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Academician S. L. Sobolev.
"The universe of academician Sobolev"
Academician Sergei L'vovich Sobolev
Founder of the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences
Sergei L’vovich Sobolev was born in Saint Petersburg on October 6, 1908.
His father Lev Aleksandrovich Sobolev, a solicitor, participated in revolutionary activities for which he was expelled from Saint-Petersburg University. His mother Natalia Georgievna Soboleva also was in her youth a revolutionary and a member of the RSDLP. She taught literature and history in a private school. Later she graduated from a medical school and occupied a position of an assistant professor at the First Leningrad Medical Institute. S. L. Sobolev was bereaved of his father in the early childhood and his mother raised him to be honest, scrupulous, and steadfast.
S. L. Sobolev was a very curious boy; he read a lot and liked mathematics, physics, philosophy, biology, and medicine. He wrote poetry and learned to play the piano. Graduating from secondary school in 1924, S. L. Sobolev was too young to enter university. Young people under the age of seventeen were admitted only according to the vouchers endorsed by the principals of the parents' places of employment. Under these circumstances, S. L. Sobolev entered the piano class of the First State Arts School in 1924. The next year he was enrolled as a student at the Physics and Mathematics Department of Leningrad State University, also continuing his studies at the Arts School. Leningrad State University was a great mathematical center that maintained remarkable traditions of the Petersburg mathematical school famous for the discoveries by P. L. Chebyshev, A. N. Korkin, A. M. Lyapunov, A. A. Markov, V. A. Steklov, and others.
At the university S. L. Sobolev attended the lectures by Professors N. M. Gyunter, V. I. Smirnov, and G. M. Fikhtengolts, who influenced considerably his uprise as a scientist. N. M. Gyunter supervised Sobolev's graduation work on analytic solutions to a system of differential equations in two independent unknowns which was published at the Doklady of the USSR Academy of Sciences. After the graduation from the university in 1929 S. L. Sobolev accepted a position in the Theoretical Department, headed by V. I. Smirnov, of the Seismological Institute of the USSR Academy of Sciences. During his stay at the Seismological Institute S. L. Sobolev carried out several profound scientific investigations. In collaboration with V. I. Smirnov he developed the method of functionally invariant solutions and applied it to solving a series of dynamical problems of elasticity. This method provided a foundation for the theory of elastic wave propagation. In particular: the famous Lamb problem of finding the displacement of an elastic half-plane subject to a concentrated impulse was solved; a rigorous theory of Raleigh surface waves was constructed; the problem of diffraction of elastic waves near a spherical surface was solved; the propagation of strong discontinuities was investigated in the problems of elasticity.
S. L. Sobolev explained in detail the results concerning the dynamical problem of elasticity in Chapter 12 (“Some Questions of the Propagation Theory of Oscillations”) of the second part of the book “Differential and Integral Equations of Mathematical Physics” by F. Frank and R. Mises (1937). These results are used in the modern mathematical methods of mineral prospecting, in the inverse problems of seismology, and in studying cracks in elastic media.
The applied problems related to wave propagation in elastic media required new approaches to partial differential equations. During this period S. L. Sobolev began to study the Cauchy problem for hyperbolic equations with variable coefficients. In 1930 at the First All-Union Mathematical Congress in Kharkov S. L. Sobolev presented the talk “The Wave Equation in an Inhomogeneous Medium”' in which he proposed a new method for solving the Cauchy problem for the wave equation with variable coefficients. The celebrated French mathematician J. Hadamard was in attendance, and he remarked to S. L. Sobolev: “My young colleague, I would be very happy if you could keep me up-to-date on your subsequent work which I found greatly interesting.”
In 1932 S. L. Sobolev started working in the Department of Differential Equations of the Steklov Mathematical Institute, and a year later for his outstanding achievements in mathematics he was elected a corresponding member of the USSR Academy of Sciences. The “Moscow period” of Sobolev's career began in 1934. Together with the Steklov Mathematical Institute he moved to Moscow and was appointed the head of a department. During this period S. L. Sobolev obtained some fundamental results in the theory of partial differential equations and functional analysis that belong to the treasure trove of the world mathematics. The ideas and methods of his articles were subsequently developed by many mathematicians.
Dealing with the Cauchy problem for hyperbolic equations and discontinuous solutions to elasticity theory equations led S. L. Sobolev to the concept of a generalized solution to a differential equation, which plays a fundamental role in the modern theory of partial differential equations. In 1934 S. L. Sobolev presented three talks on the theory of partial differential equations concerning problems of elasticity theory and the Cauchy problem for hyperbolic equations at the Second All-Union Mathematical Congress in Leningrad. The title of one of the talks was “Generalized Solutions to the Wave Equation”. It was the birth of the theory of generalized functions. In 1935-1936 S. L. Sobolev gave a detailed exposition of the results presented in those talks in two famous articles “General Theory of Diffraction of Waves on Riemann Surfaces” and “A New Method of Solving the Cauchy problem for Linear Normal Hyperbolic Equations”. In these articles the foundations of the theory of generalized functions were explained in detail for the first time.
The emergence of the theory of generalized functions had been based on the development of real analysis and theoretical physics. The available ideas of Heaviside, Dirac, Kirchhoff, and Hadamard aided in the emergence of the theory. However, the works of predecessors contained neither concepts nor constructions similar to the rigorous constructions of S. L. Sobolev. It is worth mentioning that for S. L. Sobolev his generalized functions were primarily some mathematical tools of import to applications.
During the subsequent years S. L. Sobolev advanced the theory of generalized functions in a new direction. He introduced and studied the new spaces of functions which based on the concept of a generalized derivative and became acknowledged in the literature as Sobolev spaces. For these spaces S. L. Sobolev proved the first embedding theorems, and used the spaces to investigate boundary value problems for high order elliptic equations. In 1939 he published the paper “To the Theory of Nonlinear Hyperbolic Partial Differential Equations” in which he used his theory of function spaces and solved the Cauchy problem for second order quasilinear hyperbolic equations.
The systematic exposition of the theory of function spaces, embedding theorems for these spaces, trace theorems, and applications of these results to problems in partial differential equations and equations of mathematical physics is set forth in Sobolev's famous book “Some Applications of Functional Analysis in Mathematical Physics” (1950). This book became a standard reference not only for mathematicians, but also for those working in many other areas of science. There were three editions in USSR and two editions of the English translation in the USA. This book was also translated into many other languages. The concepts of a generalized derivative and a generalized solution became universal. The theory of Sobolev spaces appeared as a new section of study in mathematics. S. L. Sobolev not only laid the foundations of the theory of generalized functions and new function spaces, but also he showed their practical applications in studying boundary value problems for differential equations.
The ideas and methods of S. L. Sobolev have been intensively developed and gained applications in differential equations, equations of mathematical physics, and computational mathematics. His embedding and trace theorems have turned into one of the most important tools of the modern real analysis.
In 1939 for his outstanding mathematical discoveries S. L. Sobolev was elected into the USSR Academy of Sciences, remaining for a long time the youngest member of the Academy. His wife Ariadna Dmitrievna Soboleva recalled that S. L. Sobolev had often said that he had been in debt to the USSR Academy of Sciences and that he would try to live up to his status of an academician. While talking to journalists many years later, S. L. Sobolev said: “As for my articles, at the time of my election no one could foresee what would grow out of them, and so my election into the Academy was a credit given to me”.
In 1941, at the very beginning of the Great Patriotic War, Academician S. L. Sobolev took up the responsibilities of the Director of the Steklov Mathematical Institute. In the difficult conditions of evacuation to Kazan S. L. Sobolev contributed much to organizing applied research in the Institute and helping effectively to the Soviet Army. When the Steklov Mathematical Institute returned back to Moscow in 1943, S. L. Sobolev was transferred to Laboratory No 2 (LIPAN, the Russian acronym for the Laboratory of Measuring Instruments of the USSR Academy of Sciences) headed by Academician I. V. Kurchatov, which was later transformed into the Kurchatov Institute of Atomic Energy. S. L. Sobolev was appointed the First Deputy Director and the Head of the Scientific Council of the Institute. Since then the name of S. L. Sobolev disappeared from the press for a long time.
The top secret laboratory worked intensely to build a nuclear shield of the USSR. This was a period of hard creative work for the members of the staff of the Institute. S. L. Sobolev collaborated with physicists, Academicians I. V. Kurchatov, I. K. Kikoin, M. A. Leontovich, and others. The physical process under study had to be understood as a whole, and the impending particular problems had to be solved using rather limited computational facilities. S. L. Sobolev faced the applied mathematical problems that took tremendous effort because it was required to calculate, optimize, and predict extremely complicated processes never studied before. The extraordinary mathematical intuition and great labor were needed in order to solve the intricate special problems thoroughly and timely. His wife Ariadna Dmitrievna recalled: “During his work at the Institute of Atomic Energy he might be absent from home for months and frequently left for long and distant trips”. For his exceptional services to the State in these years, S. L. Sobolev was awarded two State Prizes and the title of the Hero of the Socialist Labor.
In the 1950s S. L. Sobolev published his famous book “Some Applications of Functional Analysis in Mathematical Physics” (1950) and several fundamental papers on partial differential equations, functional analysis, and computational mathematics. In particular, his celebrated article “On a new problem of mathematical physics” (1954) initiated the systematic investigations of the new classes of equations and systems not solvable with respect to the highest-order derivative. Such equations are now called the “Sobolev type equations” in the literature. The problems of the kind were motivated by studying rotating fluids (1943). For this work S. L. Sobolev received a State Prize in 1986.
During the 1950s S. L. Sobolev paid great attention to the problems of computational mathematics as well. In particular, he developed the concept of the closure of a computational algorithm, studied the discrete problems that arise in approximating differential and integral equations. S. L. Sobolev recalled: “Working in the Institute of Atomic Energy, I got a taste of computational mathematics and realized its exceptional potential. Thus, I accepted with great pleasure an offer by I. G. Petrovskii to head the Chair of Computational Mathematics of Moscow State University, the first chair in this area in our country”. S. L. Sobolev headed the chair since 1952 to 1958. During this time together with A. A. Lyapunov he actively defended cybernetics, making a case for its significance.
In 1956 Academicians M. A. Lavrentiev, S. L. Sobolev, and S. A. Khristianovich came forward with a proposal to work out a plan for creating scientific centers in the eastern regions of the USSR. In 1957 the Government decided to organize the Siberian Branch of the USSR Academy of Sciences which will consist of several research institutes, including the Institute of Mathematics. Academician S. L. Sobolev was appointed the Director of the Institute. In 1958 the “Siberian period” of Sobolev's career began. S. L. Sobolev spent a year in Moscow to fill the positions in several departments of the future Institute of Mathematics and then he moved with his colleagues to the permanent residence in Novosibirsk. “Many people, even friends, could not understand what exactly had driven me”, S. L. Sobolev said, “to leave the strong chair at Moscow State University and go to Siberia which was in actuality a scientifically virgin land”. His own answer to this question was, as always, remarkably modest: “It was the natural desire of a man to live several lives, to start something new”.
Heading the Institute of Mathematics, S. L. Sobolev aimed to represent in the Institute all principal directions of the contemporary mathematics. The Department of Algebra and Logic in the Institute developed successfully under the guidance of Academician A. I. Mal’tsev; Academician A. D. Alexandrov led geometric research; Academician L. V. Kantorovich headed the Department of Mathematical Economics; Academician G. I. Marchuk did the Department of Computational Mathematics; A. A. Lyapunov, Corresponding Member of the USSR Academy of Sciences, headed the Department of Theoretical Cybernetics. S. L. Sobolev himself led research into differential equations and functional analysis. Greatly helpful to S. L. Sobolev in running the Institute was his deputy A. I. Shirshov, Corresponding Member of the USSR Academy of Sciences. In a very short time the Institute of Mathematics became a world renowned mathematical center.
S. L. Sobolev was one of the founders of Novosibirsk State University. He gave the first lecture on mathematics at the university. He headed the Chair of Differential Equations, gave a regular course in equations of mathematical physics and a special course in cubature formulas, and coordinated the work of special seminars.
During his “Siberian period” S. L. Sobolev began research into a new area: cubature formulas. He said, “After the transfer from Moscow to Novosibirsk I got preoccupied with cubature formulas. It so happened that they made me come back to the classical works of Euler. I had to investigate some properties of Euler polynomials that had not been known to the great classic of mathematics. It was like returning back to the origins”.
The approximate integration of a function is among the main problems of the theory of computation. The case of multidimensional integrals is extremely laborious in the computational aspects. As a result of studying the new problems of functional analysis, partial differential equations, and function theory, with a view towards solving the problems of computational mathematics, S. L. Sobolev expounded and created the theory of cubature formulas. In Novosibirsk S. L. Sobolev wrote the definitive monograph “Introduction to the Theory of Cubature Formulas” which was published in 1974. The book summarized the author's research into cubature formulas in the lapse of many years.
In 1983 the “Siberian period” of Sobolev's career terminated, and in 1984 he returned to Moscow to continue his work at the Steklov Mathematical Institute in the department headed by Academician S. M. Nikolskii.
The remarkable scholar and public figure, S. L. Sobolev was a great teacher who brought up a huge group of talented students and followers. He taught at Leningrad State University, the Leningrad Electrotechnical Institute, the Red Army Transport Academy, Moscow State University, the Moscow Institute of Physics and Technology, and Novosibirsk State University.
The brilliant scientific career and public activity of S. L. Sobolev that made his great reputation in the USSR received a proper international recognition. He was a foreign member of the French Academy of Sciences, the Accademia Nazionale dei Lincei in Rome, a foreign member of the Academy of Sciences in Berlin, an honorary member of the Edinburgh Royal Society, an honorary member of the Moscow Mathematical Society and the American Mathematical Society. He received honorary degrees from various universities of the world. The contributions of S. L. Sobolev had earned him many state awards. In 1989 S. L.Sobolev was awarded the highest prize of the USSR Academy of Sciences, the Lomonosov Gold Medal, for his outstanding achievements in mathematics.
Sergei L’vovich Sobolev passed away in Moscow on January 3, 1989. He is buried at Novodevichy Cemetery.