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 nikolay.g.kuznetsov@gmail.com.

Tadeusz Kulczycki and Mateusz Kwasnicki are professorsat the Institute of Mathematics and Computer Science,Wrocaw University of Technology. Tadeusz Kulczyckisemail address is Tadeusz.Kulczycki@pwr.wroc.pl andMateusz Kwasnickis email address is Mateusz.Kwasnicki@.pwr.wroc.pl.

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 al.il.nazarov@gmail.com. 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 ispoborchi@mail.ru. 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 iossif@dms.umontreal.ca.

Bartomiej Siudeja is a professor in the Department ofMathematics, University of Oregon. His email address issiudeja@uoregon.edu.

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.

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