-
The Legacy of VladimirAndreevich Steklov
Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz
Kwasnicki,Alexander Nazarov, Sergey Poborchi, Iosif Polterovich
and Bartomiej Siudeja
Vladimir Andreevich Steklov, an outstanding Rus-sian
mathematician whose 150th anniversary iscelebrated this year,
played an important role in thehistory of mathematics. Largely due
to Steklovs ef-forts, the Russian mathematical school that gave
theworld such giants as N. Lobachevsky, P. Chebyshev,and A.
Lyapunov, survived the revolution and con-tinued to flourish
despite political hardships. Steklovwas the driving force behind
the creation of thePhysicalMathematical Institute in starving
Petro-grad in 1921, while the civil war was still ragingin the
newly Soviet Russia. This institute was thepredecessor of the now
famous mathematical insti-tutes in Moscow and St. Petersburg
bearing Steklovsname.
Steklovs own mathematical achievements, albeitless widely known,
are no less remarkable thanhis contributions to the development of
science.The Steklov eigenvalue problem, the PoincarSteklov
operator, the Steklov functionthere existprobably a dozen
mathematical notions associatedwith Steklov. The present article
highlights some of
Nikolay Kuznetsov heads the Laboratory for MathematicalModelling
of Wave Phenomena at the Institute for Problemsin Mechanical
Engineering, Russian Academy of Sciences.His email address is
[email protected].
Tadeusz Kulczycki and Mateusz Kwasnicki are professorsat the
Institute of Mathematics and Computer Science,Wrocaw University of
Technology. Tadeusz Kulczyckisemail address is
[email protected] andMateusz Kwasnickis email address
is [email protected].
Alexander Nazarov is a leading researcher at the Labora-tory of
Mathematical Physics, St. Petersburg Department ofthe Steklov
Mathematical Institute, and a professor at theDepartment of
Mathematical Physics, Faculty of Mathemat-ics and Mechanics, St.
Petersburg State University. His emailaddress is
[email protected]. His work was sup-ported by RFBR grant
11-01-00825 and by the St. PetersburgUniversity grant
6.38.670.2013.
DOI: http://dx.doi.org/10.1090/noti1073
V. A. Steklov in the 1920s.
the milestones of his career, both as a researcherand as a
leader of the Russian scientific community.
Sergey Poborchi is a professor in the Department of Par-allel
Algorithms, Faculty of Mathematics and Mechan-ics, St. Petersburg
State University. His email address [email protected]. His work
was supported by RFBRgrant HK-11-01-00667/13.
Iosif Polterovich is a professor in the Dpartement
demathmatiques et de statistique, Universit de Montral. Hisemail
address is [email protected].
Bartomiej Siudeja is a professor in the Department
ofMathematics, University of Oregon. His email address
[email protected].
The authors are grateful to David Sher for proofreading
thearticle.
January 2014 Notices of the AMS 9
-
Aleksandr Mikhaylovich Lyapunov in the 1900s.
The article is organized as follows. It startswith a brief
biography of V. A. Steklov writtenby N. Kuznetsov. The next
section, written byN. Kuznetsov, A. Nazarov, and S. Poborchi,
focuseson Steklovs work related to several celebratedinequalities
in mathematical physics. The remain-ing two sections are concerned
with some recentdevelopments in the study of the Steklov
eigen-value problem, which is an exciting and rapidlydeveloping
area on the interface of spectral theory,geometry and mathematical
physics. The highspots problem for sloshing eigenfunctions
isdiscussed in the section written by T. Kulczycki, M.Kwasnicki,
and B. Siudeja. In particular, the authorsexplain why it is easier
to spill coffee from a mugthan to spill wine from a snifter. An
overview ofsome classical and recent results on
isoperimetricinequalities for Steklov eigenvalues is presented
inthe last section, written by I. Polterovich.
Many topics to which Steklov contributed in amajor way are
beyond the scope of the presentarticle. For further references, see
[14], [24], [31]and [64].
A Biographical Sketch of V.A. Steklov
Vladimir Andreevich Steklov was born in NizhniNovgorod on
January 9, 1864 (=December 28, 1863,old style). His grandfather and
great-grandfatheron the fathers side were country clergymen.His
father, Andrei Ivanovich Steklov, graduatedfrom the Kazan
Theological Academy and taughthistory and Hebrew at the Theological
Seminaryin Nizhni Novgorod. Steklovs mother, EkaterinaAleksandrovna
(ne Dobrolyubova), was a daughterof a country clergyman as well.
Her brother, NikolayAleksandrovich, was a prominent literary critic
andone of the leaders of the democratic movementthat aimed to
abolish serfdom in Russia.
At ten years of age, Steklov enrolled into theAlexander
Institute (a gymnasium that had manynotable alumni, including the
famous Russian com-poser M. Balakirev) in Nizhni Novgorod.
Steklovscritical thinking manifested itself at a very early
age. In his diaries, Steklov describes how he waschastised by
the school principal for a composi-tion deemed disrespectful
towards the Russianempress Catherine II.
I said to myself: Aha! It occurs to me thatI have my own point
of view on historicalevents which is different from that of
myschoolmates and teachers. [] It was theprincipal himself who
proved that I am, insome sense, a self-maintained thinker
andcritic. This was the initial impact that ledto my mental
awakening; I realized that Iam a human being able to reason and,
whatis important, to reason freely. [] Soon, myfree thinking
encompassed the religion aswell. [] Thus, the cornerstone was laid
formy future complete lack of faith.
After graduating from school in 1882, Stekloventered the Faculty
of Physics and Mathematics ofMoscow University. Failing to pass an
examinationin 1883, he left Moscow and the same yearentered a
similar faculty in Kharkov. There hemet A. M. Lyapunov, and this
encounter becamea turning point in his life. Steklov graduated
in1887, but remained at the university workingunder Lyapunovs
supervision towards obtaininghis Masters Degree. In the beginning
of 1890,Steklov married Olga Nikolaevna Drakina, who wasa music
teacher; their marriage lasted for 31 years.In the fall of the same
year, he was appointedLecturer in Elasticity Theory. In 1891, the
Steklovsdaughter Olga was born and, presumably, thisevent delayed
the defence of his Masters thesis,On the motion of a solid body in
a fluid, until 1893.The same year, Steklov began lecturing at
KharkovInstitute of Technology, combining it with his workat the
university; the goal was to improve hisfamilys financial situation,
given that his wife hadto leave her job after giving birth to their
child.The sudden death of their daughter in 1901 was aheavy blow to
Steklov and his wife, and caused asix-month break in his research
activities.
He was appointed to an extraordinary pro-fessorship in mechanics
in 1896. The first in aseries of full-length papers, which formed
thecore of his dissertation for the Doctor of Sciencedegree,
appeared in print the same year (not tomention numerous brief notes
in Comptes rendus).The dissertation entitled General methods of
solv-ing fundamental problems in mathematical physicswas published
as a book in 1901 by the KharkovMathematical Society [53].
At the time of completing his DSc dissertation,Steklov began to
publish his results in French.Since then, most of his papers were
written inFrench the language widely used by Russianmathematicians
to make their results accessible
10 Notices of the AMS Volume 61, Number 1
-
in Europe. Unfortunately, this did not preventsome of his
results from remaining unnoticed. Inparticular, this concerns the
so-called Wirtingersinequality which was published by Steklov in
1901in Annales fac. sci. Toulouse (see details in the nextsection).
Even before that, Steklov became veryactive in corresponding with
colleagues abroad(J. Hadamard, A. Kneser, A. Korn, T. Levi-Civita,
E.Picard, S. Zaremba, and many others were amonghis
correspondents); these contacts were of greatimportance for him,
residing in a provincial city.In 1902, Steklov was appointed to an
ordinaryprofessorship in applied mathematics and waselected a
corresponding member of the Academyof Sciences in St. Petersburg
the next year.
In 1903, the Steklovs went on a summer vacationto Europe. Some
details of this trip are described inone of Steklovs letters to
Lyapunov (see [60], letter29). In particular, the meeting with J.
Hadamard inParis:
Somehow, Hadamard found me himself;presumably, he had learned my
addressfrom A. Hermann [the well-known publisher].Once he missed
me, but the next day hecame at half past eight in the morningwhen
we had just awakened. He arrivedto Paris to stay for two days
examiningfor baccalaurat [at some lyce]; on theday of his returning
to the countryside,where he spends summer, he called on mebefore
examination. His visit lasted onlyhalf-an-hour, but he told as much
as anotherperson would tell in a whole day. He is amodel Parisian,
very agile and swift to react;he behaved so as we are old friends
whohad not seen each other for some time.
In 1908, the Lyapunovs and the Steklovs travelledto Italy
together, where A. M. and V. A. participatedin the Rome ICM. At the
Cambridge ICM (1912),Steklov was elected a vice president of the
congress(Hadamard and Volterra were the other two vicepresidents).
The Toronto ICM (1924) was the thirdand the last one for
Steklov.
Let us turn to the PetersburgPetrogradLenin-grad period of
Steklovs life. In 1906, he succeeded(after several attempts) in
moving to St. Petersburg.It is a remarkable coincidence that a
group of verytalented students entered the university the sameyear.
In the file of M. F. Petelin, who was one ofthem, this fact was
commented on by Steklov asfollows:
I should note that the class of 1910 isexceptional. In the class
of 1911 and amongthe fourth-year students who are about tograduate
there is no one equal in knowledgeand abilities to Messrs.
Tamarkin, Fried-mann, Bulygin, Petelin, Smirnov, Shohat, and
Aleksandr Aleksandrovich Friedmann in the1920s.
others. There was no such case during thefifteen years of
teaching at Kharkov Univer-sity either. This favorable situation
shouldbe used for the benefit of the University.
Steklov had done his best to nurture his students(see [66]). His
dedication as an advisor was rewardedby the outstanding
achievements of the members ofhis group, the most famous of which
is Friedmannssolution of Einsteins equations in the generaltheory
of relativity. The future fate of Steklovsstudents varied greatly;
two of them (Bulyginand Petelin) died young. Tamarkin and
Shohatemigrated to the USA and became prominentmathematicians
there. It is worth noting thatJ.D. Tamarkin has more than 1500
mathematicaldescendants, and through him the St.
Petersburgmathematical tradition had a profound impacton the
American mathematics. Tamarkins escapefrom the Soviet Union was
quite an adventure.While secretly crossing the frozen Chudskoe
Lakein order to reach Latvia, he was fired on by theSoviet border
guards. As E. Hille wrote:
One of J. D.s best stories told how he triedto convince the
American consul in Rigaof his identity: the consul attempted
toexamine him in analytic geometry, but ranout of questions and
gave up.
Friedmann and Smirnov became prominentscientists staying in
Leningrad. Together with theircolleagues and students (N. M.
Gnther, A. N. Krylov,V. A. Fock, N. E. Kochin, S. G. Mikhlin, and
S. L. So-bolev to name a few) they organized the school
ofmathematical physics in LeningradSt. Petersburg,the foundation of
which was laid by Steklov.
January 2014 Notices of the AMS 11
-
Vladimir Ivanovich Smirnov in the 1950s.
Steklovs scientific career was advancing. In1910, he was elected
an adjunct member of theAcademy of Sciences; two years later he
waselected extraordinary and then ordinary academi-cian within a
few months. After his election tothe executive committee of the
Academy in 1916,Steklov reduced his work at the university
andabandoned it completely in 1919, being electedvice president of
the Academy. It would take toomuch space to describe everything he
had accom-plished at this post by the time of his unexpectedand
untimely death on May 30, 1926. (His wifedied in 1920 from illness
caused by undernourish-ment.) Therefore, only his role in
establishing thePhysicalMathematical Institutethe predecessorof
institutes named after himwill be outlined.
In January 1919, a memorandum was submittedto the Academy in
which Steklov, A. A. MarkovSr., and A. N. Krylov proposed to
establish aMathematical cabinet as an initial stage in
furtherdevelopment of the Academys Department ofPhysics and
Mathematics (the Physical cabinetexisted in the Academy since its
foundation in1724, and it was reorganized into a laboratoryin
1912). Later the same year, the initiative wassupported, and
Steklov became the head of thenew institution named after P. L.
Chebyshev andA. M. Lyapunov. In January 1921, Steklov
submittedanother memorandum, pointing out the necessityto merge the
Physical laboratory and the recentlyestablished Mathematical
cabinet. As he writes,Mathematics and physics have now merged to
suchan extent that it is sometimes difficult to find theline that
divides them. Nowadays, this viewpointis shared by many
mathematicians; however, atthe time it was quite unusual.
The same January 1921, Steklov, S. F. Oldenburg(the Permanent
Secretary of the Academy), and V. N.Tonkov (Head of the Military
Medical Academy)visited V. I. Lenin in Moscow. In his
recollectionsabout Lenin, Maxim Gorky (the famous Russianwriter
close to the Bolsheviks), who had beenpresent at this meeting,
wrote (see also [67]):
They talked about the necessity to reor-ganize the leading
scientific institution inPetersburg [the Academy]. After seeing
offhis visitors, Lenin said with satisfaction.What clever men!
Everything is simple forthem, everything is formulated
rigorously;it is clear immediately that they know wellwhat they
want. It is a pleasure to work withsuch people. The best impression
Ive gotfrom He named one of the most prominent Rus-sian scientists;
two days later, he told meby phone.Ask S[teklov] whether he is
going to workwith us.When S[teklov] accepted the offer, this wasa
real joy for Lenin; rubbing his hands, hejoked.Just wait! One
Archimedes after the other,well gain support of all of them in
Russiaand in Europe, and then the World, willinglyor unwillingly,
will turn over!
Indeed, after the Bolshevik government declaredthe so-called New
Economic Policy, the deadlockover Academy funding was broken thanks
tothe improving economic situation in the country.This resulted in
the creation of the PhysicalMathematical Institute later in 1921,
and Steklovwas appointed its first director. In his
recollections[59] completed in 1923, he writes:
Another achievement of mine for the ben-efit of the Academy and
the developmentof science in general is establishing
thePhysicalMathematical Institute with thefollowing divisions:
mathematics, physics,magnitology and seismology. Its work isstill
in the process of being organized, thefunding is scarce and
difficult to obtain.The number of researchers is still
negligible[], but a little is better than nothing.
After Steklovs death the institute was namedafter him. In 1934,
simultaneously with reloca-tion of the Academy from Leningrad to
Moscowthe PhysicalMathematical Institute was dividedinto the
following two: the P. N. Lebedev Physi-cal Institute and the V. A.
Steklov MathematicalInstitute (even before that the division of
seismol-ogy became a separate institute). The Leningrad(now St.
Petersburg) Department of the latter wasfounded in 1940, and it is
an independent institutesince 1995. To summarize, it must be said
thatthe role of V. A. Steklov in Petrograd was similarto that of R.
Courant who organized mathematicalinstitutes, first in Gttingen and
then in New York.
In 1922 and 1923, Steklovs monograph [58]was published; it
summarizes many of his resultsin mathematical physics. Based on the
lectures
12 Notices of the AMS Volume 61, Number 1
-
The title page of Steklovs monograph [58].
given in 19181920, this book is written in Russian,despite the
fact that the corresponding papersoriginally appeared in French.
More material waspresented in the lecture course than was
includedin [58], and Steklov planned to publish the 3rdvolume about
his results concerning fundamen-tal functions (that is,
eigenfunctions of variousspectral problems for the Laplacian) and
someapplications of these functions. Unfortunately,the
administrative duties prevented him fromrealizing this project.
However, one gets an ideaabout the probable contents of the
unpublished3rd volume from the lengthy article [55], in
whichSteklov developed his approach to fundamentalfunctions (it is
briefly outlined by A. Kneser [28],section 5). It is based on two
different kindsof Greens function, and this allowed Steklov toapply
the theory of integral equations worked outby E. I. Fredholm and D.
Hilbert shortly beforethat. It is worth mentioning that [58] was
listedamong the most important mathematical bookspublished during
the period from 1900 to 1950(see Guidelines 19001950 in [45];
another itemconcerning mathematical physics published thesame year
is Hadamards Lectures on Cauchysproblem in linear partial
differential equations).
A great part of the material presented in thesecond volume of
[58] is taken from the article[54], which is concerned with
boundary value
problems for the Laplace equation. It is Steklovsmost cited
work, but, confusingly, his initials aregiven incorrectly in many
citations of this paper.Indeed, R. Weinstock found the following
spectralproblem
(1) u = 0 in D, un= u on D,
in [54], and for this reason he called it the Stekloffproblem;
here n is the exterior unit normal on Dand is a non-negative
bounded weight function.In fact, Steklov introduced this problem in
his talkat a session of the Kharkov Mathematical Societyin December
1895; it was also studied in his DScdissertation. Nowadays, it is
mainly referred to asthe Steklov problem, but, sometimes, is still
calledthe Stekloff problem. In [69], Weinstock initiatedthe study
of this problem, but, unfortunately, citing[54], he supplied
Steklovs surname with wrong ini-tials, which afterwards were
reproduced elsewhere.Weinstocks result and its later developments
arediscussed in detail in the last section.
It must be emphasized that the legacy of Steklovis multifaceted
(see [67]). He wrote biographiesof Lomonosov and Galileo, an essay
about therole of mathematics, the travelogue of his trip toCanada,
where he participated in the 1924 ICM,his correspondencepublished
(see [60] and [61])and unpublished; the recollections [59] and
stillunpublished diaries. Fortunately, many excerptsfrom Steklovs
diaries are quoted in [66] and someof them appeared in [43]. The
most expressiveis dated September 2, 1914, one month after
theRussian government declared war:
St. Petersburg has been renamed Petrogradby Imperial Order. Such
trifles are all ourtyrants can doreligious processions
andextermination of the Russian people by allpossible means.
Bastards! Well, just youwait. They will get it hot one day!
What happened in Russia during several yearsafter that confirms
clearly how right was Steklovin his assessment of the Tsarist
regime. In hisrecollections [59] written in 1923, he
describesvividly and, at the same time, critically thecomplete
bacchanalia of power preceding thecollapse of autocracy and
[Romanovs] dynasty inFebruary 1917 (old style), the shameful
Provisionalgovernment headed by Kerensky, the fast end ofwhich can
be predicted by every sane person, andhow the Bolshevik government
[] decided toaccomplish the most Utopian socialistic ideas inthe
multi-million Russia. The list can be easilycontinued.
The pinpoint characterization of Steklovspersonality was given
by A. Kneser (see [28]):
January 2014 Notices of the AMS 13
-
Everybody who maintained contact withSteklov was impressed by
his personality.He was highly educated in the traditionsof the
European culture, but at the sametime maintained distinctive
features char-acteristic to his nation. He was not only adeep
mathematician, but also a connoisseurof music and art. [] Besides,
he was askillful mediator between scientists and thenew government
in Moscow. Thus, his rolewas crucial for the survival of the
Russianscience and its restoration (predominantly,in the Academy
and its institutes) after therevolutions and the Civil War.
To conclude this section, we list some ofSteklovs awards and
distinctions. He was amember of the Russian Academy of Sciences,a
corresponding member of the Gttingen Acad-emy of Science, and a
Doctor honoris causa ofthe University of Toronto. The mathematical
So-cieties in Kharkov, Moscow, and St. Petersburg (inPetrograd, it
was reorganized into the PhysicalMathematical Society) counted him
among theirmembers, as well as the Circolo Matematico
diPalermo.
V.A. Steklov and the Sharp Constants inInequalities of
Mathematical Physics
In 1896, Lyapunov established that the trigono-metric Fourier
coefficients of a bounded functionthat is Riemann integrable on
(pi,pi) satisfy theclosedness equation. He presented this result
ata session of the Kharkov Mathematical Society,but left it
unpublished. The same year, Steklovhad taken up studies of the
closedness equationinitiated by his teacher; Steklovs extensive
workon this topic lasted for 30 years until his death.For this
reason A. Kneser [28] referred to thisequation as Steklovs favorite
formula. It shouldbe mentioned that the term closedness equationwas
introduced by Steklov for general orthonormalsystems, but only in
1910 (see brief announcements[56] and the full-length paper
[57]).
The same year (1896), Steklov [50] proved thatthe following
inequality (nowadays often referredto as Wirtingers inequality)
(2) l
0u2(x)dx
(lpi
)2 l0[u(x)]2 dx
holds for all functions which are continuouslydifferentiable on
[0, l] and have zero mean. For thispurpose, he used the closedness
equation for theFourier coefficients of u (the corresponding
systemis {cos (kpix/l)}k=0 normalised on [0, l]). Inequality(2) was
among the earliest inequalities with asharp constant that appeared
in mathematicalphysics. It was then applied to justify the
Fouriermethod for initial-boundary value problems for
the heat equation in two dimensions with variablecoefficients
independent of time. Later, Steklovjustified the Fourier method for
the wave equationas well. The fact that the constant in (2) issharp
was emphasized by Steklov in [52], wherehe gave another proof of
this inequality (seepp. 294296). There is another result proved
in[52] (see pp. 292294); it says that (2) is true forcontinuously
differentiable functions vanishing atthe intervals end-points, and
again the constantis sharp. In the first volume of his
monograph[58], Steklov presented inequality (2) along with
itsgeneralization.
The problem of finding and estimating sharpconstants in
inequalities attracted much attentionfrom those who work in theory
of functionsand mathematical physics (see, for example,
theclassical monographs [20] and [49]). It is worthmentioning that
in the famous book by Hardy,Littlewood, and Polya [20, section
7.7], inequality (2)is proved under either type of conditions
proposedby Steklov; however, the authors call it
Wirtingersinequality and refer to the book of Blaschke (whowas a
student of Wirtinger) published in 1916 [5, p.105], twenty years
after the publication of Steklovspaper [50]. This terminology
became standard.However, the controversy does not end here; werefer
to [41] for other historical aspects of thisinequality.
S. G. Mikhlin (he graduated from LeningradUniversity a few years
after Steklovs death; see hisrecollections of student years [40])
emphasizedthe role of sharp constants in his book [39]. Letus quote
the review [44] of the German version of[39]:
[This book] is devoted to appraising the(best) constantsexact
results or explicit(numerical) estimatesin various inequali-ties
arising in analysis (=PDE). [] This isthe most original work, a
bold attack in adirection where still very little is known.
In 1897, Steklov published the article [51], inwhich the
following analogue of inequality (2) wasproved:
(3)Du2 dx C
D|u|2 dx.
Here stands for the gradient operator and theintegral on the
right-hand side is called the Dirichletintegral. The assumptions
made by Steklov are asfollows: D is a bounded three-dimensional
domainwhose boundary is piecewise smooth and u is areal C1-function
on D vanishing on D. Again,inequality (3) was obtained by Steklov
with thesharp constant equal to 1/D1 , where
D1 is the
smallest eigenvalue of the Dirichlet Laplacian inD. In the early
1890s, H. Poincar [46] and [47]obtained (3) using different
assumptions, namely, u
14 Notices of the AMS Volume 61, Number 1
-
Figure 1. High spot in a coffee cup.
has zero mean over D which is a union of afinite number of
smooth convex two- and three-dimensional domains, respectively. In
the lattercase, the sharp constant in (3) is 1/N1 , where
N1 is
the smallest positive eigenvalue of the NeumannLaplacian in
D.
Ifu vanishes on D (this is understood as follows:u can be
approximated in the norm uL2(D) bysmooth functions having compact
support in adomain D Rn, n 2, of finite volume), then (3)is often
referred to as the Friedrichs inequality.In fact K. O. Friedrichs
[13] obtained a slightlydifferent inequality under the assumption
thatD R2. Namely, he proved:(4)
Du2 dx C
[D|u|2 dx+
Du2 dS
],
where dS denotes the element of length of D.Generally speaking,
(4) holds for all boundeddomains in Rn for which the divergence
theoremis true (see [38], p. 24).
Inequality (3) for functions u with a zero meanvalue over D is
equivalent to the following one (itis called the Poincar
inequality):
u uL2(D) CuL2(D),(5)where u =
D u(x)dxmeasn D ,
here measnD is the n-dimensional measure ofD. Note that the
sharp constant here is 1/
N1 .
Some requirements must be imposed on D for thevalidity of (5).
Indeed, as early as 1933 O. Nikodm[42] (see also [38], p. 7)
constructed a boundedtwo-dimensional domain D and a function
withfinite Dirichlet integral over D such that inequality(5) is not
true. Another example of a domain withthis property is given in the
classical book [10] byCourant and Hilbert (see ch. 7, sect.
8.2).
In conclusion, we consider the following bound-ary analogue of
inequality (5):
u uGL2(G) CuL2(D),where uG =
G u(x)dS
measn1 G .
Figure 2. High spot in a snifter.
HereD is a bounded Lipschitz domain inRn, n 2,whereas G is a
part of D possibly coincidingwith D. The sharp constant in this
inequality isequal to 1/
S1 , where
S1 is the smallest positive
eigenvalue of the following mixed (unless G = D)Steklov
problem:
u = 0 in D, un= u on G, u
n= 0 on D \G.
If D is a special two- or three-dimensional domainwith a
particular choice of G, then the eigenvaluesof this problem give
rise to the sloshing frequencies,that is, the frequencies of free
oscillations of aliquid in channels or containers; see, for
example,[35, Chapter IX]. The sloshing problem is discussedin
detail in the next section.
Spilling from a Wineglassand a Mixed Steklov Problem
The 2012 Ig Nobel Prize for Fluid Dynamics wasawarded to R.
Krechetnikov and H. Mayer for theirwork [37] on the dynamics of
liquid sloshing. Theyinvestigated why coffee so often spills while
peoplewalk with a filled mug. In their study, oscillationsof coffee
are modeled by an appropriate mixedSteklov problem which is usually
referred to asthe sloshing problem. They realized that withinthis
model one of the main reasons for spillingcoffee can be described
as follows. In a typical mug,the sloshing mode corresponding to the
lowesteigenfrequency of the problem tends to get excitedduring
walking.
However, there is another reason for spillingcoffee from a mug
of typical shape. Namely, a highspot is present on the boundary of
the free surface,that is, the maximal elevation of the surface
isalways located on the mugs wall (see Figure 1),provided
oscillations are free and their frequencyis the lowest one. The
latter effect (combined withthat described in [37]) makes it even
easier tospill coffee from a mug. On the other hand, in a
January 2014 Notices of the AMS 15
-
high spots
(a) (b)
Figure 3. A schematic sketch of location of highspots in a
coffee cup (a) and in a snifter (b).
bulbous wineglass both antisymmetric sloshingmodes corresponding
to the lowest eigenfrequencyare such that their maximal elevations
(high spots)are attained inside the free surface, but not onthe
wall (see Figure 2). This reduces the risk ofspilling from a
snifter. Thus, the position of ahigh spot depends on the containers
shape as isschematically shown in Figure 3 for a coffee cup(a) and
a snifter (b); theorems guaranteeing thesekinds of behavior are
proved in [34].
The natural limiting case of bulbous containersis an infinite
ocean covered by ice with a singlecircular hole (the corresponding
sloshing problemis usually referred to as the ice-fishing
problem).The question about the shape of the free surfacewhen water
oscillates at the lowest eigenfrequencyin an ice-fishing hole was
answered in [30]. Thisshape is similar to that in a snifter (see
Figures 2and 3 (b)). The highest free surface profile existingin
radial directions of an ice-fishing hole wascomputed numerically
and is plotted in Figure 4.One finds that the maximal amplitude is
attainedat some point located approximately 23r away fromthe holes
center (r is the holes radius). Thisamplitude is over 50% larger
than at the boundary.
Let us turn to the exact statement of the sloshingproblem which
is the mathematical model describ-ing small oscillations of an
inviscid, incompressibleand heavy liquid in a bounded container.
The liquiddomain W is bounded by a free surface (its meanposition F
is horizontal) and by the wetted rigidpart of W , say B (bottom).
Choosing Cartesiancoordinates (x, y, z) so that the z-axis is
verticaland points upwards, we place the two-dimensionaldomain F
into the plane z = 0.
The water motion is assumed to be irrotationaland the surface
tension is neglected on F . In theframework of linear water wave
theory, one seekssloshing modes and frequencies as
eigenfunctionsand eigenvalues, respectively, of the following
0.2 0.4 0.6 0.8 1
0.25
0.5
0.75
1
1.25
1.5
Figure 4. The highest radial free surface profilein an
ice-fishing hole.
mixed Steklov problem:
= 0 in W, z= on F,(6)
n
= 0 on B,Fdxdy = 0.(7)
The last condition is imposed to exclude the eigen-function
identically equal to a non-zero constantand corresponding to the
zero eigenvalue thatexists for the problem including only the
Laplaceequation and the above boundary conditions.
In terms of (,) found from problem (6), (7),the velocity field
of oscillations is given by
cos(t +)(x, y, z).Here is a certain constant, t stands for the
timevariable and = g is the radian frequency of os-cillations (as
usual, g denotes the acceleration dueto gravity). Furthermore, the
elevation of the freesurface is proportional to sin(t + )(x,
y,0),and so high spots are located at the points, wherethe
restriction of || to F attains its maximumvalues.
It is known that if W and F are Lipschitzdomains, then problem
(6), (7) has a sequence ofeigenvalues:
0 < 1 2 . . . n . . . , n .For all n, n H1(W), whereas their
restrictionsto F form (together with a non-zero constant) acomplete
orthogonal system in L2(F). In hydrody-namics, eigenfunctions
corresponding to 1 playan important role because the rate of their
decay(which is caused by non-ideal effects for real-lifeliquids) is
least.
Modelling a mug by the following vertical-walledcontainerW =
{(x, y, z) : x2+y2 < 1, z (h,0)},in which case F = {(x, y,0) :
x2 + y2 < 1} (cf. [37]),one finds all solutions of problem (6),
(7) explicitly.In particular, there are two linearly
independent
16 Notices of the AMS Volume 61, Number 1
-
xzy
B
F
W
r
z
F (D)
D
B(D)
Figure 5. A body of revolution (left) and its
radialcross-section (right).
eigenfunctions
1 = J1(j1,1r) sin cosh j1,1(z + h),2 = J1(j1,1r) cos cosh j1,1(z
+ h),
corresponding to 1 = 2 = j1,1 tanh j1,1h. Here(r , , z) are the
cylindrical coordinates such that is counted from the x-axis (see
the left-hand sideof Figure 5), J1 is the Bessel function of the
firstkind and j1,1 1.8412 is the first positive zero ofJ1. It is
clear that 1(x, y,0) is an odd, increasingfunction of y , and so it
attains extreme values(high spots) at the boundary points (0,1)
and(0,1); similarly, 2 has its high spots at (1,0)and (1,0).
Moreover, all linear combinations of1 and 2 have high spots on the
boundary of F .
Using the finite element method, one can obtainapproximate
positions of high spots for morecomplicated domains. We used FEniCS
[36] toimplement a trough (see [33] for the correspondingrigorous
result), which is short and has a hexagonalcross-section. Such a
trough is shown in Figure6, where several level surfaces of the
lowest-frequency mode 1 are also plotted. It is clear
Figure 6. A short trough and level surfaces of itsfundamental
eigenfunction.
Figure 7. The cross-section of a channel is givenby solid
segments. The length of the freesurface (respectively, bottom) is
equal to 2(2 (x+ 1), respectively); the depth of the
channel(respectively, its rectangular part) is equal tox+ 1 (x,
respectively). The free surface profileplotted in blue corresponds
to an isoscelestrapezoid which is 50% wider at the top than atthe
bottom; other profiles correspond to theshown hexagonal
cross-section withx = 0.1,1,10.
that the maximum of 1(x, y,0) is not on theboundary.
Since the previous example is essentially two-dimensional (see
[33]), we exploited this reductionto obtain numerically more
accurate free surfaceprofiles plotted in Figure 7. The blue curve
cor-responds to an isosceles trapezoid which is 50%wider at the top
than at the bottom; its maximumis on the boundary (as for a coffee
mug). Otherprofiles correspond to hexagons with differentslopes of
side walls. Notice that the point ofmaximum moves towards the
center for morehorizontal slopes, whereas the high spot becomesmore
pronounced.
Photos of water oscillations in bulbous andother containers were
made (examples are given inFigures 1 and 2). It proved difficult to
illustrate thehigh spot effect by photographing in a
conventionalway because of the nonlinearity caused by
relativelylarge amplitude of oscillations and non-ideal natureof
liquid. Therefore, along with photos shown inFigures 1 and 2, we
also photographed a reflectionof a dotted piece of paper on a
slightly disturbedsurface of the liquid (see Figure 8, bottom).
Imagesproduced using sufficiently long exposure timemostly consist
of blurred segments with just a fewclearly visible dots (see Figure
8, top). The reasonfor this is the fact that planes tangent to the
watersurface oscillate almost everywhere creating asegment path for
each dot. The exceptional pointsare those where the sloshing
surface has its localextrema, and so the corresponding tangent
planes
January 2014 Notices of the AMS 17
-
sloshing tank
camera
dotted paper
Figure 8. (top) Image of reflected light from thesloshing
surface of water in a fish bowl. (middle)
Similar image for a cocktail glass, showing nopoints with
vanishing gradient. (bottom) Setup
for photos.
are always horizontal, which makes these dotssharp. The image
obtained for a bulbous container(a fish bowl) has two clearly
visible points ofextrema located away from the boundary (seeFigure
8, top), which is in agreement with Figures 2and 3 (b). The similar
image for a conical tank
(a cocktail glass) consists exclusively of almostthe same
blurred segment paths for all dots (seeFigure 8, middle), and this
agrees with Figures 1and 3 (a).
Let us turn to discussing results proved rigor-ously for bodies
of revolution. If W is obtainedby rotating a two-dimensional domain
D that hasa horizontal segment on the top and is attachedto the
z-axis around this axis (see Figure 5), thenthe free surface F is a
disk in the (x, y)-plane.We assume that the fundamental modes are
an-tisymmetric; many domains have this property(see below).
Nevertheless, examples of rotationallysymmetric fundamental
eigenfunctions also exist.For example, this takes place for the
followingdomain: halves of a ball and a spherical shell joinedby a
small vertical pipe so that all of them arecoaxial.
Under antisymmetry assumption about thefundamental modes, there
are two of them
1 = (r, z) cos, 2 = (r, z) sin,that correspond to 1 = 2 and are
linearlyindependent; here (r, z) is defined on D.
In [34] (see Theorems 1.1 and 1.2), the followingis proved. If W
is a convex body of revolutionconfined to the cylinder {(x, y, z) :
(x, y,0) F, z R} (this condition was introduced byF. John in 1950),
then three assertions hold: (i)1 = 2; (ii) the corresponding
eigenfunctions 1and 2 are antisymmetric; (iii) the high spots
ofthese modes are attained on F .
On the other hand (see [34], Proposition 1.3), ifthe angle
between B and F is bigger than pi2 andsmaller than pi then 1(x,
y,0), 2(x, y,0) attaintheir extrema inside F as is shown in Figure
8, top.
The proof of (ii) and (iii) is based on the techniqueof domain
deformation used by D. Jerison andN. Nadirashvili [26], who studied
the hot spotsconjecture. The latter was posed by J. Rauchin 1974
(see a description by I. Stewart in hisNature article [62], and
Terence Taos Polymathproject [65] for current developments).
Roughlyspeaking, the hot spots conjecture states thatin a thermally
insulated domain, for typicalinitial conditions, the hottest point
will movetowards the boundary of the domain as timepasses. The
mathematical formulation of the hotspots conjecture is as follows:
every fundamentaleigenfunction of the Neumann Laplacian in an
n-dimensional domain D attains its extrema on D. Itwas proved for
sufficiently regular planar domains(see, for example, [3] and
[26]), disproved for somedomains with holes (see, for example, [7])
and isstill open for arbitrary convex planar domains.
There is a remarkable relationship between highand hot spots
(see, for example, [32], Proposition3.1). If 1, . . . ,k are the
sloshing eigenfunctions
18 Notices of the AMS Volume 61, Number 1
-
corresponding to an eigenvalue in a verticalcylinder W = {(x, y,
z) : (x, y) F, z (h,0)},then is an eigenvalue of the Neumann
LaplacianinD = F , if and only if = tanhh. Moreover,every
eigenfunction (x, y) corresponding to isdefined by some j , j = 1,
. . . , k, in the followingway: j(x, y, z) = (x, y) cosh(z + h).
Thisrelationship between the two problems implies,in particular,
that the results obtained in [3]and [26] for planar convex domains
with twoorthogonal axes of symmetry (e.g., ellipses) canbe
reformulated as follows. In a vertical-walled,cylindrical tank with
such free surface, high spotsof fundamental sloshing modes are
located on thefree surfaces boundary.
Isoperimetric Inequalitiesfor Steklov Eigenvalues
It was mentioned above that Steklovs major contri-bution to
mathematical physics appeared in 1902,namely, the article [54], in
which he introducedan eigenvalue problem (1). This problem withthe
spectral parameter in the boundary conditionturned out to have
numerous applications. More-over, the Steklov problemas it is
referred tonowadaysprovides a new playground for
excitinginteractions between geometry and spectral
theory,exhibiting phenomena that could not be observedin other
eigenvalue problems. For the sake ofsimplicity, it is assumed
throughout this sectionthat 1 in formula (1).
In two dimensions, problem (1) can be viewedas a cousin of the
Neumann problem in abounded domain D. Indeed, the latter
problemdescribes the vibration of a homogeneous freemembrane,
whereas the Steklov problem modelsthe vibration of a free membrane
with all its massconcentrated along the boundary (see [2], p.
95).Steklov eigenvalues
0 = 0 < 1(D) 2(D) 3(D) correspond to the frequencies of
oscillations. Asin the Neumann case, the Steklov spectrum
startswith zero, and in order to ensure discreteness ofthe spectrum
it is sufficient to assume that theboundary is Lipschitz.
Isoperimetric inequalities for eigenvalues isa classical topic
in geometric spectral theorythat goes back to the ground-breaking
results ofRayleighFaberKrahn and SzegoWeinberger onthe first
Dirichlet and the first non-zero Neumanneigenvalues. The problem is
to find a shapethat extremizes (minimizes for Dirichlet
andmaximizes for Neumann) the first eigenvalueamong all shapes of
fixed volume. In both cases,the unique extremal domain is a ball,
similarly tothe classical isoperimetric inequality in
Euclideangeometry.
Szegos proof of the isoperimetric inequalityfor the first
Neumann eigenvalue on a simply con-nected planar domain D is based
on the Riemannmapping theorem and a delicate construction oftrial
functions using eigenfunctions on a disk[63]. In 1954 (the same
year Szegos paper waspublished), R. Weinstock [69] realized that
thisapproach could be adapted to prove a sharp isoperi-metric
inequality for the first Steklov eigenvalue.Weinstock showed that
the first nonzero Stekloveigenvalue is maximized by a disk among
all sim-ply connected planar domains of fixed perimeter.Note that
for the Steklov problem, the perimeteris proportional to the mass
of the membrane,like the area in the Neumann problem. In
fact,Weinstocks proof is easier than Szegos, becausethe first
Steklov eigenfunctions on a disk are justcoordinate functions, not
Bessel functions as inthe Neumann case. In a way, Weinstocks
argumentis a first application of the barycentric methodthat is
being widely used in geometric eigenvalueestimates.
The analogy between isoperimetric inequalitiesfor Neumann and
Steklov eigenvalues is far frombeing complete, which makes the
study of Stekloveigenvalues particularly interesting. For
instance,as was shown by Weinberger [68], Szegos inequalityfor the
first Neumann eigenvalue can be generalizedto arbitrary Euclidean
domains of any dimension.At the same time, Weinstocks result fails
fornon-simply connected planar domains: if one digsa small hole in
the center of a disk, the firstSteklov eigenvalue of the
corresponding annulus,normalized by the perimeter, is bigger than
thenormalized first Steklov eigenvalue of a disk [19].
Another major distinction from the Neumanncase is that for
simply connected planar domains,sharp isoperimetric inequalities
are known forall Steklov eigenvalues. It was shown in [16] thatthe
inequality n(D)L(D) 2pin, n = 1,2,3 . . . ,proved in [23] is sharp,
with the equality attainedin the limit by a sequence of domains
degeneratingto a disjoint union of n identical disks; hereL(D)
denotes the perimeter of D. For Neumanneigenvalues, a similar
result holds for n = 2 [15],but the situation is quite different
for n 3 (see[1] and [48]).
In 1970, J. Hersch [22] developed the approachof Szego in a more
geometric direction. He provedthat among all Riemannian metrics on
a sphereof given area, the first eigenvalue of the corre-sponding
LaplaceBeltrami operator is maximalfor the standard round metric.
Note that the firsteigenspace on a round sphere is generated
bycoordinate functions, which allows one to proveHerschs theorem in
a similar way as Weinstocksinequality [17]. The result of Hersch
stimulated a
January 2014 Notices of the AMS 19
-
whole direction of research on extremal metricsfor
LaplaceBeltrami eigenvalues on surfaces.
Recently, Fraser and Schoen (see [11] and[12]) extended the
theory of extremal metrics toSteklov eigenvalues on surfaces with
boundary.They have studied extremal metrics for the firstSteklov
eigenvalue on a surface of genus zerowith l boundary components,
and proved theexistence of maximizers for all ` 1.
Weinstocksinequality covers the case ` = 1, but alreadyfor ` = 2
the result is quite unexpected. Themaximizer is given by a critical
catenoid; it is acertain metric of revolution on an annulus
suchthat the first Steklov eigenvalue has multiplicitythree.
Interestingly enough, this is the maximalpossible multiplicity for
the first eigenvalue onan annulus (see [12], [27] and [25]). The
criticalcatenoid admits the following characterization:it is a
unique free boundary minimal annulusembedded into a Euclidean ball
by the first Stekloveigenfunctions [12].
Maximizers for higher Steklov eigenvalues onsurfaces, as well as
sharp isoperimetric inequalitiesfor eigenvalues on surfaces of
higher genus, arestill to be found. Some bounds were obtained
in[11], [29], [18], [21] in terms of the genus and thenumber of
boundary components. At the sametime, it was shown in [9] that
there exists asequence of surfaces of fixed perimeter, such thatthe
corresponding first eigenvalues of the Steklovproblem tend to
infinity.
Apart from the two-dimensional vibrating mem-brane model
discussed above, there is anotherphysical interpretation of the
Steklov problem,which is valid in arbitrary dimension. It
describesthe stationary heat distribution in a body D, un-der the
condition that the heat flux through theboundary is proportional to
the temperature. Inthis context, it is meaningful to use the volume
ofD as a normalizing factor. Weinstocks inequalitycombined with the
classical isoperimetric inequal-ity implies that the disk maximizes
the first Stekloveigenvalue among all simply connected planar
do-mains of given area. Generalizations of this resultwere obtained
in [6] and [4]. In particular, it wasshown that in any dimension,
the ball maximizesthe first Steklov eigenvalue among all
Euclideandomains of given volume.
Yet another interpretation of the Steklov spec-trum involves the
concept of the Dirichlet-to-Neu-mann map (sometimes called the
PoincarSteklovoperator), which is important in many
applications,such as electric impedance tomography, cloak-ing, etc.
The Dirichlet-to-Neumann map acts onfunctions on the boundary of a
domain D (or,more generally, of a Riemannian manifold), andassigns
to each function the normal derivative ofits harmonic extension
into D. The spectrum of
this operator is given precisely by the Steklov eigen-values.
Since the Dirichlet-to-Neumann map actson D, it is natural to
normalize the eigenvalues bythe volume of the boundary. If the
volume of Dis fixed, the corresponding Steklov eigenvalues nof a
Euclidean domain D can be bounded in termsof n and the dimension
[8]. However, no sharpisoperimetric inequalities of this type are
knownat the moment in dimensions higher than two.
Mathematical notions often lead a life of theirown, independent
of the will of their creators.When Steklov introduced the
eigenvalue problemthat now bears his name, he was motivated
mainlyby applications. It is hard to tell whether hecould foresee
the interest in the problem comingfrom geometric spectral theory.
There is no doubt,however, that the past and future work of
manymathematicians on isoperimetric inequalities forSteklov
eigenvalues owes a lot to Steklovs insight.
References[1] P. Antunes and P. Freitas, Numerical optimization
of
low eigenvalues of the Dirichlet and Neumann Laplac-ians, J.
Optim. Theor. Appl., 154 (2012), 235257.
[2] C. Bandle, Isoperimetric Inequalities and
Applications,Pitman, 1980, 228 pp.
[3] R. Bauelos and K. Burdzy, On the hot spotsconjecture of J.
Rauch, J. Funct. Anal., 164 (1999),133.
[4] Binoy and G. Santharam, Sharp upper bound anda comparison
theorem for the first nonzero Stekloveigenvalue, Preprint
arXiv:1208.1690v1 [math.DG] 8Aug 2012.
[5] W. Blaschke, Kreis und Kugel, Verlag von Veit &Comp.,
1916, x+169 pp.
[6] F. Brock, An isoperimetric inequality for eigenvaluesof the
Stekloff problem, Z. Angew. Math. Mech. 81(2001), 6971.
[7] K. Burdzy, The hot spots problem in planar domainswith one
hole, Duke Math. J., 129 (2005), 481502.
[8] B. Colbois, A. El Soufi, and A. Girouard, Isoperimet-ric
control of the Steklov spectrum, J. Func. Anal., 261(2011),
13841399.
[9] B. Colbois and A. Girouard, The spectral gap ofgraphs and
Steklov eigenvalues on surfaces. PreprintarXiv:1310.2869 [math.SP]
10 Oct 2013.
[10] R. Courant, D. Hilbert, Methoden der mathemati-schen
Physik. II, Springer, 1937, 549 pp.
[11] A. Fraser and R. Schoen, The first Steklov eigen-value,
conformal geometry, and minimal surfaces, Adv.Math., 226 (2011),
40114030.
[12] , Sharp eigenvalue bounds and minimal sur-faces in the
ball, Preprint arXiv: 1209.3789 [math.DG]3 Apr 2013.
[13] K. O. Friedrichs, Eine invariante Formulierung
desNewtonschen Gravitationsgesetzes und des Gren-zberganges vom
Einsteinschen zum NewtonschenGesetz, Math. Ann., 98 (1927),
566575.
[14] B. G. Galerkin, N. M. Gnther, R. O. Kuzmin, I.
V.Meshcherski, V. I. Smirnov, Pamyati V. A. Steklova(In Memoriam of
V. A. Steklov), Leningrad, Acad. Sci.USSR, 1928, 92 pp. (In
Russian.)
20 Notices of the AMS Volume 61, Number 1
-
[15] A. Girouard, N. Nadirashvili and I.
Polterovich,Maximization of the second positive Neumann eigen-value
for planar domains, J. Diff. Geom., 83 (2009),637662.
[16] A. Girouard and I. Polterovich, On the HerschPayneSchiffer
inequalities for Steklov eigenvalues,Funk. Anal. Prilozh., 44, no.
2 (2010), 3347. (In Rus-sian, English transl. Func. Anal. Appl., 44
(2010),106117.)
[17] , Shape optimization for low Neumann andSteklov
eigenvalues, Math. Meth. Appl. Sci., 33 (2010),501516.
[18] , Upper bounds for Steklov eigenvalues on sur-faces,
Electron. Res. Announc. Amer. Math. Soc., 19(2012), 7785.
[19] , Spectral geometry of the Steklov problem
(inpreparation).
[20] G. H. Hardy, J. E. Littlewood and G. Plya,Inequalities,
Cambridge, University Press, 1934,324 pp.
[21] A. Hassannezhad, Conformal upper bounds for theeigenvalues
of the Laplacian and Steklov problem,J. Func. Anal., 261 (2011),
34193436.
[22] J. Hersch, Quatre proprits isoprimtriques demembranes
sphriques homognes, C. R. Acad. Sci.Paris, Sr. A-B, 270 (1970),
A1645A1648.
[23] J. Hersch, L. Payne and M. Schiffer, Some inequal-ities for
Stekloff eigenvalues, Arch. Rat. Mech. Anal.,57 (1974), 99114.
[24] V. A. Ilin, On solvability of mixed problems for
hy-perbolic and parabolic equations, Uspekhi Mat. Nauk15, no. 2
(1960), 97154. (In Russian, English transl.Russian Math. Surveys 15
(2) (1960), 85142.)
[25] P. Jammes, Prescription du spectre de Steklov dans
uneclasse conforme, Preprint arXiv:1209.4571v1 [math.DG] 20 Sep
2012.
[26] D. Jerison and N. Nadirashvili, The hot spots con-jecture
for domains with two axes of symmetry, J.Amer. Math. Soc., 13
(2000), 741772.
[27] M. Karpukhin, G. Kokarev, and I. Polterovich,Multiplicity
bounds for Steklov eigenvalues onRiemannian surfaces, Preprint
arXiv:1209.4869v2[math.DG] 20 Mar 2013.
[28] A. Kneser, Wladimir Stekloff zum Gedchtnis,Jahresber.
Deutsch. Math. Verein., 38 (1929), 206231.
[29] G. Kokarev, Variational aspects of Laplaceeigenvalues on
Riemannian surfaces, PreprintarXiv:1103.2448v2 [math.SP] 21 May
2011.
[30] V. Kozlov and N. Kuznetsov, The ice-fishing problem:the
fundamental sloshing frequency versus geometryof holes, Math. Meth.
Appl. Sci., 27 (2004), 289312.
[31] J. Krl, Integral Operators in Potential Theory. Lect.Notes
Math. 823, 1980, i+171 pp.
[32] T. Kulczycki and N. Kuznetsov, High spots theo-rems for
sloshing problems, Bull. Lond. Math. Soc., 41(2009), 495505.
[33] , On the high spots of fundamental sloshingmodes in a
trough, Proc. R. Soc., A467 (2011), 14911502.
[34] T. Kulczycki and M. Kwasnicki, On high spots ofthe
fundamental sloshing eigenfunctions in axiallysymmetric domains,
Proc. Lond. Math. Soc., 105 (2012),921952.
[35] H. Lamb, Hydrodynamics, Cambridge, University Press,1932,
xv+738 pp.
[36] A. Logg, K.-A. Mardal and G.Wells, Automated So-lution of
Differential Equations by the Finite ElementMethod. The FEniCS
book. Lecture Notes in Comput.Sci. and Engin., 84, Berlin,
Springer-Verlag, 2012,xiii+723 pp.
[37] H. Mayer and R. Krechetnikov, Walking with coffee:Why does
it spill? Phys. Rev. E, 85 (2012), 046117(7 pp.)
[38] V. G. Mazya, Sobolev Spaces with Applications to El-liptic
Partial Differential Equations, Springer-Verlag,2011, 866 pp.
[39] S. G. Mikhlin, Constants in Some Inequalities ofAnalysis,
Wiley-Intersci., 1986, 108 pp.
[40] , On the history of mathematics at LeningradState
University at the end of the 1920s, Proc. St. Pe-tersburg Math.
Soc., 2 (1993), 300308. (In Russian,English transl. AMS
Translations, Ser. 2, 159 (1994),207212.)
[41] D.S. Mitrinovic and P.M. Vasic, An integral
inequalityascribed to Wirtinger, and its variations and
general-izations, Public. fac. lectrotech. Univ. Belgrade, no272,
(1969), 157170.
[42] O. Nikodm, Sur une classe de fonctions considresdans ltude
du problme de Dirichlet, Fund. Math.,21 (1933), 129150.
[43] J. J. OConnor and E. F. Robertson, VladimirAndreevich
Steklov, MacTutor History of Mathematics.Available online at
http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Steklov.html
[44] J. Peetre, MR655789 (84a:46076).[45] J.-P. Pier, (ed.)
Developments of Mathematics 1900
1950, Birkhuser, Basel, 1994, xviii+729 pp.[46] H. Poincar, Sur
les quations aux derives partielles
de la physique mathematique, Am. J. Math., 12 (1890),211294.
[47] , Sur les quations de la physique mathematiqueRend. Circ.
Mat. Palermo, 8 (1894), 57156.
[48] G. Poliquin and G. Roy-Fortin, WolfKeller theoremfor
Neumann eigenvalues, Ann. Sci. Math. Qubec, 36(2012), no. 1,
169178.
[49] G. Plya and G. Szego, Isoperimetric Inequalitiesin
Mathematical Physics, Princeton University Press,1951, xvi+279
pp.
[50] V. A. Steklov, The problem of cooling of an heteroge-neous
rigid rod, Communs Kharkov Math. Soc., Ser. 2,5 (1896), 136181. (In
Russian.)
[51] , On the expansion of a given function into aseries of
harmonic functions, Communs Kharkov Math.Soc., Ser. 2, 6 (1897),
57124. (In Russian.)
[52] W. Stekloff, Problme de refroidissement dune barrehtrogne,
Ann. fac. sci. Toulouse, Sr. 2, 3 (1901),281313.
[53] V.A. Steklov, General methods of solving funda-mental
problems in mathematical physics, KharkovMathematical Society
(1901) (in Russian). Availableonline at
http://nasledie.enip.ras.ru/unicollections/list.html?id=42034006
[54] W. Stekloff, Sur les problmes fondamentaux de laphysique
mathematique, Annales sci. ENS, Sr. 3, 19(1902), 191259 and
455490.
[55] , Thorie gnrale des fonctions fondamentales,Annales fac.
sci. Toulouse, Sr. 2, 6 (1904), 351475.
[56] W. Stekloff [V. A. Steklov], Sur la condition de ferme-ture
des systmes de fonctions orthogonales, Comp.rend. Acad. sci. Paris,
151 (1910), 11161119. (Russianversion Bull. Imp. Acad. Sci., Sr. 6,
5 (1911), 754757.)
January 2014 Notices of the AMS 21
-
Forum of MathematicsHigh Quality Open Access
Comes to Mathematics
PiThe finest general-interest research in mathematics, edited
and peer-reviewed by a community of experts
journals.cambridge.org/pi
SigmaA ground-breaking journal, publishing specialised research
in mathematics, edited and peer-reviewed by subject experts
journals.cambridge.org/sigma
Content now online
Find out more here journals.cambridge.org/forumofmathematics
[57] W. Stekloff, Sur la thorie de fermeture des sys-tmes de
fonctions orthogonales dpendant dunnombre quelconque des variables,
Mm. Acad. Sci.St. Ptersbourg, Cl. Phys. Math., Sr. 8, 30, no. 4
(1911),187.
[58] V. A. Steklov, Osnovnye Zadachi MatematicheskoyFiziki
(Fundamental Problems of Mathematical Physics),Vol. 1, Petrograd,
1922, iv+285 pp., Vol. 2, Petrograd,1923, ii+ 285 pp. (In
Russian.)
[59] , Vospominaniya (The Recollections), NauchnoeNasledstvo
(Scientific Legacy), 17, Nauka, Leningrad,1991, pp. 235299. (In
Russian.) Available online
athttp://mathedu.ru/memory/steklov.djvu
[60] , Perepiska s A. M. Lyapunovym, A. N. Krylovym,N. M.
Krylovym, A. A. Markovym (Correspondencewith A. M. Lyapunov, A. N.
Krylov, N. M. Krylov, A. A.Markov Sr.), Nauchnoe Nasledstvo
(Scientific Legacy),17, Nauka, Leningrad, 1991, pp. 19234. (In
Russian.)
[61] V. A. Steklov and A. Kneser, Nauchnaya Pere-piska
(Scientific Correspondence), 19011925, Nauka,Moscow, 1980, 81 pp.
(in Russian).
[62] I. Stewart, Mathematics: Holes and hot spots, Nature,401
(1999), 863865.
[63] G. Szego, Inequalities for certain eigenvalues of amembrane
of given area, J. Rat. Mech. Anal., 3 (1954),343356.
[64] P. K. Suetin, V. A. Steklovs problem in the theory
oforthogonal polynomials, J. Soviet Math., 12 (1979),631682.
[65] T. Tao et al., Polymath7 research threads 4: theHot Spots
Conjecture. Available online at
http://polymathprojects.org/2012/09/10/polymath7-research-threads-4-the-hot-spots-conjecture/
[66] E. A. Tropp, V. Ya. Frenkel, and A. D. Chernin, Alek-sandr
Aleksandrovich Fridman: Zhizn i Deyatelnost,Nauka, Moscow, 1988,
304 pp. (In Russian, Englishtransl. Alexander A. Friedmann: the Man
Who Made theUniverse Expand, Cambridge University Press, 1993,277
pp.)
[67] V. S. Vladimirov and I. I. Markush, AcademicianSteklov:
Mathematician Extraordinary, Translatedfrom Russian by C. A.
Hainsworth and R. N. Hain-sworth, Mir, Moscow, 1983, 126 pp.
[68] H. F. Weinberger, An isoperimetric inequality for
theN-dimensional free membrane problem, J. Rat. Mech.Anal., 5
(1956), 633636.
[69] R. Weinstock, Inequalities for a classical
eigenvalueproblem, J. Rat. Mech. Anal., 3 (1954), 745753.
22 Notices of the AMS Volume 61, Number 1