Table Of ContentLecture Notes ni
Mathematics
Edited by ~ Dold dna .B Eckmann
942
Theory dna snoitacilppA
fo Singular snoitabrutreP
sgnideecorP of a Conference
Held ni Oberwolfach, August 16-22, 1891
Edited by .W Eckhaus and E.M. de regaJ
galreV-regnirpS
Berlin Heidelberg New York 1982
Editors
W. Eckhaus
Mathematisch Instituut
De Uithof, Utrecht, Netherlands
EM. de .lager
Mathematisch Instituut
Roeterstr. 15, Amsterdam, Netherlands
AMS Subject Classifications (1980): 34 E15, 34 E20, 35 B 25, 35 C20,
65 L05, 65 L10, 76D30
ISBN 3-540-11584-6 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?-11584-6 Springer-Verlag New York Heidelberg Berlin
yrarbiL of Congress Cataloging in Publication .ataD Main entry under title: Theory dna
applications of singular .snoitabrutrep (Lecture notes in ;scitamehtam 942) .:yhpargoilbiB .p
Includes index. .1 Differential .sessergnoC--snoitauqe .2 Differential ,snoitauqe
.sessergnoC--laitraP .3 Perturbation .sessergnoC--)scitamehtaM( .I ,suahkcE .rotkiW
,regal..1I .E .M de sudraudE( Marie )ed .11I Series: Lecture notes in scitamehtam
;)galreV-regnirpS( 942. OA3.L28 .on 942 073A.O 510s 515.3'5 87601-28
ISBN 6-48511-783-0 ).S.U(
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PREFACE
This volume contains lectures presented at a meeting on
singular perturbations, held in Oberwolfach, Aug. 16 - 27,
1981. In organizing the meeting we have attempted to bring
together, and confront with each other, various different
types of activities in the field of research in singular
perturbations. There were 36 participants (by invitation),
from 7 countries, presenting 28 lectures ranging in subject
from pure analytic to very applied considerations. The
mathematical techniques include classical, functional, non-
standard and numerical methods. We wish to thank the
authors for the careful presentation of their work.
It is a pleasure to record our gratitude to prof. Martin
Barner, director of the Mathematisches Forschungsinstitut
Oberwolfach, for ~is invitation to organize the conference
and for the hospitality of his institute.
Wiktor Eckhaus
Eduard M. de Jager
April 1982
CONTENTS
PART :I THEORY OF SINGULAR PERTURBATIONS
H.J.K. Moet: Asymptotic analysis of the free boundary
in singularly perturbed elliptic varia-
tional inequalities.
C.M. Brauner and B. Nicolaenko: Regularization and
bounded penalization in free boundary
problems. 19
W.M. Greenly: Singular perturbation of nonselfadjoint
elliptic eigenvalue problems. 43
L.S. Frank and W. Wendt: Coercive singular pertur-
bations: reduction and convergence. 54
B. Kawohl: A singularperturbation approach ot non-
linear elliptic boundary value problems. 65
J. Mika: Singular-singularly perturbed linear equa-
sno~:t in Banach spaces. 72
R.E. Meyer: Wave reflection and quasiresonance. 84
R. Lutz and T. Sari: Applications of nonstandard
analysis ot boundary value problems in
singular perturbation theory. 113
A. Troesch: Etude macroscopique de l'~quation de Van
der Poi. 136
S. Kamin: On elliptic singular perturbation problems
with several turning pdints. 145
J. Lorenz: Non linear boundary value problems with
turning points and properties of difference
schemes. 150
J.E. Flaherty and R.E. O'Malley, Jr.: Singularly per-
turbed boundary value problems for nonlinear
systems including a chalanging problem for
a nonlinear beam. 170
P.W. Hemker: An accurate method without directional bias
for the numerical solution of a 2-D
elliptic singular perturbation problem. 192
H.J. Reinhardt: Analysis of adaptive FEM's for
-eu" + ku' = f based on a-posteriori error
estimates. 207
V
228
PART II: APPLICATIONS
G.C. Hsiao and R.C. MaeCamy: Singular perturbations
for the two~dimensional viscous flow
problem. 229
F.A. Howes: The asymptotic solution of singularly
perturbed Dirichlet problems with appli-
cation ot the study of incompressible
flows at high Reynolds number. 245
S.V. Parter: On the swirling flow between rotating coaxial
disks: a survey. 258
A.J. Hermans: The wave pattern of a ship sailing at
low speed. 281
A. van Harten: Applications of singular perturbation
techniques ot combustion theory. 295
D. Hilhorst: A perturbed free boundary problem arising
in the physics of ionized gases. 903:
B. Matkowsky and Z. Schuss: Kramers' diffusion problem
and diffusion across characteristic
bounderies. 318
L.S. Frank and W.D. Wendt: On a singular perturbation
in the kinetic theory of enzymes. 64;2
ASYMPTOTIC ANALYSIS OF THE FREE BOUNDARY
IN SINGULARLY PERTURBED ELLIPTIC VARIATIONAL INEQUALITIES
H.J.K. Moet
Mathematisch Instituut
Rijksuniversiteit Utrecht
Postbus 80.010
3508 TA Utrecht.
.1 Introduction
Singularly perturbed elliptic variational inequalities arise in
the study of dynamical systems with small stochastic perturbations
(see Bensoussan and Lions 1). A typical problem related to
variational inequalities in general is the occurrence of an implicit
unknown, the so-called free boundary. In this paper we survey some
recent results on the asymptotic behavior of the free boundary in
singularly perturbed elliptic variational inequalities. Detailed
proofs of all results mentioned below will appear in 9.
There exists a relatively small number of papers on the asymptotic
behavior of the free boundary in the above type variational in-
equalities. The first paper dealing with one-dimensional problems
is by Eckhaus and Moot 5. In 7 Moot also deals with one-
dimensional problems. More recently Moet 8 has given a method for
the analysis of the free boundary in higher dimensions for symmetric
bilinear forms.
Let ~ C A N be open, bounded and simply connected with a smooth
boundary ~.
Let
{v 6 H~(~) : v ~ 0 in ~}.
U
Consider the pmoblem of finding a solution u e of the variational
inequality
~u S
(1.1) u s e ~: e ~S grad ue.grad(v-ug)dx + ~S b~2(v-ug)dx +
+ ~ ue(v-ue)dx > ~ f(v-u )dx, for all v e ~,
where f is a given element in LZ(e), and b is a parameter which is
is either 0 or .I Of course, s is positive and small.
The general existence and uniqueness theorem for elliptic varia-
tional inequalities of G. Stampacchia i0 guarantees the existence of
a unique solution u S of (I.1) for all s > 0. We note that the first
paper on singularly perturbed elliptic variational inequalities is by
D. Huet (see 6).
By a regularity result of H. Br4ziS and G. Stampacchia 3 we know
that~ if f belongs to LP(~) with p > N > 2, then u S is an element of
H2'P(~) A CI'H(~) with D = 1 - N/p. In general, this is the best
degree of smoothness of u S one can expect, regardless of the smooth-
ness of the data; the regularity of the solution of a variational in-
equality may be impeded by a constraint in the set of competing
functions ~.
Now assume f 6 LP(~)~ p > N > 2, then by the regularity of us, it
is easily verified that (1.1) is equivalent to
~U
-eAu S + ~x 2
~u
(1.2) (-eAu S + b e~---3x 2 + u S f)u e : 0 in ~, u s 0 on ~.
u s ~ 0
The set of conditions (1.2) is called the complimentarity form of
(1.1). By continuity of u e the set a s defined by
: {x e m us(x) < 0}
is open. From (1.2) we obtain the following boundary value problem for
u S ,
~u
-eAu + + u S : f in a
(1.3) ~x 2 e
u = 0 on ~ .
S S
The set ~e' which is unknown, is called the free boundary. We note
that ~ue/~x i = 0 on ~e ~ ~, since u e E CI(~) and u e assumes its
maximum on 3~ . Hence, for smooth enough ~e we have ~ue/~n = 0
on ~
E
For the sake of simplicity of exposition we will assume f to be in
C=(~) (this assumption will be relaxed at appropriate places) and we
will only consider the most elementary geometrical situations. Here,
as usual in singular perturbation theory, the emphasis will be on the
method of analysis rather than on obtaining the most general result
for the most general situation.
In an easy way we can derive some information from (1.2). For
instance, if f > 0 in ~ one easily checks by substitution that lhe
solution u e is identically zero. Or, if the set ~_ = {x E ~: f(x) < }0
has positive measure, then one immediately sees from the first con-
dition in (1.2) that u e cannot be identically zero on any open subset
of ~_.
The following lemma contains some further useful information about
U (cid:12)9
e
Lemma 1.1. Let ~_ = {x E ~: f(x) < O} be nonempty and let u E be the
solution of (1.1). Then u < 0 in ~_. In particular, if ~ U ~_ C
then ue < 0 in ~ U ~_. Furthermore, u possesses no nonzero local
minima outside ~_. Finally, if for some open subset ~ of
~+ = {x E ~: f(x) ~ O} we have leu ~ ~ = ,O then u e i8 identically
zero in ~ .
Now, if f < 0 in all of ~, then Lemma 1.1 yields u e < 0 in ~.
Hence, by the second condition in (1.2) we have a s = ~ for all e > 0,
which shows that 3~ = ~ for all e > 0.
e
Clearly, the above observations show that in order to have a non-
trivial problem f must have different signs on S.
Below we shall deal with the problem of approximation u E and
as ~ ~ 0.
2. Asymptotic analysis of u s and 3~e by upper and lower approximations
In this section we intend to describe the method of upper and
lower approximations, given in 8, to determine the asymptotic behav-
ior, as e ~ 0, of the solution'u E and the free boundary of the varia-
tional inequality (1.1). This method is an amalgamation of variational
inequality techniques and standard results from the theory of matched
asymptotic expansions (see Eckhaus and de Jager 4).
Lemma 2.1. Let ~1 C ~2 be open smoothly bounded sets in ~N~ Let f
be given in L2(~2 .) Let ~ : {v E H~(~2):_ v ~ 0 in ~2 } and let u be the
solution of
u 6 ~: a(u,v-u) > (f,v-u) for all v 6 ~,
1
where a(.,.) is a coercive continuous bilinear form on H0(~2). Next,
let w be the solution of
1(~1 )
w E H (~1): a(w,v-w) : (f,v-w) for all v 6 H 0 (cid:12)9
Then (w is extended to be zero in ~2 )
u ~ w in ~2"
We note that the w in the cast of this Lemma satisfies a Dirichlet
boundary value problem. In fact, with the bilinear form given in (1.1)
w satisfies
I
-tAw + bSW2 + w : f in n l
(2.1)
w = 0 on 321.
Problem (2.1), being amenable to the method of matched asymptotic
expansions, provides us, as we shall see below, with an excellent
means to find upper approximations of u .e
A lower approximation Qe of u e is obtained in the following way.
First we construct a function 0 6 C1(~) N H2(~) such that
~U
-eAQ + ~x 2 + Qe ~e < 0
8Qe ^
(2.2) (-EAQe + b~'~2 + ue - fe)Qe = 0 in ~, Qe = 0 on ~,
Qc ~ 0
where fe 6 L2(~) satisfies
^
fe ( f in ~.
Then we apply Lemma 2.2 below to get
QE ~ u~.
Lemma 2.2 (Br&zis 2). Let ~ be an open smoothly bounded set in
~ Let f,f be elements of L2(~) such that f ~ f and let u,Q
be the respective solutions of
u E :K a(u,v-u) > (f,v-u) for all v 6 ,<17
Q e ~: a(Q,v-Q) > (~,v-Q) for all V E ,<~
where a(.,.) is a coercive continuous bilinear form on H~(~). Then
Q ~ u in ~.
First we shall treat the case b = .0 In this case (1.3) becomes
-e&u e + u e = f in ~c, ue = 0 on ~e"
Now, the maximum principle points in the direction of ~_ as a suitable
choice for ~1 in Lemma 2.1. Application of classical singular pertur-
bation techniques to the thus found boundary value problem for an
upper approximation we, that is
-e&w e + w e = f in ~_, we = 0 on ~_,
yields
w e ~ f + Me, as e $ 0,
