3.123)
Taking now
E.
0, , = IOIm, pl < 0
s2
we get (3.121) from (3.123) by applying Theorem 3.4 and by
t < 0
taking the limit of (3.105) when 3.35 follow by choosing
c
0.
when
01 < 0
Similarly (3.122) and the Theorem
t < 0, S2 > 0
when
t > 0
and by
passing to the limit in (3.123). Remark 3.36.
where
ul
It is clear from (3.121) that the mapping
+ + u1($) = ul
is the solution of
ul E KO, > 0
_ (v E H01 (S2)Iv(x) > 4(x)
Vv E K
a.e. on Wl'm
is continuous at
On + 0
in
0
W1'W (52)
on
(3.124) S2}
W1'm
(0)
- i.e., if n E
then ul (+n) * 0
in H (ft) n
is not generally continuous in W(S2)
when
[30] we easily see that in ]Rn (n > 1) and for
(S2), On < 0 W1'm
n > 1.
(S2) .
But
on
r,
+ + ul (#)
Indeed following
60
THE OBSTACLE PROBLEMS:
3.
SZ= {xE]Rnl
A REGULARITY THEORY
lxi 0)
lxl
the solution of (3.124) is given by (use similar arguments that those right after (3.45))
if r
Vv E K
(3.126)
3.7. Wl'PCS2)-Regularity (2 0
> 0 > 0 v
equal to
and
T($') > O'
thus
T(0') + (0-0')>0,
and the above functions are all in
with
f = 0.
T($') +
HI(S2).
Thus using the inequalities
Vv E KO Vv E KOI
in the first and
v = T(0) + (0'
-0)
in the second one leads to > 0 . 1
1
we deduce:
vllun-ul1I1,2 < + - -+ 0 with
n
and the result follows.
Remark 3.44.
As a consequence of Theorem 3.42 and under the same assump-
tions we obtain also (see [24])
ul($n) -r u1(0) for all
q
in Wl,q(p)
satisfying
strong
2 < q < p.
Indeed, as a consequence of Holder's inequality, we have
u(0))IIq < IIV(ul(@n) -
< CIIO(u1n) - ul(4))II2 with
r =
2(p-q) q(p-2)
and
s =
p(q-2) q(p-2)
uI(O))IIP (by (3.127))
Appendix
LP-Estimates for the Solution of the Dirichlet Problem
In this section ® will be a bounded open set of IRn
m L (®)
be functions in
vj with
V > 0.
f0,...,fn
a..
will
1J
Vx E® , VE E,n
2
(A. 1)
A will be the operator defined by (3.2). Under the above assumptions, if
(Stampacchia [100]).
Theorem AI.
and
satisfying the usual coerciveness assumption
Lp(®)
are functions in
with
p > 2
and
u
the solution of
the Dirichlet problem n
2f
-Au = f0 +
.
in
2x1
i=1
1
(A.2)
Then
Le(®) (i)
If
u E
p > n, we have
and
u
satisfies an estimate
of type n jul.. < C
Ifil
E
i=0 (ii)
If
2 < p < n, then we have
by P =
and
-
P
n
u
u E L
(®)
where
P*
is defined
satisfies an estimate of type
n
Ifilp
lulp* I 0 i= 0 (iii)
If
p = n = 2, then we have
67
u E Lq(®)
for all
q
and
APPENDIX
68
n
lulq < C lIo 1fil2' The constants of the
f i
;
First, let us assume that
Proof:
u
and
LP(Q).
LP-norms in
denotes
lp
i
in (i), (ii), and (iii) are independent of
C
p = 2.
Then from (A.1), (A.2), and
(2.7), we deduce that
2
n
t
r vllvul12 < J
0
a..u u ij x) xi
f u - f.u 0 1 xi
e
< C l IfiJ I1Du(A.3) 2112' i=0
Hence, it follows that n iiVu112 < v
(A.4)
lfil2
I
'
i=0
Thus, if
p = 2 = n, (iii) follows immediately from Sobolev's inequality
(see for instance [82]).
Now if
n = 1, &= (x0,x1), we have
lu(x)i < fx lu'(t)ldt < lu'l2ioil/2
-
x0
and this combined with (A.4) gives
n
nn
lulp < C
E i= 0
n
ifil2 < C'
L
i =0
which proves (i) in this case.
ifilp
Thus we now assume
n > 2
and that we are
not in the case (iii). k > 0, E
Let us consider, for
E= (u-k)+- (u+k)
defined by
.
As the sum of two functions of H1('), it is clear (see Theorem 2.4) that
E
equal to
is a function in H(&) which ±(lul
- k)
on
Jul > k.
vanishes on
Let us denote by
Jul < k A(k)
and is
the set
A(k) _ {x E Ol lu(x) l > k}. An application of the same arguments as in (A.3) with Holder's inequality leads to
viivEii2
k, we have
p.
p' = n,
which is the case (iii).) and so for such a
A(h) c A(k)
h, we have k)P'*j1P
(h-k)IA(h)l"p '* < Lj
(Jul
-
A(h) (Jul
< If
-
2
n
k)P ICI
A (k)
* P
C I If i=0
I
1 P
IA(k)IP,
1
This can be written as
n /(h-k))p*
IA(h)I < (C
n
i0
Ifil
IA(k)IP
'*( 2r
-1) (A.7)
P
Let us now use the following lemma which will be proved later on. Let : [0,+.0) -+]R+
Lemma.
$(h) < (h'k)a'$(k)o where
c,a,S
(i)
If
be a nonincreasing function satisfying
h>k
are positive constants.
0 > 1, we have (d) = 0
(A.8) Then
for
d = c. (0)S-1/a
2S/S-l
70
APPENDIX
/2 1/1-S
a < 1, we have
if
(ii)
(h)
h
u
cl
l
with
u = a/1 - s
for
\
h > 0.
More precisely, if we put (t) = IA(t)l, then we have by (A.7) an inequalNow if
ity of type (A.8).
is greater than
(1 - P)/(1 - P - n)
(d) = 0
that
p > n, the exponent
0-1
E
IfilP 'I ®Ipi*
and by (i) of the lemma we deduce
1
a.e.) with
Jul < d
(i.e.
n
1) =
0 =
0 25-1
d =
C
i=0
Now if
Hence (i) results.
p < n, we have
and by the
$ = p'*(P, - 1) < 1
part (ii) of the lemma and since
p'*
1-P
=
-
1
=
1
1
-l) p,-*-P,+1 P, - n - P, + 1 pn
2 (__
1*
2
1
1
1
2
1
1
= P,
we deduce that
n C 11Olfilp P* I[Jul > h] I
(A.9)
h
i = 0,...,n
Consider now for
the mapping
Ti: f -+ ui = Ti(f)
where
ui
is the solution of
-Au. = of 1 axi
ui E H1(®) (with
of
Then by (A.9) for all
f).
-gx
2 < q < n, Ti
is a linear map
0
from
Lq(B)
Lq (®)-weak.
into
for instance, 2 < p < q < n,
Choose,
into L2*(®) and Lq(&) into Lq*(&)-weak.
then Ti maps L2(Q)
Therefore by the Marcinkiewicz Theorem (see [20], p. 6) T.
maps
LP(®)
*
into
LP (B)
continuously and we have by (A.9)
ITi(f)Ip* < Clflp n Since now
u =
T(f.), (ii) follows.
I
1
i=0
What remains now is to prove
1
the lemma.
Proof of the Lemma:
By (A.8), we have for
hn = d - n (n E AI) 2
(hn+l) < (h
c - h
n+1
n
)0'0(hn)S =
ca 2(n+l)a,(hn)g d
Appendix
71
na
$(hn) < ,(0)21-8
Now we prove by induction that
.
If
n = 0, it is clear.
n, from the inequality above we deduce
If it is true at order
nab 21-8
ca (
(hn+l) < d
But by definition of
this leads to
d
nsa -0 t(0)2(n+1)a.21-8.28-1
$(hn+l)
1
p =
18
> 0.
By
h > k > 0
(h)up(h) < (A.10) hu
(h)
0
in
in & (in a weak sense)
i
(A.10)
'
u E H1(0) then (i), (ii), and (iii) hold without any change. Proof:
Let us call
v
the solution of (A.2).
Then from (A.10), we have
-Au < -Av.
By considering
(A. 11)
(u - v)+, we have from (A.11)
vI iV(u-v)+I Iz < < 0 and so the result follows from the Theorem A
since we have 1
0 0].
p
is analytic in
[p > 0].
Ap = 0
This will be
used later.
Let us now point out the properties of the set Theorem 4.9.
Let
(p,X)
then there exists an cc: = {(x,y) E QI
lies in
[p > 0].
be a solution of (P).
e > 0
If
such that the cylinder
Ix-xoI < e, y < yo + E)
[p > 0].
First:
(x0,y0) E [p > 0]
in
4.2.
Some Properties of (p,X) Solution of (P)
If
Proof;
85
and since this set is open, for
(x0,y0) E [p > 0]
small
E
enough the square QE = {(x,y) E )1
is also included in
tonicity of CE
Thus we have
[p > 0].
X, X =
ly-y0I < s}
1x-x01 < e,
CE
on
1
is connected by vertical segments). Op = 0
obtain
CE, and if
on
maximum principle
p
on
y = 1
QE
and, by mono-
(we have used here the fact that by (4.2),
p
But from Theorem 4.5 (i) we now
takes the value
is identically
on
0
by the
CE
CE, which contradicts
on
0
Q. c
[p>0]. As a consequence we have: Corollary 4.10.
Let
p(x0,y) = 0
then Proof:
be a solution of (P).
(p,X)
for all
(x0,y) E S)
Otherwise, we would have
with
If
p(x0,y0) = 0,
y > y0.
p(x0,y) > 0
and, from
y > y0
for some
Theorem 4.9, a contradiction. In the general case (see Remark 4.6) Theorems 3.9 and Cor-
Remark 4.11.
ollary 4.10 are replaced by the following: [p > 0], then for
a
QE
ment coming down, starting in and
p(x0,y0) = 0
y > y0
and included in
p(x0,y) = 0
for all
(x0,y0) E
on each straight seg-
p > 0
S.
(x0,y0) E 0
If
(x0,y) E SZ
such that
The proof is the same as above.
Physically the meaning of Theorem 4.9 is clear:
Remark 4.12.
point
then
{x0} x [y0,y] c 52.
and
choose a point
small enough, we have
is wet, then by
(x0,y0)
when a
gravity acting, all is wet below.
How-
ever note that this is no longer the case when the permeability is not constant (see [17]).
Concerning the wet set, under the assumptions (4.2)-(4.5) we can prove: Theorem 4.13.
Let
p(x,y) > 0
V(x,y) E n,
in other words below Proof:
If
be a solution of (P), then
(p,X)
S3
x E Trx(S3)
the dam is wet.
(x,y) E S3, we have
p(x,y) = hi - y > 0.
is strictly positive in a neighborhood of from Theorem 4.9.
(x,y)
(In the general case we have only
up to the first point of
SI U S2
By Theorem 4.7, p
and the result follows p > 0
vertical lines.)
About the set
[p = 0]
below
S3
that we encounter by coming down along
we have in complete generality:
86
4.
Theorem 4.14.
Let
ball of center
be a solution of (P).
(p,X)
(x0,y0)
r
and radius
THE DAM PROBLEM
Denote by
included in
52.
B
an open
r
If p = 0
in
Br, then
in Br.
p=X=0
By Corollary 4.10 and Remark 4.11 we have
Proof:
on the part
of
C
Choose a strip For
of points connected to
S2
included in
E = R x (h,++)
a E 9(R), a > 0,
Br
p = 0
Br U (x0-r,x0+r) x (y0,+n).
is a test function for
=
Br, i.e.,
above
by vertical segments.
P.
Thus
we get
f Vp.V
+ X.
=
Y
C
Since
a > 0
X = 0
a.e. on
0.
I
11 C(lZ
is arbitrary in _9(t)
and
and thus on
E fl C
At this stage, let us note that if are able to define the free boundary the assumptions (4.2)-(4.5) set for Sup{yjp(x,y) > 0}
X > 0, this implies clearly
C.
0
(p,X)
is a solution of (P) we
of our problem.
Indeed under
x E nx(S2)
if this set is not empty
0(x) _
(4.23)
otherwise.
IS -(x)
By Theorem 4.9, Corollary 4.10 this definition makes sense and we have: For all solutions
Theorem 4.15.
semi-continuous (l.s.c.) on Thus
(p,X)
nx(0)
of (P), the function
except perhaps on .5.'
is lower
(see (4.10)).
is measurable and we have:
0
(4.24)
[p > 0] = { (x, Y) E D y < @(x) } = [y < '(x) ] . Proof:
The l.s.c. of
not in .Y S (x0)
for
is clear (by (4.10)) on the points which are
O(x) = S (x) .
e > 0, let
y0
p(x0,y0) > 0
0
(S2)
and
O(x0) >
O(x0) > y0 > O(x0) - c
with
and Corollary 4.10 we have of course
p(x,y) > 0
and thus
This means that for
a.
If now x0 E 'r
be such that
By definition of
(x0,y0) E Q.
and radius
0
and where
on a ball
Ba
of center
(x0,y0)
x E (x0-a,x0+a), '(x) > y0 > 41(x0)-e.
Hence the result follows since (4.24) is easy to check. Remark 4.16.
One of the advantages of assumptions (4.2)-(4.5) is that in
the decomposition of one 0
52j, namely
Sa
52
into
itself.
2j
(see Remark 4.6) we actually have only
So if one wants to extend the definition of
to the general case it is easily done by defining one
4) }
in each
0
87
Some Properties of (p,X) Solution of (P)
4.2.
The formula is the same as in (4.23) with now
ferent
in place of
Std
0, S (x)
Under the assumptions of the Remark 4.6 the dif-
being replaced by
are l.s.c. on their domain of definition except perhaps on
(D )
(J =
'p) -
Let us now state a result which will play a fundamental role in the sequel:
Theorem 4.17. C
Let
be a solution of (P), h
(p,X)
be a connected component of the set is an interval.
wrx(C) = (x0,x1)
Zh = SZ
fl
Clearly
Set
we have
+X2< fZ pY +X h] = {(x,y) E Sty > h}).
[y > h]
Let us assume that we have proved the following inequality for all
Proof:
{ E H1(Zh) fl C(Zh), C
positive, vanishing on
y
Zh
1
(x0 + e,x1 - E), e > 0
on
IZ py + X = h
(
h) +
JZh
(4.26)
(x0,x1)
aE a function in -9((x0,x1)), between
and choose equal to
[y = h]:
C(x,O(x))dx
< 1
X([p > 0])C
f
( = JZ
and
0
1, and which is
We have:
being small.
h)y X.[ac(Y-h)]y + lZ
h
h +
Zh fl S3 = 0, clearly
Since
Let
a real number. [y > h].
fl
(x0,x1) x (h,+-).
If Zh fl S3 = 0 Z
[p > 0]
X(Zh)ac.(y - h)
E)(y-h)]y.
is a test function for (P)
and the first integral in the above formula is negative.
Applying (4.26)
to the second leads to j
p y
X < J
Zh Y
Letting
(1-a (x0,x1)
a - 0
that (4.26) holds.
E
Zh
(X-X([p > 0]))(I-a E ).
the result follows by Lebesgue's Theorem, provided (The first inequality is clear since
X2 < X
and
88
THE DAM PROBLEM
4.
Xpy = py.)
To prove (4.26) let us first remark that for
C(Zh), with
C > 0, and
= 0
on
f1
[y = h], the function
(E > 0)
= X(Zh)min(e,C)
is a test function for (P).
(Note that
p(xi,y) = 0
This follows from the definition of
i = 0,1.
C E HI(Zh)
C
V(xi,y) E 1, y > h,
and from Theorem 4.9.)
Thus from (P)(iii) we get:
Ej
Since
f
IVPI2 + j
Zhn [P<EC]
Zhft [P>eC]
[min(E,C)]y = 0
f Zh
4-0
h
almost everywhere on
this leads to
[p = 0]
Y([P>eC])OP'VC + X([P > 0])[min(E,C)]y < 0
X([P > 0])[C - min(e,C)] y
X([P>EC])OP'OC + )(([P > 0])Cy 1Z
0.
1Z
Z
h
h (4.27)
JZhX([P>EC])OP'OC + )(([p > O])Cy < JZhX([P > 0])(C - e)Y
By Fubini's Theorem the last integral is equal to (C - P)Y(x,y)dy dx, (h,O(x))
(xO,xl)
where for simplicity we have still denoted by S (x).
(S1
can indeed intersect
continuity in (x0,x1)
all
of
y
such that
(C - E)+
y = h.)
(C
Now letting
6
the maximum of
d
h
and
But now using the absolute
(see [98], p. 57) for almost all
(D(x) > h, and for
f
h
x
in
small enough we have
E)Y(x,y)dy < (C - e)+(x,Vx)-6) < C(x,(P (x) d). go to zero, by continuity of
C
we get for almost
x E (x0,x1)
(C - E)y(x,Y)dY '< C(x,O(x)) (h,'(x))
and clearly (4.26) follows.
(Note that in (4.26) the right-hand side makes
sense by Theorem 4.15.) Remark 4.18.
In the general case of Remark 4.6 the result is as follows.
As above let
C
be a connected component of
[p > 0]
11
[y > h].
4.2.
Some Properties of (p,X) Solution of (P)
If
denotes the decomposition of
(EL )
S2 = {(x,y) ESL x E 7Tx(C
fl S2)),
89
in
S2
52j, set:
Y>h}
Zh= (Note that each
S2
Roughly speaking
is a connected open set which is vertically convex.
0
is obtained as follows:
Zh
the vertical segment from counters
through
S, then
Zh
pick
is the union of such segments
then (4.25) holds.
C, draw
running
(x,y)
Now the result of Theorem 4.16 is the same:
C.)
in
(x,y)
up to the first points where it en-
(x,y)
If
Zh fl S3 = 0,
First, with the same proof as above, (4.26) is
replaced by: P
J OP'V + X([P > 0])C < y Zh
J1
f,x(52'.) 7r J
C E H1(Zh) fl C(Zh), C > 0, C = 0
for all
boundary defined in each
on
[y = h].
as in the Remark 4.15.
12j
tained by splitting (4.27) on the different function
on each
aE
nx(Q')
is the free
(0)
The above sum is ob-
Introducing then a
the result follows as above.
Once more
the simplification under the assumptions (4.2)-(4.5) lies in the fact that Zh
is, in this case, comprised of only one
Q'.
Finally, to conclude this section let us prove the following result valid for any
0:
Theorem 4.19.
Let
and radius
(x0,y0)
(p,X)
r
be a solution of (P), B
r
included in
0.
an open ball of center
Then the following cannot occur:
P(x0,Y) = 0
V(xO,Y) E Br
p(x,y) > 0
V(x,y) E Br,
(i)
(ii)
Proof:
x # x0
P(x,Y) = 0
(x, y) E Br n [x < x0]
(Resp. Br n [x > x0])
P(x,Y) > 0
(x,y) E Br fl [x > x0]
(Resp. B. n [x < x0])
+l
Let
E 9(Br).
Since
E
and - are test functions for (P)
we have: (4.28)
0. J
B
r
Under the assumption (i) we have
X = 1
a.e. on
Br
and thus
90
4.
JB
Ey = 0.
))IB
r
r
Under the assumption (ii) by Theorem 4.14 we have [x < x0]
f / BrX
X = 0
a.e. on
Br fl
and thus
= 0.
v
Ey ='Brfl [x>x0] Ey
B (Brfl [x>x0])
denotes the component in
(v
THE DAM PROBLEM
Y
of the outward unit normal
y
v
to
Y 8(Br ll
[x > x0])). 0
J
Thus in both cases by (4.28)
C E 9(Br) - Ap = 0
in
Br.
But (i) and (ii) are now in contradiction with the maximum principle since p
achieves its minimum inside
4.3.
B
S3-Connected Solutions
An important point, that we have not yet discussed is of course: does (P) have a unique solution?
The answer is no.
The simplest example
is the following:
Assume that we are in the case of Figure (4.29) with noting the regions indicated in (4.29). (hl - y,l)
on
C
and
C'
de-
Then clearly
C
4.30)
(P,X)
I(0'0)
outside
is a solution of (P), but so too is
C
S3-Connected Solutions
4.3.
(P,X)
91
(hl - y,l)
on
C
(k - y,l)
on
C'
(0,0)
elsewhere.
Of course, the choice of
C'
is not unique.
One of our goals will now be to show that the Figure (4.29) provides us in fact with the only pathology which leads to nonuniqueness.
More
precisely (4.30) seems to be the only physically relevant solution of (P). It will be called, for obvious reasons, the
S3-connected solution of (P).
Then we will show that all other solutions are obtained by adding to the S3-connected solution pairs defined by
on sets of type
(k - y,l)
C'.
As we will see such a splitting of the solutions of (P) is quite general, no matter the shape of the dam considered. Let us first begin with a precise definition.
We shall say that a solution
Definition 4.20.
connected if for all connected components
(Recall that
positive.
denotes the closure of
C
Assume that
Remark 4.21. P
we have
[p > 0]
0.
Cns3
a point
of
C
of (P) is S3-
(p,)()
of Thus
which touches
S
3,j' C
S3,j.
c n S3,j
$
C
in
1R2
for some
Then C contains
j.
p is strictly
Around this point (see Theorem 4.7)
contains all the connected component of This shows also that if
[p > 0]
then
c n S3 = 0
wet without supply of water from the different reservoirs.
is
C
This, of
course, seems to be incorrect from a physical point of view.
Although the formulation (P) does not rule out completely the possibility of connected components of
[p > 0]
which do not touch
S3,
(see (4.29)) the shape of such a component as well as the pressure inside is easily described.
Indeed, as claimed in the beginning of this section,
we have: Theorem 4.22.
ponent of
Let
[p > 0]
(p,X)
be a solution of (P) and
such that
C n S3 = 0.
C
a connected com-
Then there is some
h
such
that: C
is a connected component of
Proof:
Recall that
y < h.
Set
C
[y < h]
[y < h],
p = h - y
is the set of points
k = Inf{yl(x,y) E C}
which is a connected component of
in
(x,y)
and consider the set [p > 0] n [y > k]
here holds for a general dam, see Remark 4.18).
on
Zk
C.
0
such that
associated to
(the proof given
By Theorem 4.17 we have
92
THE DAM PROBLEM
4.
< 0.
X2
I
1Zk On the other hand (see Remark 4.18) which are in
f0p12
Zk
(Recall that
This leads to:
Zk fl S3 = 0.)
J
is a test function for (P).
±X(Zk)p
and
52
on all boundary points of
p = 0
+
0.
Zk
Adding this to the above inequality we get p2 + (X + py) 2 < 0. 1
Zk
Since
X =
on
1
on
p = (h - y)
this leads to
[p > 0]
for some
C
the bottom of such a
plies that
in the remainder of
(p,X) = 0
Definition 4.23.
Op = (0,-1)
on
is surrounded by
C
C
and that
SI
Zk.
Let us call a "pool" a pair of functions
on a connected component of
and thus
Note that this im-
The result follows.
h.
(p,X)
defined
by
[y < h]
(P,X) = (h - y,l).
Then we have:
Any solution
Theorem 4.24.
(p,X)
of (P) is the sum of an
S3-connected
solution and "pools". Let
Proof:
(p,X)
be a solution of (P).
Denote by
the connected
C i
component of
such that
[p > 0]
(P',X') = (P,X)
C. fl
fl S3 # ID.
f 0p'.V 52
for all
and set
(i E I)
(X(C1)P,X(Ci))
-
Then clearly all connected components C'
S3 = 0
C'
of
are such that
[p' > 0]
Moreover, by Theorem 4.22 we have
X -
+ Y
152
E E HI(S2), E > 0
i
C.
on
S2, E = 0
on
< 0,
+
-
f
Y
Y
S3.
y Thus
(p',X')
is a
S3-connected solution of (P) which concludes the proof.
This theorem allows us to restrict our attention to solutions.
S3-connected
S3-Connected Solutions
4.3.
93
First let us assume that the numbering of the S3 i (i = 1,...,n)
reservoirs with bottom
is chosen so that
hn < hn-1 < ... < hl.
(4.31)
Then the first natural result is to show that for an
S3-connected
solution the level of the wet set cannot exceed the level
hl, and in
fact, we have more precisely: Theorem 4.25.
Let
be an
(p,X)
S3-connected solution of (P) then:
(i) P=X=0 in
[Y>h1] 0 < p < (h1 - y)+ in 0.
(ii)
Set
Proof: (P)
E = (p -
(hl-y)+)+
Clearly
C
and by
S2 U S3
vanishes on
(iii) we get:
(hl-y)+)y < 0
1 Op-0(P - (hl-Y)+))+ +
f OP-0(P - (hl-Y)+)++ X (P - (hl-Y)+)+ + J 1
[Yh11
y
0.
we have almost everywhere
X-(P - (hl-y)+)Y = V - (hl-y)+.V(P - (hl-y)+)+. (This results from the fact that is equal to 0
when
p
(hl-y)+)y
-(hl-y)+ = 0, and since
(p - (hl-y)+)y = -(hl-y)y(p - (h1-y)+)y
where
p > 0
and to
vanishes.)
Thus the above inequality becomes: (h1-y)+)+12
1V(p -
f
+
IVP12 + X.Py < 0.
f
[Yh1]
In the second integral it is enough to integrate on the different con-
nected components of
[p > 0]
fl
[y > hl]
or still on the different
Zh 1
generated by these components (see Remark 4.18).
But on these sets we
have by Theorem 4.17
0]
containing
S3-connected, such a component must touch k > hI.
Thus
p = 0
on each con-
Op = (0,-1)
p = k - y
This implies that
[y > h1].
fl
But
on all
p = k - y Since
C1.
hl.
is
(p,X)
which is impossible since
S3
[y > hl], and (i) results from Theorem 4.14.
on
(ii) results from the fact that (4.32) can now be written Y)+)+I2
0.
f jv(p - (hl -
Remark 4.26.
The necessity of assuming that
(p,X)
is
S3-connected is
clear from the Figure (4.29).
Before describing more completely the properties of S3-connected solutions let us take a look at the case of Baiocchi's dam, that is to say the rectangular one (see Figure (4.1)(B)).
In this case the main tool is to integrate along vertical lines (hence the necessity of vertical walls).
More precisely, let
(p,X)
be a solu-
tion of (P) in the case of the Figure (4.1)(B) set
u(x,Y) = j I p(x,y)dy.
(4.33)
y (This transformation is called the Baiocchi Transform), then we have: Theorem 4.27.
(i)
(ii)
u
defined by (4.33) satisfies:
[u > 0] = [P > 0] -Au +X= 0 in Q.
Proof:
For (i), u(x,y) > 0
(x,y')
with
p(x,y) > 0.
if and only if
p
is positive for some point
But by Theorem 4.9 it is the case if and only if
y' > y.
For (ii) now, we first deduce from 4.33 that
p = -u
Thus Y'
by Theorem 4.5(i) (-Au + X)y = 0 and
-Au + X
in
SZ
depends only on
x.
But on a level greater than
(i) in the previous theorem) we have Remark 4.28.
p = u = X = 0
hl
(see
and thus (ii) follows.
At this stage (ii) allows us to answer an important ques-
tion (this technique applies to more general domains than rectangular ones but not in all generality).
Indeed from regularity theory of elliptic
problems we deduce from (i) that
2
u E wloc(O)
for all
s, and thus (see
4.3.
S3-Connected Solutions
Remark 2.5) Au = X = 0 (i) proves that
95
almost everywhere on
[u = 0].
This combined with
is a characteristic function, i.e., that we have
X
X=X([P> 0]).
(4.34)
To obtain a variational inequality, it is enough to note that we have now
u > 0,
-Au > -1,
Moreover, 4.33 allows us to compute the boundary
(compare with 3.41).
value of
We have:
u.
u=0
(-Au + 1).U = 0
on
S2 lnl - YJ
rnl
-
U
(hl - y)dy =
J
y
on
S3 3,1
on
S3,2.
(4.35)
2
ln2 - YJ
In n2
U
2
2
(h2 - Y)dY =
2
y
value of u
Now to get the i.e., that
S1
p > 0
around
on
S1, assume that the dam is wet around
(this can be shown easily).
S1
Then we
have
Au = around
1
S1.
Moreover, by (4.33) and (4.15)
uyy = -py = 1 and from
Au = 1
on
S1
it follows that
that is to say, u
is linear on
uxx S1.
has to be equal
to
0
on
Sl
This leads with the notation of
(4.1) (B) to h2
h2 on
u = (a - x) 2a + x. 2a
Thus, introducing
(4.36)
as the set
K
K = {u E H1(f2)ju > 0, u
S1.
satisfy (4.35), (4.36)}
u
has to be a solution of >
Vv E K
uE K. This problem was investigated for the first time by Baiocchi in [9], [10].
The uniqueness of
in this case.
u
of course leads to the uniqueness of
Note that the knowledge of u
gives us
p
as
-uy
(p,X)
(see
96
4.
THE DAM PROBLEM
For extensions of this method to higher dimensions
[10], [15], [76]).
the reader is referred to [76] and [102].
To prove (4.34) in all cases, as well as to establish some monotonicity results about the free boundary of our problem, the next two theorems will be very convenient tools. Theorem 4.29.
k = 1,...,n
Let
(p,)()
and all
h
First we have in all generality:
be a
S3-connected solution of (P).
satisfying
y)+}
ph = { (x,Y) E 01 p(x,y) > Ch has at most
satisfies
connected components.
k
we denote by Ch
S3
i =
More precisely, if for
,
i = 1,...,k
whose closure in
the connected component of Ph
Ch,i
For all
hk+l < h < hk, the set
one has:
i
Ph = Ch,I U Ch,2 U ... U Ch,k.
(4.37)
(Of course in the above formula some of the Chi also that when to be read
k
can be the same.
Note
not being defined the assumption has simply
n, hn+I
h < hn
IR2
In the case
h > h1
it has already been proved that
See Theorem 4.25.)
Ph = (d.
k
Set
Proof:
Ch = a -
U
Ch i.
i=1
'
Then
k (h-y)')+
= (I -
= X(Ch)-(P - (h-y)+)+
L
i=1
is a test function for (P) (see [49] for a complete justification.
that in the above sum the same Chi Thus we get:
once).
(h-y)+)+ + X-(P - (h-y)+)Y < 0 f Ch'
ab
j
Vp.V(p - (h-y)+)+ + X-(p - (h-y)+)+ + Y
IOpI2
+
Crif[y>h]
But, (see the proof of Theorem 4.25) on X-(p - (h-y)+)Y = V and thus the above inequality becomes:
Note
are assumed to be taken into account
[y < h]
(h-y)+)+,
+
0.
Y
one has
4.3.
S3-Connected Solutions
97
IVp12
(h-y)+)+I2
IV(p 1
+
f
+
(4.38)
0.
C,ir1 [y>h]
Chn [y 0]
fl
which are not included
[y > h]
Of course, this is still the same as integrating over the
Ch i.
generated by these connected components, (see Remark 4.18) and since
such a
satisfies h fl S3 = 0
Zh
we have by Theorem 4.17
X2 < 0. JZ] h
Combining this with (4.32) we get
of
By analytic continuation containing
[p > 0]
eluded in some
Z
i
Thus on the connected components
Chi
If
C..
P2 +
f
+
Chfl[y h.
(p y
C
(h-y)+)+I2
IV(p -
f
C3
X)2 < 0.
Y
x
h
we must have
p = k - y, where
p = k - y
on all the connected component
(p,X)
S -connected and
is
this is impossible.
3
So all the
C.
not in-
C.
are empty and
(4.38) becomes (h-y)+)+I2
< 0.
IV(P -
f C
h
Thus
p < (h - y)+
and (4.37) follows.
Ch
on
Let us now point out an interesting feature of these first that (4.2)-(4.5) holds. Theorem 4.30. (xl,h)
of
Ph.
and
Let
(p,X)
Assume
Ch i.
Then we have:
be an
S3-connected solution of (P).
Let
be two points in the same connected component
(x2,h)
Then if the segment
[xl,x2] x (h}
does not intersect
S2
Ch
i
one has
for (x,y) E 2: p(x,y) > 0 Proof:
Vx E [xl,x2],
Let us assume
By definition of
p(x0)y0) = 0
Vy < h.
for some point below
[xl,x2] x (h}.
Ph
(4.39)
98
THE DAM PROBLEM
4.
one has
Thus, by Theorem 4.9, one can assume
p(xi,h) > 0.
Now since Chi
(x1,x2) x (--,h).
is connected and open, it is arcwise
Thus, one can find a path
connected.
(x0,y0) E
joining
r
(x1,h)
to
(x2,h)
(see Figure (4.39)) which is built with straight line segments and moreover has no double point. Thus since
being in
P
p(x0,y) = 0
has to cut the vertical line
[y < h]
below
x = x0
the connected component of
w0
is in particular in
Ch,i (x0,y) E S2, y > y0
for
Let us denote by
(x0,y0).
in the complement of
(x0,y0)
[p > 0].
(see Corollary 4.10), P
r
in
and set:
_ -X (w0) (P - (h - y) ) Clearly
tion of
= 0
[x1,x2] x {h}
and in a neighborhood of
[y = h]
on
Moreover, E = 0
Ph.
on the
since for such a
r, by defini-
S3,j, which eventually intersects
we have necessarily
S3,j
h. > h.
Thus,
is a test function for (P) and we get
Op-V[-(p - (h-y))
]
+ X-[-(p - (h-y)) ]y < 0.
w0
w0 c [y < h]
Since
fw0f[P>Ol Vp.V[-(p Now since on X > 0
w0 fl
- (h-y))
]
+ X'[-(P - Ch-y)) -]
[p > 0]
we have
- (h-y))
I2 < 0.
Thus
+
(O,X) = V - [(h-y)]
X < 0. ( w0f[P=0]
and since
Since
is constant on all connected components of
(h-y))
(p -
w0 n [p > 0].
Let
be the connected component whose boundary contains
C
p > (h - y)
is equal to
w0.
on
r
we have
Indeed otherwise
and at this point we would have both have
y
we have
fw0fl [p>0] IV(p
P.
this inequality becomes
p > h - y
in
w0
C
p > h - y
on
C.
But clearly
could have a boundary point in p = 0
which contradicts
and
p > h - y > 0.
p(x0,y0) = 0
C
w0
Thus we
and concludes
the proof.
Remark 4.31. ture:
The general case can be illustrated by the following pic-
4.3.
99
S3-Connected Solutions
(4.40)
In this case clearly the result is the following (and the proof is the same as above):
Let
(p,y)
be an
S3-connected solution of (P) and
(xi,h); i = 1,2, be two points in the same
ponents of in
s nIR x (-o,h)
S1 U S3
then
first point of
S
[xl,x2] x {h}
is strictly positive below
p
are included
[xl,x2] x {h}
up to the
encountered by coming down along vertical lines.
particular, this is the case when cluded in
If all connected com-
Ch i.
with end point on
In
are both in a ball in-
(xl,h), (x2,h)
Q.
To conclude this section and assuming first that we are in the case (4.2)-(4.5) we have: Theorem 4.32.
Let
(p,y)
by (4.23) is continuous on
The function
be a solution of (P). wx(7)
except perhaps on 3"
c
defined
U Y+. Moreover
_V+ '
has right and left limits on
limit at the endpoints of
U 3`
as well as a right or a left
ax(p).
Clearly by Theorem 4.24 one can assume that
Proof:
(p,x)
is
S3-connected.
Now let us first prove the following: (x,y) E 2, y > O(x), then
If
neighborhood of
Indeed assume that (4.41) fails. 0
of center (i)
(x,y)
and of radius
For all ball
BE,
x < x, p(x,y) > 0 For
(Resp.
c'
vanishes in a (4.41)
Let us denote by E < y - cD(x).
centered at
there exist two points
(ii)
p
(x,y).
(x,Z)
and
a ball included in
and of radius c' < E
(x,y)
and
BE
Two cases could occur:
(x,y)
in
BE,
x < x, p(x,y) > 0.
small enough, less than
a
one has
p(x,y) = 0
V(x,L) E Be
x < x
p(x,y) = 0
V(x,y) E BE
x < x)
such that
100
THE DAM PROBLEM
4.
and for all
e" < e'
centered at
(x,y))
(x,v) E Be
there exists a point
(a ball
such that
X < X. P(X,y) > 0 (Resp.
x < x, P(x,y) > 0).
Let us consider for instance case (i).
We thus can find two se-
quences (see Figure (4.42)).
(xn,Yn) (4.42)
(Xn.Y-E)
and
(x ,yn)
(xn,yn)
in
such that
B.
x
< x,
P(xn,,vn) > 0,
(x
yn) - (x, y)
when
n -+ +m
x
<xn,
P(xn,yn) > 0,
(xn,yn) - (x,y)
when
n -+ +m.
Moreover, we can assume () nondecreasing and us then consider the sequence
(xn,y - E).
(xn)
(If we choose
one can assume without loss of generality that this the straight segment
[x1,x] x {y - e}
P(xn,y - E) > 0
deduce that
for all
nonincreasing.
are in
small enough,
a
sequence as well as From Theorem 4.9 we
0).
(x y - e)
Thus the points
n.
Let
belong to
P which (by Theorem 4.29) has only a finite number of cony-E nected components. Thus there is an infinity of (xn,y - e) (starting
for instance from
ponent of {y - E}.
Py-E.
(xl,y - e))
which belong to the same connected com-
By Theorem 4.30 this leads
to
p > 0
below
But clearly the same applies on the other side of
are in the case (i) of Theorem 4.19 (recall that e])
and thus a contradiction.
x
p(x,y') = 0
(x1,x) x
and we for
y' E
It is not difficult to see that
case (ii) would lead to case (ii) of Theorem 4.19, which proves (4.41) in all cases.
To conclude the proof, let us first remark that by Theorem 4.15 we already know that -V-.
is l.s.c. (lower semicontinuous) except perhaps on
So let us prove that
perhaps on it is clear that
0
is u.s.c. (upper semicontinuous) except
If we are on a point where '
is u.s.c. since
0 < S+
O(x) = S+(x) and
S+
and
x t 5ol+
is continuous at
x.
4.4.
Uniqueness of S3-Connected Solutions
If now
the u.s.c. of
4'(x) < S+ (x)
the proof of Theorem 4.15).
Thus
at
0 0
101
x
results from (4.41) (see
is continuous except perhaps on
To prove the existence of a left (or right) limit on a point of 5
or So+ (as well as the end of
wrx(S2))
assume:
1 = lim inf O(x) < lim sup '(x) = L. x+xO X')-X0
x<XO
x<x0
Then choose a sequence infinity.
For
a
x
such that O (x ) - L
< x0
n
small enough such that
L - e > t
n
when
argument as above that an infinity of terms of the sequence belongs to the same connected component of V(x',y') E 0, y' < L - e $(x') > L - E > t
near
for
near
x'
PL-e'
goes to
we get by the same
Thus
(x
L - E)
p(x',y') > 0
This clearly implies that
x0.
and a contradiction since then
x0
lim inf '(x) > L - e > L. x<x0 Remark 4.33.
In the general case, and with the same proof as above, the
Theorem 4.32 has simply to be replaced by: perhaps on
x(S2
is continuous on each
(see Remark 4.6).
limits exists on
points of
cj
Moreover, right and left
as well as right or left limit at the end
U-91"+
irx (S2.) .
As a consequence we have now in complete generality: Theorem 4.34.
Let
be a solution of (P), then
(p,X)
(4.43)
X = Up > OD. Proof:
This follows directly from the previous theorem and from Theorem
4.14 according to the fact that union of the sets
{(x,o.(x))Ix E 7rx(S2j))
is of measure zero.
Let us now complete Theorem 4.24 by proving that there is only one S3-connected solution of (P).
Uniqueness of S3-Connected Solutions
4.4.
This time let us argue directly in the general case. be two S3-connected solutions of (P).
a decomposition of 0 denote
in
s2j,
(D ,0 the function
0j
j = 1,...,p
Let
(p 1,X1),
Let us assume to be chosen
(see Remark 4.6) and let us
(see Remark 4.16) corresponding to
respectively in this decomposition.
pl'p2
102
THE DAM PROBLEM
4.
Now set p0 = min(P1,P2),
the functions
being defined in
00
= min('
XO = min(Xl,X2), irx(S2.),
First we have:
For all
Lemma 4.35.
C E Hl (2)
C(C), c > 0 we have for i = 1 or
fl
(Pi-PO)-VC +
(4.44)
JDi c(x,oj(x))dx
JSIV
2
J
where
D _ {x First note that the sets D are clearly measurable and the inte-
Proof:
grals on the right side of (4.44) make sense. instance for
=
i = 1.
min(P1
Clearly
Thus for
Let us give the proof for
as above and
C
e > 0
consider
a p0
E = 0
on
S2 U S3
and
±
is a test function for (P).
Applying (P) (iii) corresponding respectively to
pl
and
p2
we get by
subtraction: j 0(P1 - P2)OE + (X1 -
0.
But in this integral it is enough to integrate on the set
[p1 - p0 > 0]
p2 = pO, thus we have
where
f 0(P1-PO).V
1
0
+
IV(pl-p0) I2 +
J
0(pl-p0) Vc
ec]
[pl-pO<Ec] + J
(X1-XO)
[min(Pi
PoC)ly
This implies that >ec]O(P1 - PO )-VC +
f
jr)
P1 PO
(Xl-XO)[C - min(pl IS2
s
pO,c)]y
= 0.
Uniqueness of S3-Connected Solutions
4.4.
But now splitting
Sl
into the different
hand side we get (note that on
103
in the integral on the right
92. J
[p0 > 0], )(1 = X0 = 1
and use (4.43)) 7,
f (Xl-X0)[
-
min(p1Ep0,C)]y =
52
f
j-1
0
(L
1
-)y
(4.45)
where (x)] _ {(x,y) E 01lD0(x) < y < @(x)
[D0(x) < y
Ec]
f
f
on the right hand side of (4.45) we get
ID
[p1-p0
D.
j=l l
and the lemma follows by letting
go to zero.
e
We are now able to prove:
There is one and only one
Theorem 4.36.
The existence results obviously from Theorem 2.42.
Proof:
is always that of the preceding lemma. S3 k
S3-connected solution of (P).
for some
ponent of
such that
k
[p1 > 0]
Choose
Br
The notation
a ball centered on
is included both in the connected com-
Br
and in those of
[p2 > 0]
touching
S3,k.
(See
Theorems 4.7 and 4.13 and the figure below.)
(4.46)
Let
and
rl
r0
be an open part of
the complement of
r0
BBr in
Br) outside
0
Moreover, let us denote by
a
(the bonddary of
BBr.
a smooth function such that Aa = 0
in
Br,
a = 1
on
r1,
By Green's formula we have for
0 < a < 1 i = 1
or
on
2
r0.
(4.47)
104
THE DAM PROBLEM
4.
f
rlno
(pi-p ). 2a __ 2V 0
r
V(Pi-P )'Va +
(p i-p
f
0
f2fBr
0
c2fBr
v(pi-po) Va.
2nB
If we still denote by
which agrees with the fact that
o
and is equal to
Br
in
Xi = XO =
on
1
r
the function (clearly in
a
H1(D) n C(?))
outside, we get using
1
Br:
nc(pi-p0).
(4.48)
2v = 12
!P 1
Let us now denote by
A0
be a smooth function in IR aE(x) = 0
when
suitable.)
satisfying
d(x,AO) > e
0 < aE < 1, aE = 1
on
A0,
would be
(1 - d(x,AO)/c)+
(for instance
aE
let
0 < a < 1.
is a test function for (P) since it is a positive func-
tion which vanishes on
This leads to:
A0.
aa)a]y < 0
10 VPi.0[(1 - aE)a] +
Since
e > 0
For
[p0 > 0].
By the maximum principle we deduce from (4.47) that
(1 - aa)a
Thus
the set
on
(1 - a,)a = 0
and also
A0
(i = 1,2).
p0 = XO = 0
outside
A0
we
have:
aE)a]y = 0.
f VP0'O[(1 - aE)a] +
By subtracting from the above inequality we thus get: I0v(pi-pO)'va + (Xi-XO)'ay < f
(Pi-PO)'V(aca) + (Xi-XO)(aca)y.
Thus using (4.48) and (4.44) we obtain: P
P 1no Now letting on
(pl-p0)'W
0
on
i = 1,2
F1.
and
So the above
4.4.
Uniqueness of S3-Connected Solutions
105
Repeating the procedure for balls included in
Br, we get
By analytic continuation we obtain that
Br n Q.
connected component of
pl = p2
p1 = p2
C, the
in
which contains
[p1 > 0] n [p2 > 0]
in
And
Br n 0.
so we easily obtain that
C=C where S3,k C1.
1 =C2 denote the connected component of
Ci
in its boundary.
Now if
p2(x) > 0
xn + x and thus
in C
Indeed, C c C1 C1
from
we deduce that
pI(xn) = p2(xn)
is closed in
C1.
Hence
One should note that the
(see Theorem 2.24).
solution of [5]
(see [54]).
k'
k = 1,...,n.
S3-connected solution
(P) is also the minimal one, i.e., any other solution
pl > p, X1 > X
S3
pl(x) _
and the
C = C1 = C2
result follows since we can do the same proof for all Remark 4.37.
which contains
[pi > 0]
and is an open nonempty subset of
of
(p,X)
(p',X')
satisfies
It coincides also with the minimal
Moreover, when the shape of
0
is such that
no pools can appear (see Theorem 2.24) then clearly the solution of (P) is unique.
This is the case for the rectangular dam and more generally
for a dam with horizontal bottom as studied in [11]
(see Figure (4.49)(A)).
We have drawn on the Figure (4.49)(B) a case of a more complicated
0
where such uniqueness holds too, since no pool can appear!
(4.49)
(A)
(G)
The reader is referred to [49] for more results in this direction.
106
THE DAM PROBLEM
4.
Let us conclude this section by a theorem which is useful in deciding what part of the dam is wet as well as whether uniqueness holds or not. Physically, the result is simply that the pressure of the
S3-connected
solution increases if the levels of the reservoirs increase. Theorem 4.38.
Let
(p1)X1), (P2,X2)
be two
associated respectively to the functions
viously
S1
data of
01,02.)
S3-connected solutions of (P)
1 and
Then if
we have
02 > 01
as in (4.13).
02
S3,S2
is assumed to be prescribed and
p2 > pl, X2 > X1
From (4.43) it is obviously enough to prove that
Proof:
denote by
p
E:
1
, p
e
the solutions of (P e)
En -+ 0
p2 > pl.
If we
(see Section 4.1) associated with
2
6
respectively, by Theorem 4.2 we have 4.3) we can find a sequence
(Ob-
are defined by the
6
p2 > pi.
Now (see Theorem
such thate pin -. p!
L2(O)
in
E
(i = 1,2).
Thus letting
(since the
S3-connected solution is the minimal one).
clearly that
p2 > pl
have
and thus
to
p2 > 0
in both cases
S3
Remark 4.39.
cn
go to zero in
p2n > p1n
we get
p2 > P1 > pl But this implies
since on the connected components of p2 = p2.
[p1 > 0]
we
(Note that such a component is connected
i = 1,2.)
As a consequence of this theorem consider for instance a dam
as in the Figure (4.1)(A).
S3 = S3,2), then clearly the tion is given by
If we assume the above reservoir empty (i.e.,
S3-connected solution of (P) in this situa-
((h2 - y)+, X([y < h2]).
Thus, if we assume now that
both reservoirs are full, then the Theorem 4.38 implies that the
S3-
connected solution of (P) satisfies p > (h2 - Y)+.
Since no pools can appear on the part of set
[y < h2]
S1
which is outside the
this implies also that the solution of (P) in this case is
unique.
4.5.
Some Monotonicity Results for the Free Boundary In this section we shall study only the case of one or two reservoirs. 4.5.1.
The Case of One Reservoir
The notation will be that of (4.50).
4.5.
Some Monotonicity Results for the Free Boundary
107
(4.50)
0
is here a general domain satisfying, for simplicity, (4.2) and
For all
such that
(x,y), (x',y) E 52
we have
y < hl
(4.51)
(x,x') x {y}
fl S2 = 0.
Then we have: Theorem 4.40.
Let
be the
(p,X)
the function defined by (4.23).
S3-connected solution of (P) and
t
Then under the assumption (4.51) we have:
0
is a non-decreasing function on the interval
(--,x1) fl trx([p > 0])
0
is a non-increasing function on the interval
(x2,+-) 0 ax([p > 0]).
Proof:
Let
a point
which is below
p(x0,y0) = 0, we would have in 4.7 one can find a point S > h, p(a,s) > 0. Theorem 4.29).
h < O(x) < hl.
If there exists
(x2,x) x {h}, and such that
S2, p(x0,y) = 0
Vy > y0.
Now by Theorem
near
S3
such that
Clearly this point is in
Ph
which is connected (see
in
(a,s)
So since
(-tl,x0) x {h}
52
is also in
(x,h)
Vy > y0, an arc connecting half line
and
x E (x 2,+m) 0 nx([p > 0])
(x0,y0) E 0
(x,h)
to
(a,B)
Ph
and since
a < x0,
p(x0,y) = 0
must intersect in
at some point which would be thus in
S2
Ph.
the But
using (4.51), Theorem 4.30 and Remark 4.31 this leads to a contradiction of
p(x0,y0) = 0.
(D(x') > h
for
Thus in
x' E (x2,x).
52
(x2,x) x {h}
below Since
h
we have
p > 0
is arbitrary we have
and
4(x') > O(x)
and the result follows as the proof is the same on the other side. Remark 4.41.
In fact, except for some trivial cases where
(h1 - y)+, the free boundary can never be stationary in also true for more than one reservoir.
52.
p
is given by
This fact is
A proof is given in [49].
Note
that this leads to some nontrivial examples where the uniqueness of the solution of (P) fails.
(See also [49].)
108
4.
THE DAM PROBLEM
The Case of Two Reservoirs
4.5.2.
The notation is that of (4.52) and we assume that (4.2), (4.51) hold.
(4.52)
Let
Theorem 4.42.
be the
(p,X)
the function defined by (4.23). (i)
0
S3-connected solution of (P) and
Then under the assumption (4.51) we have:
is a non-decreasing function of the interval (--,xI) q
Yx([P > 0]).
Moreover, if we assume
is connected (otherwise it would
[p > 0]
be the case of one reservoir) then: (ii)
(iii)
If
O(x) > h2
on
(x2,x3)
for all
If there exists an
Here for
(ail$i) E Ch,i.
First let (x0,y0) E 0
have in
al > x0.
h < hi
x < x1
below
and
Assume
is
(xm,x3).
a point near an end
dy > y0.
(x0,+o) x {h)
Clearly by definition of and
(x,h)
(a1,R1) to
h < h2
0
we have
as above with moreover
(ail$1)
at some point.
by considering the arc connecting [x,x1) x {h}
If there exists a point
p(x0,y0) = 0, then we would
such that
(x,h) E Ch 1
we would necessarily have
below
(ai,Si)
0
p(ai,$i) > 0, that is to say
h < 4D(x) < h1.
(x,xl) x {h}
and Remark 4.31 this contradicts
p > 0
and
(Clearly such a point exists by Theorem 4.7.)
Then any arc binding
the half line
O(x) < h2, then
O(xm) < h2
is non-decreasing on
we shall denote by
0, p(x0,y) = 0
(x,h) E Ph.
such that
(x2,xm), 0
and which satisfies
S3 i
is non-increasing
0
such that
x E (x2,x3)
xm E (x2,x3)
non-increasing on
point of
then
(see 4.52).
there exists
Proof:
x E (x2,x3)
must intersect in
0
But by (4.51), Theorem 4.30
p(x0,y0) = 0.
By assuming
(x,h) E Ch
,2
and a contradiction would be obtained (x,h)
to
and consequently
(a2,s2).
O(x') > h,
Thus we have dx' > x, but
h
4.5.
Some Monotonicity Results for the Free Boundary
being arbitrary leads to
and to (i).
(D(x)
with the same proof by considering a point to prove (iii) let us denote by 4P(xm) =
xm
so let us assume
First if
c(x
)
< @(x)
m
(x,h) E Ch,l
the arc connecting
to
(x,h)
(--,xm] x {h}
Vx' > x
(x,h)
to some
with
(a1,Sl)
h < h2. (a2,82)
'(x)
such that
al < xm
If
Vx' > x,
O(x ) < h < m
p(xm,y) = 0
Vy > h
must intersect the
But by (4.51) and Theorem 4.30
p(xm,O(xm)) = 0, thus
(x,h) E Ch,2'
By considering as previously an arc we deduce then that
and the proof is complete since
Remark 4.43.
h
we have
Sl
x E (xm,x3).
xm, (D(x') >
and consider
at some point.
which implies in particular
Let
[xm,x3).
since in
we would have a contradiction with
connected
Now
such that
Let us prove then that, for instance,
O(x) = @(xm), then clearly by definition of
half line
O(x) > h2.
(x2,x3)
@(x) < h2. inf xE(x2,x3)
is nondecreasing on the interval
cD(x).
Clearly (ii) holds
such that
x
a point in
Such a point exists by Theorem 4.15. $
109
h
4P(x1) > h
is arbitrary.
The Figure (4.53) gives an example where (iii) occurs.
(4.53)
Here O(x)
hl = h2 hl
and thus
fi(x) < hl.
But if between the reservoirs
we have a contradiction with the Remark 4.41.
case (iii), see [49].
Thus we are in
COMMENTS
110
Comments
The presentation here follows Brezis [32] and Carrillo-Menendez-Chipot [49].
When the problem reduces to a variational inequality a nice exposi-
tion is given in Kinderlehrer-Stampacchia [76]. [15].
See also Baiocchi-Capelo
Other approaches are given in Alt [5] and Visintin [109].
Recently
Alt and Gilardi [8] studied the manner in which the free boundary meets the fixed boundary of the domain; they give also other proofs of some of our results.
Most of the matter exposed here has an extension to of course, the physical case, the dimension in canals, St
IRP; p = 3
is,
being a good model for flow
2
being the plane section of the porous medium (see [49]).
Finally, we did not study the evolution problem.
Details on this subject
as well as complete references can be found in the proceedings [13]. Our model studies the case where the pressure is prescribed on the pervious boundary.
Some other models are also valuable.
In particular,
the interested reader will find in [13] the derivation of such a model from the macroscopic point of view.
We note in conclusion that the prescription of
p
on the pervious
boundary leads to realistic prediction on location and shape of the wet set (see Theorem 4.38, Section 4.5 and [49]). in hydrodynamics.
This should be of interest
References
Sobolev Spaces, Academic Press, New York (1975).
[1]
R. A. Adams:
[2]
D. R. Adams: LP-capacitary integrals with some applications Proc. Symp. Pure Math. 35, Part I - A.M.S., Providence (1979), 359367.
[3]
D. R. Adams: Capacity and the obstacle problem. 8 (1981), 39-57.
Appl. Math. Optim.
[4] S. Agmon, A. Douglis and L. Nirenberg: Estimates near the boundary for solution of elliptic partial differential equation satisfying general boundary conditions. Comm. Pure Appl. Math. 12 (1959), 623-727. [5]
H. W. Alt: A free boundary problem associated with the flow of ground water. Arch. Rat. Mech. Anal. 64 (1977), 111-126.
[6]
H. W. Alt: Regularity of The fluid flow through porous media. the free boundary. Manuscripta Math. 21 (1977), 255-272.
[7]
H. W. Alt: Stromungen durch inhomogene poroue medien mit freiem Rand. J. Reine Angew. Math. 305 (1979), 89-115.
[8]
H. W. Alt and G. Gilardi: dam problem, preprint.
[9]
Sur un problbme a frontiere libre traduisant le filC. Baiocchi: trage de liquides a travers des milieux poreux. C.R. Acad. Sc. Paris Se'rie A 273 (1971), 1215-1217.
The behavior of the free boundary for the
[10]
C. Baiocchi: Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. 92 (1972), 107-127.
[11]
C. Baiocchi: Free boundary problems in the theory of fluid flow through porous media, Proc. Intern. Congress Math. Vancouver (1974) II, 237-243.
[12]
C. Baiocchi: Studio di un problems quasi-variazionale conesso a problemi di frontiera libera. Boll. U.M.I. (4) 11 (Suppl. fasc. 3), (1975), 589-613.
[13]
C. Baiocchi: Free boundary problems in fluid flow through porous media and variational inequalities - In: Free Boundary Problems Proc. of a Seminar held in Pavia, Sept.-Oct., 1979, Vol 1, (1980), 175-191. ill
112
REFERENCES
[14]
C. Baiocchi: 173-187.
[15]
C. Baiocchi, A Capelo: Disequazioni Variazionali e Quasivariazionali, Vol. 1 and 2, Pitagora Editrice (1978) Bologna.
[16]
C. Baiocchi, V. Comincioli, E. Magenes and G. A. Pozzi: Free boundary problems in the theory of fluid flow through porous media: Existence and uniqueness theorem. Am. Mat. Pura Appl. 97 (1973),
[17]
C. Baiocchi and A. Friedman: A filtration problem in a porous medium with variable permeability. Ann. Mat. Pura Appl. 114 (1977), 377-
Disequazioni variazionali, Boll. U.M.I., 18-A(1981),
1-82.
393. [18]
A. Bensoussan and J. L. Lions: Nouvelle formulation de problbmes de contr6le impulsionnel et application, C. R. Acad. Sc. Paris, Serie A, 276 (1973).
[19]
A Bensoussan and J. L. Lions: Application des Inequations Variationnelles en Contr6le Stochastique, Dunod, Paris (1978).
[20]
J. Bergh and J. Ldfstrom: Springer (1976).
[21]
M. F. Bidaut-Veron: Variational inequalities of order 2m in unbounded domains, Nonlinear Anal. Th. Meth. Appl. 6 (1982), 253-269.
[22]
M. Biroli: A De Giorgi-Nash-Moser result for a variational inequality, Boll. U.M.I. 16-A (1979), 598-605.
[23]
L. Boccardo: unilateral.
[24]
Interpolation Spaces:
An Introduction.
Rdgularite W1,P (2 < p < +-) de la solution d'un problbme Ann. Fac. Sc. Toulouse 3 (1981), 69-74.
L. Boccardo and F. Murat: Nouveaux resultats de convergence dans des problbmes unilateraux - Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, Vol. II, Research Notes in Math., Pitman, London (1982) - H. Brezis and J. L. Lions, Ed.
[25]
J. M. Bony: Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sc. Paris, 265 (1967), 333-336.
[26]
H. Brezis: Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies 5, North Holland (1975).
[27]
H. Brezis: Equations et inequations nonlineaires dans les espaces vectoriels en dualite, Am. Inst. Fourier 18 (1968), 115-175.
[28]
H. Brezis:
Problbmes unilateraux, J. Math. Pures Appl. 51 (1972),
1-168. [29]
H. Brezis: Nouveaux The`orbmes de Regularit6 pour les problbmes Unilateraux Rencontre entre Physicien Theoriciens et Mathematicien, Strasbourg, Vol. 12 (1971).
[30]
H. Brezis:
[31]
H. Brezis: Remarque sur Particle de F. Murat, J. Math. Pures Appl. 60 (1981), 321-322.
[32]
H. Brezis, The dam problem revisited. Pitman, London (1983).
[33]
H. Brezis, L. C. Evans, A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators, Arch. Rat. Mech. Anal. 71 (1979), 1-13.
Personal communication.
Proc. Sem. Montecatini,
References
[34]
113
H. Brezis and D. Kinderlehrer: The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1974), 831-844.
[35]
H. Brezis, D. Kinderlehrer and S. Stampacchia: Sur une nouvelle formulation du probleme de d'ecoulement a travers une digue, C. R. Acad. Sc., Paris, Serie A 287 (1978), 711-714.
[36]
H. Brezis and M. Sibony: Equivalence de deux inequations variationelles et applications. Arch. Rat. Mech. Anal. 41 (1971), 254265.
[37]
H. Brezis and G. Stampacchia: Sur la regularite de la solution d'inequations elliptiques, Bull. Soc. Math., France 96 (1968), 152-
[38]
H. Brezis and G. Stampacchia: Remarks on some fourth order variational inequalities, Ann. Scuola Norm. Sup. Pisa 4 (1977), 363-371.
[39]
P. L. Butzner and H. Berens: tion, Springer (1967).
[40]
L. A. Caffarelli: The regularity of free boundaries in higher dimension, Acta Math. 139 (1978), 155-184.
[41]
L. A. Caffarelli: Further regularity in the signorini problem, Comm. in P.D.E., 4 (1979), 1067-1076.
[42]
L. A. Caffarelli and A. Friedman: The free boundary for elasticplastic torsion problems, Trans. Amer. Math. Soc. 252 (1979), 65-97.
[43]
L. A. Caffarelli and A. Friedman: Reinforcement problems in elastoplasticity, Rocky Mount. J. of Math. 10 (1980), 155-184.
[44]
L. A. Caffarelli and A. Friedman: The obstacle problem for the biharmonic operator. Ann. Scuola Norm. Sup. Pisa 6 (1979), 151-184.
[45]
L. A. Caffarelli and G. Gilardi: Monotonicity of the free boundary in the two dimensional dam problem, Anal. Scol. Norm. Sup., Pisa, 7 (1980), 523-537.
[46]
L. A. Caffarelli and D. Kinderlehrer: Potential methods in variational inequalities, Journal d'Anal. Math. 37 (1980), 285-295.
[47]
L. A. Caffarelli and N. Riviere: Smoothness and analyticity of free boundaries in variational inequalities. Ann. Scuola Norm. Sup. Pisa 3 (1976), 289-310.
[48]
J. Carrillo-Menendez and M. Chipot: Sur l'unicite de la solution du probleme de 1'6coulement a travers une digue, C. R. Acad. Sc., Paris, Serie A, 292 (1981) 191-194.
[49]
J. Carrillo-Menendez and M. Chipot: Eqns. 45 (1982), 234-271.
[50]
M. Chipot: Sur la regularite Lipschitzienne de la solution d'inequations elliptiques, J. Math. Pures Appl. 57 (1978), 69-76.
[51]
M. Chipot: Sur la regularite de la solution d'inequations variationelles elliptiques, C. R. Acad. Sc., Paris, Serie A 288 (1979), 543-546.
[52]
Free Boundary Problems, M. Chipot: On the two obstacle problem in: Proc. of a Seminar held in Pavia, Sept.-Oct. 1979, Vol. 2, Roma
[53]
M. Chipot: Some Results about an elastic-plastic torsion problem, Nonlinear Anal., Th. Meth. Appl., 3 (1979), 261-270.
180.
Semigroup of Operators and Approxima-
On the dam problem, J. Diff.
(1980).
REFERENCES
114
[54]
Sur quelques inequations variationnelles, problbme de M. Chipot: 1'ecoulement a travers une digue, These d'Etat, Universite de Paris
VI (1981). Methods of Mathematical Physics, Inter-
[55]
R. Courant and D. Hilbert: science.
[56]
R. DeVore and K. Scherer: Interpolation of linear operator on Sobolev Spaces, Ann. of Math., 109 (1979), 583-599.
[57]
G. Duvaut and J. L. Lions: que, Dunod, Paris (1972).
[58]
it G. Fichera: Probemi elastostatici con vincoli unilaterali: problema di Signorini con ambigue condizioni al contorno. Atti Acad. Naz. Lincei Mem., 8 (1963-64), 91-140.
[59]
J. Frehse: On the regularity of the solution of a second order variational inequality, Boll. Un. Mat. Ital., 6 (1972), 312-315.
[60]
J. Frehse: On Signorini's problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup., Pisa 4 (1977), 343-362.
[61]
Partial Differential Equations of Parabolic Type, A. Friedman: Prentice Hall (1964), 196-197.
[62]
Convexity of the free boundary in the A. Friedman and R. Jensen: Stefan problem and in the dam problem, Arch. Rat. Mech. Anal. 67 (1978), 1-24.
[63]
The free boundary for elastic-plastic A. Friedman and G. A. Pozzi: torsion problem, Trans. Amer. Math. Soc. 257 (1980), 411-425.
[64]
Regolarita Lipschitziana per la soluM. Giaquinta and G. Modica: zione di alcuni problemi di minimo con vincolo, Ann. Matematica, 106 (1975), 95-117.
[65]
D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer (1977).
[66]
Regularity of solutions of nonlinear variational C. Gerhardt: inequalities, Arch. Rat. Mech. Anal., 52 (1973), 389-393.
[67]
J. K. Hale: Ordinary Differential Equations, Wiley (1969), Revised edition, Krieger (1980).
[68]
On some nonlinear elliptic differP. Hartman and G. Stampacchia: ential functional equations, Acta Math., 115 (1966), 153-188.
[69]
Boundary regularity for variational inequalities, R. Jensen: Indiana Univ. Math. J., 29 (1980), 495-504.
[70]
Variational inequalities with lower dimensional D. Kinderlehrer: obstacles, Israel J. Math., 10 (1971), 339-348.
[71]
D. Kinderlehrer: Variational inequalities and free boundary problems, Bull. Amer. Math. Soc., 84 (1978), 7-26.
[72]
The coincidence set of solutions of certain D. Kinderlehrer: variational inequalities, Arch. Rat. Mech. Anal., 40 (1971), 231-
Les Inequations en Mecanique et en Physi-
250. [73]
D. Kinderlehrer: The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl., 60 (1981), 193-212.
[74]
Regularity in elliptic D. Kinderlehrer, L. Nirenberg and J. Spruck: free boundary problems, I. J. Anal. Math., 34 (1978), 86-118.
References
[75]
115
D. Kinderlehrer, L. Nirenberg and J. Spruck: Regularity in elliptic free boundary problems, II. Ann. Scuola Norm. Sup., Pisa, 6 (1979), 637-683.
[76]
D. Kinderlehrer and G. Stampacchia: An Introduction to Variational Inequalities and their Applications, Academic Press, 1980.
[77]
H. Lanchon: Torsion elastoplastique d'un arbre cylindrique de section simplement on multiplement connexe, These Universit6 de Paris VI (1972).
[78]
H. Lanchon: Torsion elastoplastique d'un arbre cylindrique de section simplement on multiplement connexe, Journal de Mecanique, 13 (1974), 267-320.
[79]
L. D. Landau and E. M. Lifshitz: Press 1959.
[80]
H. Lewy and G. Stampacchia: On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), 153-188.
[81]
H. Lewy and G. Stampacchia: On the smoothness of superharmonics which solve a minimum problem, J. Anal. Math., 23 (1970), 224-236.
[82]
J. L. Lions: Problemes aux limites dans les equations aux derivees partielles, Presses Universitg de Montreal (1962).
[83]
J. L. Lions: Quelques methodes de resolution des problemes aux limites nonlineaires, Dunod-Gauthier-Villars, Paris (1969).
[84]
J. L. Lions: Sur quelques questions d'analyse de mecanique et de controle optimal, Universite` de Montreal (1976).
[85]
J. L. Lions and E. Magenes: Problemes aux limites non Homogenes, Vol. I,II,III, Dunod, Paris (1968-70).
[86]
J. L. Lions and G. Stampacchia: Variational inequalities. Pure Appl. Math., 20 (1967), 493-519.
[87]
P.
[88]
0. Mancino and G. Stampacchia: Convex programming and variational inequalities, J. Opt. Theory Appl., 9 (1972), 3-23.
[89]
F. Mignot and J. P. Puel: Inequations devolution paraboliques avec convexes dependant du temps. applications aux inequations quasivariationnelles d'e"volution, Arch. Rat. Mech. Anal., 64 (1977),
Theory of Plasticity, Pergamon
Comm.
L. Lions: Problemes elliptiques du 2eme ordre non sous forme divergence, Proc. Roy. Soc. Edinburgh, 84 (1979), 263-272.
59-91. [90]
Proximite' et dualite dans un espace hilbertien, Bull. J. J. Moreau: Soc. Math. France, 93 (1965), 273-299.
[91]
Implicit variational problems and quasi variational inU. Mosco: equalities, Summer School, Bruxelles 1975, in Lecture Notes in Math. No. 543.
[92]
U. Mosco and G. M. Troianiello: On the smoothness of solutions of unilateral Dirichlet problems, Boll. U.M.I., 8 (1973), 57-67.
[93]
H-I L'injection du cone postif de dans W-l,q est compacte F. Murat: pour tout q < 2. J. Math. Pures Appl., 60 (1981), 309-321.
[94]
M. K. Murthy and G. Stampacchia: A variational inequality with mixed boundary conditions, Israel J. Math., 13 (1972), 188-224.
[95]
Necas: Les Methodes directes en theorie des equations elliptiques, Masson (1967).
REFERENCES
116
[96]
M. H. Protter and H. F. Weinberger: A Maximum principle and gradient Bounds for linear elliptic equations, Indiana U. Math. J., 23 (1973), 239-249.
[97]
M. Schechter: On LP estimates and regularity I, Amer. J. Math., 85 (1963) 1-13; II, Math. Scand., 13 (1963), 47-69. Hermann, Paris (1966).
[98]
L. Schwartz:
[99]
G. Stampacchia: Formes bilinearies coercitives sur les ensembles convexes, C. R. Acad. Sc., Paris, 258 (1964), 4413-4416.
The'orie des distributions.
[100]
G. Stampacchia: Equations Elliptiques du Second Ordre a Coefficients Discontinus, Presse de 1'Universite de Montreal (1965).
[101]
G. Stampacchia: Regularity of solutions of some variational inequalities, Nonlinear functional analysis (Proceedings Symp. Pure Math., Vol. 18, Part I, 271-281).
[102]
G. Stampacchia: On the filtration of a liquid through a porous medium, Usp. Mat. Nauk., 29 (1974), 89-101.
[103]
E. M. Stein: Singular Integrals and Differentialibility Properties of Functions, Princeton University Press (1970).
[104]
L. Tartar: Interpolation nonlineaire et regularite. Anal., 9 (1972), 469-489.
[105]
L. Tartar: Inequations quasivariationnelles abstraites, C. R. Acad. Sc., Paris, Serie A, 278 (1974), 1193-1196.
[106)
T. W. Ting: Elastic-plastic torsion of simply connected cylindrical bars, Indiana Univ. Math. J., 20 (1971), 1047-1076.
[107]
T. W. Ting: Elastic-plastic torsion problem over multiple connected domains, Ann. Scuola Norm. Sup., Pisa, 4 (1977), 291-312.
[108]
H. Triebel: Interpolation Theory, Function Spaces, Differential Operators, North Holland (1978).
[109]
A. Visintin: Study of a free boundary filtration problem by a nonlinear variational equation, Boll. U.M.I., 5 (1979), 212-237.
J. Funct.
Index
Baiocchi transform, 94
Elastic-plastic torsion problem, 17, 52
Bellman-Dirichlet problem, 32 Bilinear form, 8
Elliptic equation existence of solutions, 14, 1S
Chain rule for piecewise smooth functions, 13 Coercive operator, 4
Lp-estimates, 67-72 nonlinear, 15
Exterior sphere condition, 53
Coincidence set, 31
Compact embedding, 13
Fixed point theorem, 1
Continuous on finite dimensional subspaces, 5
Fourth order variational inequality,
Convex function, 3
Free boundary, 31, 86, 106, 107
21
Convex set, 3 Hilbert space, 8 Dam problem, 74-110
Holder continuous functions, 14
existence of solutions, 79, 81 monotonicity of free boundary,
Interpolation spaces, 60-62
106
nonuniqueness of solutions,
properties of solutions, 82-90 strong formulation, 76
uniqueness of S3-connected solutions, 10l weak formulation, 78
Laplacian, 14 Lax-Milgram theorem, 8 Lp-spaces, 10 Minty's Lemma, 6 Monotone operator, 5 strictly, 5
Darcy's law, 76
surjectivity of, 7
Dirichlet problem, (see elliptic equation)
117
INDEX
118
Negative part of a function, 13 Norms, 10 Numerical analysis, 21
Variational inequality, 2, (see also dam problem, elastic-plastic torsion problem, and obstacle problems)
existence of solutions, 3, 4, 6 Obstacle problems, 16, 18, 22-66 boundary estimates, 3S comparison principle, 22
fourth order, 20
parabolic, 21
uniqueness of solutions, 7
continuity of solutions, 30 existence of solutions, 16, 18 free boundary, 31
W1'p regularity, 60-66 W1'° regularity, 50-60 W2'P regularity, 27-30
W2'- regularity, 33-50 Parabolic variational inequalities, 21
Penalization, 79 Penalty method, 24-26, 73 Permeability, 76 Poincarg's inequality, 11 Pool, 92
Positive part of a function, 13 Porous medium, 74 Prescribed mean curvature operator, 20
Projection on a convex set, 3 Quasi-variational inequality, 9 Regularity, (see obstacle problems)
Reservoir, 75
Schauder fixed point theorem, 1 Sobolev embedding theorems, 12, 13
Sobolev spaces, 10-13, 21
Spaces of continuous functions, 14
Wet set, 76, 84, 85
Applied Mathematical Sciences cont. from page ii
36. Bengtsson/Ghil/Kallen: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. 39. Piccinini/Stampacchia/Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.
43. Ockendon/Tayler: Inviscid Fluid Flows.
44. Pazy:
Semigroups of Linear Operators and Applications to Partial Differential Equations.
45. Glashoff/Gustafson: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47. Hale et al.: An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory.
48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. I. 52. Chipot: Variational Inequalities and Flow in Porous Media.
53. Majda:
Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables.