# the legacy of vladimir steklov

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The Legacy of Vladimir SteklovTRANSCRIPT

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:

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