Olga Arsen
evna Oleı˘nik
(1925–2001)
Willi Jäger, Peter Lax, and Cathleen Synge Morawetz
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Her Life
1
On October 13, 2001, Olga Arsen evna Oleı˘nik, one
of a handful of truly exceptional women mathe-
maticians of the twentieth century, died in Moscow
at the age of seventy-six, succumbing, after a long
struggle, to cancer.
Olga Oleı˘nik was born in the Ukraine on July 2,
1925. In 1941 the Ukraine was invaded by Germany,
and the machine factory where Olga’s father was
bookkeeper was evacuated to Perm in the Urals. The
sixteen-year-old Olga accompanied him and finished
high school there. Her mother, her sister, and her
nephew remained in the Ukraine. Olga then attended
the University of Perm, to which the mathematics
and mechanics faculty of the Moscow State Univer-
sity had also been evacuated. There her talents came
to the attention of professors Sof ya Yanovskaya
and Dynnikov, who arranged in 1944 for her to be-
come a student in Moscow at the university. She was
married briefly and had one son, to whom she was
devoted and who predeceased her.
Oleı˘nik continued at the university, receiving her
doctor’s degree in 1954 with a thesis in partial dif-
ferential equations (PDEs) under the guidance of
I. G. Petrovskiı˘. The topic, partial differential equa-
tions with small coefficients multiplying the
highest-order terms, heralded some of the under-
lying approaches to much of her future work.
Although she also made contributions to algebraic
geometry, to Hilbert’s sixteenth problem, and to
other topics, her forte and lifetime contributions
were focused on PDEs that arise in very important
applications such as boundary-layer theory and
elasticity. In fact, she constantly emphasized the
role of PDEs in applications [O1].
She remained a devoted student and disciple of
Petrovskiı˘ and his memory for her whole life. She
succeeded him to the chair of differential equations,
and despite the political difficulties of the seven-
ties and eighties, she steered the department for-
ward successfully throughout her tenure. Oleı˘nik
received many prizes and awards in her lifetime;
she was named a member of the Russian Academy
of Sciences in 1990 but had earlier been made an
honorary member of too many foreign academies
and societies to list here. She was also awarded
many honorary degrees as well as prizes both in
her native land and abroad. In all, she published
more than three hundred papers and eight mono-
graphs.
She was deeply involved in improving and ex-
tending contacts between Russian and Western
scientists. She was particularly eager to bring the
Western world to Russian mathematicians and was
therefore anxious to see Western mathematical lit-
erature translated into Russian. An early beneficiary
of such a translation was the new edition of
Courant-Hilbert, Vol. 2, which appeared in Russian
in 1962, the same year it appeared in English. When
Courant died in 1972, Oleı˘nik and Paul Aleksandrov
wrote a long obituary in Uspekhi
2
emphasizing
Willi Jäger (WJ) is a professor in the Institute for Applied
Mathematics at the University of Heidelberg, Germany. His
email address is jaeger@iwr.uni-heidelberg.de .
Peter Lax (PL) is professor emeritus at the Courant
Institute, New York University. His email address is
lax@cims.nyu.edu.
Cathleen Synge Morawetz (CSM) is professor emeritus
at the Courant Institute, New York University. Her email
address is morawetz@cims.nyu.edu .
1
CSM. The author (CSM) is indebted to Tatiana A. Sha-
poshnikova of Moscow University for much of this infor-
mation.
2
P. S. Aleksandrov and O. A. Oleı˘nik, Uspekhi Mat. Nauk
30 (1975), no. 4 (184), 205–26; MR 52 #7786.
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Courant’s contributions to American mathemat-
ics through his institute at New York University.
Oleı˘nik loved to travel. In 1960 she met Courant
and Lax in Moscow on their visit to Alexandrov,
an old friend of Courant’s. Thus when she was in-
vited to a women’s congress in California shortly
afterwards, she was able to accept Courant’s invi-
tation to visit New York. It is hard now to remem-
ber how unusual such a visit was in the decade
following Stalin’s death. She retained an abiding
affection and interest in those working in Courant’s
group: K. O. Friedrichs, F. John, P. D. Lax, and
L. Nirenberg. I met her then for the first time, but
I became close to her only when, at G. Fichera’s in-
vitation, she and I spent a month in 1965 together
at the University of Rome. The university was
“under siege” by the student body, and it was only
Fichera’s political skill that made it possible for
Oleı˘nik to give her talks. We were excluded from
the campus much of the time, so together we hap-
pily toured the wonderful sights of Rome instead.
Oleı˘nik was a very private person. In the latter
part of her life she suffered badly from knee trou-
ble, which prevented her from walking properly and
sometimes kept her hospitalized. During one such
long stay she kept occupied by writing another
book. For a woman who had been so active as a
mathematician and with her students (she super-
vised fifty-six dissertations), these spells of inac-
tivity were trying. She told me once that as a young
student she had worked in what we would call a
lumber camp and how much she, in contrast to the
other girls, had enjoyed doing the hard physical
work.
The life span of Olga Oleı˘nik was from the early
days of the sovietization of Russia to the complete
collapse of communism. I never heard her express
a political opinion nor make a complaint about
how things were going. She clearly tried to live
within the system she had been raised in, and her
model was always her teacher, Petrovskiı˘. Uprooted
from the Ukraine by the war, she identified herself
as a Russian, not as a Ukrainian. Of one thing I am
sure: she believed a larger picture of her country
was the right one.
Oleı˘nik was devoted to mathematics. She drove
her theorems to their absolute limits. She never
seemed happier than when she was doing mathe-
matics or working with her students, the most im-
portant elements in her life. She left a great deal
of unfinished work, although she had published so
much. We may hope that this work will be com-
pleted by her students and colleagues.
Oleı˘nik was one of the major figures in the
study, during the fifties and sixties, of elliptic
and parabolic equations. This study was the major
field of partial differential equations at the time,
and many mathematicians such as Agmon, John,
Ladyzhenskaya, Morrey, Nirenberg, and Vishik
were involved in it. It would
make this article too tech-
nical to describe Oleı˘nik’s
most significant achieve-
ments in this area, and we
will confine the scientific
description to some of the
many other areas to which
she made even more signif-
icant contributions.
Nonlinear Hyperbolic
Equations
3
During the years 1954–61,
Olga Oleı˘nik studied the
theory of nonlinear hyper-
bolic conservation laws and
the propagation of shock
waves. Her contributions
were basic and extremely
original. Her 1957 paper in
the Uspekhi [O2] was par-
ticularly influential. The starting point of her work,
like much in this field, was Eberhard Hopf’s fun-
damental paper of 1950 [H].
Oleı
˘nik’s results deal mainly with the existence,
uniqueness, and properties of solutions of the
initial-value problem for single conservation
laws. She showed that solutions satisfy one-sided
Lipschitz conditions and formulated for flux
functions that have points of inflection what today
is called the Oleı˘nik entropy condition.
Her principal tool was the parabolic perturba-
tion of a conservation law. She proved that as the
coefficient of viscosity tends to zero, the solution
of the initial-value problem for the parabolic
equation tends to a solution, in the sense of
distributions, of the conservation law, and that
this limit satisfies the Oleı˘nik entropy condition.
In 1957 Oleı˘nik and Vvedenskaya considered a
discretized form of a conservation law and proved
the convergence of their solutions to solutions
of the conservation law, as the discretization
parameter tends to zero [OV]. This provided
another approach to the problem.
Oleı˘nik also studied special systems of pairs of
conservation laws, which are derivable from second-
order equations; in 1966 she proved uniqueness of
the initial-value problem in the class of solutions
that satisfy a one-sided Lipschitz condition [O4].
Boundary Layer Theory
4
In their simplest form, Prandtl’s steady 2D bound-
ary-layer equations are obtained from the Navier-
Stokes equations by stretching the variables
appropriately. If
x is distance along the boundary,
3
PL.
4
CSM.
Olga Oleı
˘nik.
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and y , a depth variable, is distance from the
boundary, the equations become after rescaling:
u
x
+ v
y
= 0
(mass) ,
uu
x
+ vu
y
=
−p
x
+ µu
xx
(x-momentum),
p
y
= 0
( y-momentum) .
The standard boundary conditions are u = v = 0 on
y = 0 and u
→ U(x) as y → ∞. The function U is
the speed of the flow “along the body” given by
inviscid incompressible flow and is related to the
pressure by Bernoulli’s law,
p +
1
2
U
2
= constant.
Upstream, say x = 0 , the incoming flow is pre-
scribed. Experimental observations confirm the
scaling.
Thus we have a nonlinear parabolic equation for
u with the role of time derivative played by the
Lagrangian along the particle path. The coefficient
v is to be found from the mass equation.
The crucial question of how the separation or
breaking away of the boundary layer from the
boundary takes place is still not resolved. Oleı˘nik’s
work concentrated on where and why separation
does NOT take place by proving that there exists
a unique boundary flow and hence one without
separation, provided in our simple case that p
x
= 0
[O3]. In real life on an airplane wing this is
achieved by artificial suction, and Oleı˘nik exam-
ined this question in detail (see [O4]). In 1997, after
many papers with others from 1963 on concern-
ing many different kinds of flow from magneto-
hydrodynamics to non-Newtonian fluids, she
wrote a book on the subject summarizing and
proving many background results and her own key
theorems [O5].
Singular Elliptic and Parabolic Theory
5
Equations of elliptic or parabolic type enjoyed a
great deal of attention in the decades following
World War II, and Oleı˘nik’s thesis was in this area.
She followed it with many other results, but she may
well be best remembered for her work on degen-
erate problems where an elliptic equation becomes
parabolic at points or segments on the boundary
or even in whole patches of a domain. One of
the simplest examples is for the solutions u of
the wave equation depending only on x/t. Thus
with r =
|x|, ρ = r/t, the equation for w = ru is
(ρ
2
− 1)w
ρρ
+ ρw
ρ
−
Λw = 0, where Λ is the angu-
lar part of the Laplacian, and the degeneracy, not
surprisingly, occurs on the light cone ρ = 1 .
Oleı˘nik determined the conditions for well-
posedness in many cases and in 1964 generalized
and completed the problems posed by G. Fichera
[F] (see [O6]). The technique was to add a term
so that the equation remains elliptic and to obtain
estimates for the limiting case when this term
vanishes. By 1977, however, Oleı˘nik was directly
using a priori energy estimates [O7].
Homogenization of Differential Equations
6
Oleı˘nik’s contributions to homogenization and her
impact on this field were very broad. Mathemati-
cal modelling leads naturally to systems with mul-
tiple scales, for example in electromagnetism, me-
chanics, material science, and flows through porous
media or biological tissues. The resulting underlying
structures may be highly oscillatory in space and
time. To analyze the transition between the dif-
ferent scales and to derive effective model equa-
tions are great challenges to mathematical research
and imply many practical consequences. The main
problem lies in identifying the proper scaling and,
if possible, deriving effective equations for limits
when a “scale” goes to 0 or
∞. The random situa-
tion especially, the most important in real life,
poses a lot of difficult questions.
The challenge is to derive quantitative results,
including estimates for the approximations. Ho-
mogenization, a special kind of averaging, started
as a mathematical discipline only about thirty years
ago, although in physics and engineering such av-
eraging methods had been used for many years to
determine effective properties and effective laws
for heterogeneous media.
The theory of homogenization is strongly linked
to Oleı˘nik’s name. An impressive group of mathe-
maticians from the Department of Differential
Equations at Moscow State University formed a
leading team, and with them Oleı˘nik developed a
fruitful cooperation with France, Germany, and
Italy, especially after the fall of the iron curtain. The
long list of contributions of Oleı˘nik and her cowork-
ers, covering all of the main areas of homogeniza-
tion, can be found in the monographs by Oleı˘nik,
Shamaev, and Yosifian [OSY] and by Jikov, Kozlov,
and Oleı˘nik [JKO]. These books are important
sources of information and original ideas, and
cover important topics in theory and applications.
Oleı˘nik’s contributions were mainly in devel-
oping the necessary tools to control the scale limit
for initial and boundary value problems for systems
with oscillatory coefficients or in domains with
complex structures, holes, or oscillatory bound-
aries. Essentially two methods are available: energy
methods, based on proper estimates of the solu-
tions and compactness results; and multiscale
expansions.
Introducing fast and slow (microscopic and
macroscopic) variables, one starts from a formal
expansion with respect to the scale parameter,
obtaining a recursive system for the coefficients
5
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of the expansion. Oleı˘nik and her coworkers devel-
oped a systematic technique for determining
approximations and validating the expansion. In
the case of periodic structures, the approximation
problem in the interior of the domain is reduced to
solving a system with respect to fast variables in the
scaled periodicity cell coupled with slow equations
in the whole domain. Effective equations can often
be obtained by such averaging. The cited monographs
contain many examples.
The analysis at the boundary or at interfaces is
more complicated, and here the contributions of
Oleı˘nik have been especially important. She and her
coworkers did pioneering work on the approximation
near the boundary. The analysis of boundary layers
is a nontrivial problem in the case of periodic struc-
tures. Oleı˘nik and her coworkers obtained optimal
results when the boundary is flat and in rational
position with respect to the periodicity lattice. In
this situation an unbounded boundary-layer
cell has to be considered. Ideas developed in the
papers of Oleı˘nik and Yosifian can be used to study
problems in domains where the scale changes across
an interface. Typical examples are processes in
partially porous or perforated domains where the
derivation of effective transmission conditions
and the estimate of the errors at the interface are
the main aims. Flow and transport through filters
are practical, important examples. During her last
years, despite her serious health problems, Olga
Oleı˘nik was also involved with such multiscale
problems, strongly motivated by the many possible
applications. Oleı˘nik also developed a spectral
theory adapted to homogenization.
Stochastic homogenization, mainly in the case
of random coefficients, is the main topic in [JKO].
Several important contributions were obtained in
Oleı˘nik’s group. Homogenization of stochastic
processes, also in random geometry, is a crucial
topic of ongoing research, and here Oleı˘nik’s in-
fluence is clear.
Problems in Elasticity
7
Traditionally the links between mathematics and
mechanics are very strong in Russia, and Oleı˘nik’s
mathematical research reflected this. Prominent
examples are Korn-type inequalities, basic in prov-
ing existence and estimating the solutions of
the main boundary-value problems in elasticity.
Consider the tensor e(u) defined by
e
ij
(u) = ∂
j
u
i
+ ∂
i
u
j
.
Korn inequalities relate the L
2
-norm of e(u) with
the L
2
-norm of
∇ u, for instance in the form of the
inequality
u
H
1
(Ω)
≤ C( u
L
2
(Ω)
+ e( u)
L
2
(Ω)
) .
In general, assumptions on the domain are nec-
essary: for example, the assumption that Ω is
bounded and a Lipschitz domain. Also, an optimal
constant C is wanted, with the constant depending
on Ω. Kondratev and Oleı˘nik obtained the estimate
by a rather simple proof, with asymptotically sharp
constants in the case of star-shaped domains. As
Oleı˘nik liked to say, the original proof was nineteen
pages, Friedrichs’s proof was nine pages, and they
had reduced it to four pages [OSY, Chapter I, §2].
They also proved further Korn-type inequalities
that are even more useful.
References
[F] G. F
ICHERA
, Sulle equazioni differenziali lineari ellittico-
paraboliche del secondo ordine, Atti Accad. Naz. Lin-
cei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. (8) 5 (1956), 1–30.
[H] E
BERHARD
H
OPF
, The partial differential equation
u
t
+ uu
x
= µu
xx
, Comm. Pure Appl. Math. 3 (1950),
201–30.
[JKO] V. V. J
IKOV
, S. M. K
OSLOV
, and O. A. O
LE
ı˘
NIK
, Homog-
enization of Differential Operators and Integral Func-
tionals, translated from the Russian by G. A. Yosifian,
Springer-Verlag, 1994.
[O1] O. A. O
LE
ı˘
NIK
, The place of the theory of differential
equations in contemporary mathematics and its appli-
cations (Russian), Differential Equations and Their Ap-
plications, Moskov. Gos. Univ., Moscow, 1984, pp. 4–17.
[O2] ——— , Discontinuous solutions of non-linear differ-
ential equations, Uspekhi Mat. Nauk 12 (1957), no. 3
(75), 3–73; English translation in Amer. Math. Soc.
Transl. (2) 26 (1963), 95–172.
[O3] ——— , On the equations of a boundary layer, Semi-
nari 1962/63 Anal. Algebra Geom. e Topol., Vol. 1, Ist.
Naz. Alta Mat. Ediz. Cremonese, Rome, 1962/1963,
pp. 372–87.
[O4] ——— , On the mathematical theory of boundary
layer for an unsteady flow of incompressible fluid,
Prikl. Mat. Mekh. 30, 801–21; English translation in
J. Appl. Math. Mech. 30 (1966), 951–74 (1967).
[O5] O. A. O
LE
ı˘
NIK
and V. N. S
AMOKHIN
, Mathematical Meth-
ods in Boundary-Layer Theory, Fizmatlit “Nauka”,
Moscow, 1997; translated into English as Mathemati-
cal Models in Boundary Layer Theory, Chapman &
Hall, 1999.
[O6] O. A. O
LE
ı˘
NIK
, On a problem of G. Fichera, Dokl. Akad.
Nauk SSSR 157 (1964), 1297–1300 (Russian).
[O7] O. A. O
LE
ı˘
NIK
and G. A. I
OSIF JAN
, Removable boundary
singularities and uniqueness of solutions of boundary
value problems for second order elliptic and parabolic
equations, Funktsional. Anal. i Prilozhen. 11 (1977),
no. 3, 54–67, 96 (Russian).
[OSY] O. A. O
LE
ı˘
NIK
, A. S. S
HAMAEV
, and G. A. Y
OSIFIAN
, Math-
ematical Problems in Elasticity and Homogenization,
North-Holland, 1992.
[OV] O. A. O
LE
ı˘
NIK
and N. D. V
VEDENSKAYA
, The solution of
Cauchy problem and boundary value problem for
nonlinear equations in the discontinuous functions
class, Dokl. Akad. Nauk SSSR (N.S.) 113 (1957), 503–6
(Russian).
7
WJ.
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