# Solomon Grigoryevich Mikhlin

### Born: 23 April 1908 in Kholmech, Belarus

Died: 29 August 1990 in St Petersburg, Russia

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**Solomon Grigoryevich Mikhlin**was born in Kholmech, a Belorussian village, into a Jewish family of modest means: his real name was Zalman Girshevich Mikhlin, and he was the youngest of five children. In the document [12] he states that his father was a petty trader. However, this could be untrue since people of his generation sometimes lied about the profession of their parents in order to avoid limitations in their access to higher education. A different version, reported in [10], states that S G Mikhlin's father was a melamed (a teacher of religion) at a primary religious school (kheder). His first wife was Victoria Isaevna Libina. In 1940 they adopted a son, Grigory Zalmanovich. (Details of S G Mikhlin's family and his student years can be found in reminiscences of Grigory Z Mikhlin [11].) Victoria died of peritonitis in 1961 during a boat trip on the Volga - apparently, there had been no doctor on board. The book [1] is dedicated to her memory. Later S G Mikhlin married Eugenia Yakovlevna Rubinova, born in 1918, who became his companion throughout the rest of his life.

He graduated from a secondary school in Gomel (Belorussia) in 1923 and entered the Leningrad State Pedagogical Institute, named after Herzen, in 1925. In January 1927 he became a second year student in the Department of Mathematics and Mechanics (MatMekh) of Leningrad State University after passing all the first year examinations without attending any lectures. Sergei Lvovich Sobolev studied in the same class as Mikhlin. Among their university professors were Nikolai Maximovich Günther and Vladimir Ivanovich Smirnov. The latter became Mikhlin's master thesis supervisor: the topic of the thesis, defended in 1929, was the convergence of double power series. In 1930 he started his teaching career, working for short periods in several Leningrad institutes. In 1932 he obtained a position at the Seismological Institute of the USSR Academy of Sciences, where he worked till 1941. He was awarded the degree of "Doktor nauk" in Mathematics and Physics in 1935 (equivalent to the Doctor of Science), without having to earn the "Kandidat nauk" degree (equivalent to a Ph.D.), and finally in 1937 he was promoted to the rank of professor. During World War II he was a professor at the State Alma Ata University (Kazakhstan). In 1944 Mikhlin returned to Leningrad State University as full professor. From 1964 to 1986 he headed the Laboratory of Numerical Methods at the Research Institute of Mathematics and Mechanics of the same university. From 1986 until his death Mikhlin continued as a senior researcher at that laboratory. (An interesting source of information about Mikhlin's laboratory is the article by M Anolik [2].)

In 1966 Miklin and Maz'ya launched a Tuesday seminar on integral and partial differential equations at MatMekh. Both beginners and well established mathematicians gave two-hour talks. Among invited speakers were Mark Krasnoselsky, Israel Gohberg, Alexander Dynin, Igor Simonenko, Mark Freidlin, Olga Oleinik, and also foreign visitors Günther Wildenhain, Lars-Inge Hedberg, Siegfried Prössdorf and Gaetano Fichera. When the Department of Mathematics and Mechanics moved to the Leningrad suburb of Peterhof in 1978, the seminar became more specialized, turning into the seminar of Mikhlin's laboratory on numerical methods.

In 1961 Mikhlin received the State Order of the Badge of Honour. He was awarded the Laurea honoris causa by the Karl-Marx-Stadt (now Chemnitz) Polytechnic University in 1968. He was also elected to membership of two academies, the German Academy of Sciences Leopoldina in 1970 and the Accademia Nazionale dei Lincei in 1981, but, according to Gaetano Fichera [6], in the USSR Mikhlin never received honours befitting his scientific stature, mainly due of the anti-Semitic policy of the communist regime.

He lived during the period of contemporary Russian history when the Communist party ideology ruled academic life. Local administrators and party functionaries interfered with scientists on either ethnic or ideological grounds. Every citizen's ethnicity was specified in his/her passport and always taken into account by academic authorities. Frequently, ethnic Jews could not enter prestigious departments of universities to study for scientific degrees. They were not allowed to travel and publish abroad, they were not included (with a few exceptions) in official delegations for international conferences and congresses. Even their publications were discriminated by the editorial boards of the best Soviet journals.

As a matter of fact, before and during World War II, as well as in the first decade after it, Mikhlin did not experience difficulties on the same scale as younger Jewish Soviet mathematicians did from the mid-1960s. He could travel to countries of the Eastern European block and even was a member of the Soviet delegation at the 1958 International Congress of Mathematicians in Edinburgh, Scotland. He was a full professor of the University, a permanent member of the Scientific Board at MatMekh, and the head of a laboratory. Nevertheless, he strongly felt the general anti-Semitic atmosphere. "They have power but we have theorems. In them is our strength", Mikhlin once said, as reported by Vladimir Maz'ya.

Fichera, examining Mikhlin's life in [6], finds it surprisingly similar to the life of Vito Volterra under the fascist regime. He notes that anti-Semitism in communist countries took different forms compared to its Nazi counterparts. The communist regime did not resort to brutal mass homicide of Jews, but imposed on them a number of restrictions, making their life difficult. During the period from 1963 to 1981, Fichera met Mikhlin at several conferences in the Soviet Union, and realised that he was in a state of isolation, later describing several episodes revealing this. Perhaps, the most illuminating example is the election of Mikhlin into the Accademia dei Lincei. As already mentioned, in June 1981, Solomon G Mikhlin was elected a Foreign Member of the class of Mathematical Sciences of the Academy. However, at first he was nominated to receive the Antonio Feltrinelli Prize. The anticipated likely confiscation of the prize money by Soviet authorities spurred Italian academicians "to honour him in a different way, which could not be expropriated by any political authority", as Fichera [6] reports. Thus, Mikhlin was elected as a Foreign Member of the Academy. Mikhlin was subsequently not allowed to visit Italy by the Soviet authorities (this episode is also remembered by Vladimir G Maz'ya, see [9]), so Fichera and his wife brought the tiny golden lynx badge, the symbol of the Accademia dei Lincei membership, directly to Mikhlin's apartment in Leningrad on 17 October 1981. The only guests of that "ceremony" were Vladimir Maz'ya and his wife Tatyana Shaposhnikova.

The circumstances of Mikhlin's death are described by Gaetano Fichera in [6], who refers to a conversation with Mark Vishik and Olga Oleinik. On the 29

^{th}of August 1990, Mikhlin left home to buy medicines for his wife Eugenia. On public transport, he suffered a lethal stroke. He had no documents with him, therefore he was identified only some time after his death - this may be the cause of different dates of death reported in several biographies and obituary notices. Mikhlin's wife Eugenia survived him by only a few months.

Comprehensive descriptions of Mikhlin's work appear in the papers [6], [7] and in the references cited therein. Considering his research activity, perhaps, one of the first things to note is that he was the author of monographs and textbooks which became classics for their style. His main contributions belong to elasticity theory and elliptic boundary value problems, singular integrals and Fourier multipliers, as well as numerical mathematics.

In mathematical elasticity theory, Mikhlin was concerned with three themes: the plane problem (mainly from 1932 to 1935), the theory of shells (from 1954) and the Cosserat spectrum (from 1967 to 1973). Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains. The first one is based upon the so-called complex Green's function and the reduction of the related boundary value problem to integral equations. The second method is a certain generalisation of the classical Schwarz algorithm for the solution of the Dirichlet problem in a given domain by reducing it to simpler problems in smaller domains whose union is the original one. Mikhlin studied its convergence and gave applications to special applied problems. He proved existence theorems for the fundamental problems of plane elasticity involving inhomogeneous anisotropic media. Concerning the theory of shells, there are several Mikhlin's articles dealing with it. He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress. As a result of his study of this problem, Mikhlin also gave a new invariant form of the basic equations of the theory. He also proved a theorem on perturbations of positive operators in a Hilbert space which let him to obtain an error estimate for the problem of approximating a sloping shell by a plane plate. Mikhlin studied also the spectrum of the operator pencil of the classical Navier-Cauchy operator i.e. the Cosserat spectrum. The full description of this spectrum and the proof of the completeness of the system of eigenfunctions are also due to Mikhlin, and partly to Vladimir G Maz'ya in their only joint work.

Perhaps, his most important contributions are his works on the theory of singular integral operators and singular integral equations: he is one of the founders of the multi-dimensional theory, jointly with Francesco Tricomi and Georges Giraud. Such integral operators are called "singular" since the singularity of their kernel is so strong that the integral does not exist in the ordinary sense, but only in the sense of the Cauchy principal value. Mikhlin was the first to develop a theory of singular integral equations as a theory of operator equations in function spaces

*L*

_{2}. In 1936 he found a rule for the composition of double singular integrals (i.e. in the Euclidean plane) and introduced the fundamental notion of the symbol of a singular integral. This enabled him to prove that the algebra of bounded singular integral operators is isomorphic to the algebra of either scalar or matrix-valued functions. He established Fredholm's theorems for singular integral equations and systems of such equations under the hypothesis of non-degeneracy of the symbol. He also proved that the index of a single singular integral equation in the Euclidean space is zero.

In 1961 Mikhlin developed a theory of multidimensional singular integral equations on Lipschitz spaces which are widely used in the theory of one-dimensional singular integral equations. However, a direct extension of the related theory to the multidimensional case meets some technical difficulties, and Mikhlin suggested another approach to this problem. Namely, he obtained basic properties of this kind of singular integral equations as a by-product of the

*L*

_{p}-space theory of these equations. Mikhlin also proved a now classical theorem on multipliers of Fourier transform in the

*L*

_{p}-space, based on an analogous theorem of Józef Marcinkiewicz on Fourier series. A complete collection of his results in this field up to 1965, as well as contributions by Francesco Tricomi, Georges Giraud, Alberto Calderón and Antoni Zygmund, is contained in the monograph [1]. Fichera [6] remarks that Mikhlin, with his discoveries in singular integral operators, was a forerunner of the theory of pseudo-differential operators developed by Joseph J Kohn, Louis Nirenberg, Lars Hörmander and others. Mikhlin's multiplier theorem is widely used in different branches of mathematical analysis, particularly in the theory of differential equations.

Four Mikhlin papers, published in the period 1940-1942, deal with applications of the method of potentials to the mixed problem for the wave equation. In particular, he solved the mixed problem for the two-space dimensional wave equation in the half-plane by reduction to the planar Abel integral equation. For a planar domain with a sufficiently smooth curvilinear boundary, he reduced the problem to an integro-differential equation, which he is able to solve when the given boundary is analytic. In 1951 Mikhlin proved the convergence of the Schwarz alternating method for second order elliptic equations. He also applied methods of functional analysis, at the same time as Mark Vishik but independently of him, to the investigation of boundary value problems for degenerate second order elliptic partial differential equations.

He was, again citing Fichera [6], one of the pioneers of modern numerical analysis together with Boris Galerkin, Alexander Ostrowski, John von Neumann, Walter Ritz and Mauro Picone. His work in the field of numerical analysis can be divided into several branches. The first branch deals with the study of convergence of variational methods for problems connected with positive operators, in particular, for some problems of mathematical physics. Both "a priori" and "posteriori" estimates of the errors concerning the approximation given by these methods are obtained by him. The second branch deals with the notion of stability of numerical processes introduced by Mikhlin himself. When applied to the variational method, this notion enabled him to state necessary and sufficient conditions in order to minimise errors in the solution of the given problem when the error arising in the numerical construction of the algebraic system resulting from the application of the method itself is sufficiently small, no matter how large is the system's order. The third branch is the study of variational-difference and finite element methods. Mikhlin investigated the completeness of the coordinate functions used in these methods in the Sobolev space

*W*

^{{1, p}}, deriving the order of approximation as a function of the smoothness properties of the functions to be approximated. He also characterised the class of coordinate functions which give the best order of approximation, and studied the stability of the variational-difference process and the growth of the condition number of the variation-difference matrix. Mikhlin also studied the finite element approximation in weighted Sobolev spaces related to the numerical solution of degenerate elliptic equations. He found the optimal order of approximation for some methods of solution of variational inequalities. The fourth branch of his research in numerical mathematics is a method for solving Fredholm integral equations which he called the "resolvent method". Its essence relies on the possibility of substituting the kernel of the integral operator by its variational-difference approximation, so that the resolvent of the new kernel can be expressed by simple recurrence formulae. This eliminates the need to construct and solve large systems of equations. During his last years, Mikhlin contributed to the theory of errors in numerical processes, proposing the following classification of errors.

1.

2.

3.

4.

This classification is useful since it enables one to develop computational methods aiming to diminish the errors of each particular type, following the "divide et impera" principle.
*Approximation error*: is the error due to the replacement of an exact problem by an approximating one.2.

*Perturbation error*: is the error due to the inaccuracies in the computation of the data of the approximating problem.3.

*Algorithm error*: is the intrinsic error of the algorithm used for the solution of the approximating problem.4.

*Rounding error*: is the error due to the limits of computer arithmetics.An active teacher, Mikhlin was the "Kandidat nauk" advisor of a number of mathematicians. A partial list of them includes Yuri K Dem'yanovich, Vladimir L Fomin, Lyudmila N Gagen-Torn (Dovbysh), Joseph A Itskovich, Anatolii E Kosulin, Arno Langenbach, Natalia M Mikhailova-Gubenko, Boris A Plamenevsky, Siegfried Prössdorf, R K Radeva, Vera D Sapozhnikova, Tatyana O Shaposhnikova, and Irina V Tsaritsina.

Mikhlin was also a mentor and friend of Vladimir Maz'ya. He did not play the role of official supervisor, but his friendship with the young undergraduate Maz'ya had a great influence on Vladimir's mathematical style.

**Article by:** Vladimir G Maz'ya, Tatyana O Shaposhnikova, Daniele Tampieri

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