Presently the Sobolev spaces play a very special role at least in our Institute of Applied Mathematics and Mechanics. Sobolev spaces are the special language of mathematicians, which distinguishes them from all others. I am very proud to offer the special course of Sobolev spaces to our students. But let’s talk about the man who give his name to these spaces – Sergey L’vovich Sobolev.
In fact, three cities, the three largest Russian scientific centers, can lay claim to be the hometowns of Sergey Sobolev. These cities are St. Petersburg, Moscow and Novosibirsk. But Sergey Sobolev is undoubtedly a Petersburger, since he was born in Petersburg, received an education and began a scientific career here. So, here is another story of a great Petersburger.
Sergey Sobolev was born on October 6, 1908 in St. Petersburg. During the Civil War from 1918 to 1923 he lived with his mother in Kharkov, where he studied at a technical school. Sergey Sobolev mastered the secondary school program on his own, especially being carried away by mathematics. Having moved from Kharkov to Petrograd (presently St. Petersburg) in 1923, Sergey entered the last grade of high school and graduated from it in 1924. Since childhood, Sergey Sobolev was distinguished by great curiosity, read a lot, was fond of mathematics, physics, philosophy, biology, medicine. He wrote poetry, studied piano. After graduating from school in 1924, Sergey, due to his “insufficient age”, could not enter the university. Therefore, in 1924, Sergey. entered the First State Art Studio in the piano class. A year later, he became a student of the Physics and Mathematics Faculty of Leningrad State University, while continuing to study in an art studio. Leningrad University was the largest mathematical center that preserved the remarkable traditions of the St. Petersburg Mathematical School. At the Leningrad University S. Sobolev wrote his thesis on analytical solutions of a system of differential equations with two independent variables. This work was published in the Reports of the Academy of Sciences of the USSR.
In 1929, after graduating from the university, Sergey Sobolev was hired by the theoretical department of the Seismological Institute of the USSR Academy of Sciences. During his work at the Seismological Institute, S.L. Sobolev carried out a number of deep scientific studies. Together with V.I. Smirnov, he developed a method of functionally invariant solutions, which was then applied to solving a number of dynamic problems in the theory of elasticity. The theory of propagation of elastic waves was constructed on the base of this method.
Sergey Sobolev begins to study the Cauchy problem for hyperbolic equations with variable coefficients. In 1930, at the Mathematical Congress in Kharkov, S.L. Sobolev makes a report “Wave equation in an inhomogeneous medium”, in which he proposes a new method for solving the Cauchy problem for a wave equation with variable coefficients. The famous French mathematician Jacques Hadamard, who was present at the congress, said to Sergey Sobolev: “I will be very glad, young colleague, if you keep me informed of your further work, which has greatly interested me.”
Since 1932 S.L. Sobolev works in the Department of Differential Equations of the Steklov Mathematical Institute. and a year later he was elected a corresponding member of the USSR Academy of Sciences for outstanding achievements in mathematics. Since 1934, the “Moscow period” of S. Sobolev has started. Together with the entire Steklov Mathematical Institute he moved to Moscow and was appointed as a Head of the department. At this time, S.L. Sobolev obtained the fundamental results in the theory of partial differential equations and functional analysis, which entered the heritage of world mathematics.
The study of the Cauchy problem for hyperbolic equations and discontinuous solutions of equations of elasticity theory has led Sergey Sobolev to the concept of a generalized solution, which plays a fundamental role in the modern theory of partial differential equations. In 1934, at the Mathematical Congress in Leningrad, Sergey Sobolev gives three talks on the theory of partial differential equations, concerning problems of elasticity theory and the Cauchy problem for hyperbolic equations. The title of one of the reports is “Generalized solutions of the wave equation”. This was the beginning of the theory of generalized functions. In 1935-36 Sergey Sobolev describes in details the results presented by these reports in two famous papers “General theory of wave diffraction on Riemann surfaces” and “A new method for solving the Cauchy problem for linear normal hyperbolic equations”. In these works, for the first time, the foundations of the theory of generalized functions are presented in detail.
The emergence of the theory of generalized functions was prepared by the development of mathematical analysis and theoretical physics. The well-known ideas of Heaviside, Dirac, Kirchhoff, Poincaré and Hadamard contributed to its emergence. However, in the works of predecessors, there were no concepts and constructions similar to the strict constructions of S.L. Sobolev. It should be noted that for S.L. Sobolev, generalized functions primarily used as tools important for applications.
In subsequent years, S.L. Sobolev develops the theory of generalized functions in a new direction. On the basis of the concept of a generalized derivative, he introduces and studies new function spaces, which in the literature have come to be called Sobolev spaces. For these spaces S.L. Sobolev proves the first embedding theorems; he uses these spaces in the study of boundary value problems for higher order elliptic equations. A systematic presentation of the theory of function spaces, embedding theorems for these spaces, theorems on traces and applications of these results to problems of partial differential equations and equations of mathematical physics is contained in the famous book by Sergey Sobolev “Some applications of functional analysis in mathematical physics” (1950). This book has become a tabletop not only for mathematicians, but also for representatives of many other sciences. It was reprinted three times in USSR, twice in the United States, and translated into many languages of the world. The concepts of a generalized derivative and a generalized solution became widespread, a new direction of research was formed in mathematics, which was called the “theory of Sobolev spaces”. S.L. Sobolev not only laid the foundations of the theory of generalized functions and the theory of new function spaces, but also showed their practical application in the study of boundary value problems for differential equations.
In 1939, for outstanding mathematical discoveries, S.L. Sobolev was elected a full member of the USSR Academy of Sciences, for a long time remaining the youngest Academician in the country. In 1941, at the very beginning of the Great Patriotic War (as we call WW2 in Russia), the Academician Sobolev becomes the director of Steklov Mathematical Institute.
In the 1950s Sergey Sobolev also pays a lot of attention to the issues of computational mathematics. In particular, he develops the concept of closure of a computational algorithm, investigates the discrete problems arising in the approximation of differential and integral equations. Sergey Sobolev says: “While working at the Institute of Atomic Energy, I acquired a taste for computational mathematics and realized its exceptional possibilities. Therefore, I gladly accepted the proposal to head the first in our country Department of Computational Mathematics at Moscow University. ” S.L. Sobolev headed the department from 1952 to 1958.
In 1956 S.L. Sobolev, together with other academicians, came up with a proposal to develop a roadmap for the creation of scientific centers in eastern Russia. In 1957, it was decided to establish the Siberian Branch of the USSR Academy of Sciences consisting of several research institutes, including the Institute of Mathematics. Academician Sobolev was appointed as the director of this Institute. Since 1958, the “Siberian period” of S.L. Sobolev has started. Together with his colleagues, he moved to a permanent job in Novosibirsk. He says: “Many did not understand, even my friends, what actually made me to leave a strong department at Moscow University and go to Siberia, which was essentially a scientific desert.” The answer of S.L. Sobolev to this question, as always, is extremely modest: “The natural desire of a person to live several lives, to start something new.” In 1984 he returned to Moscow and continued to work at the Mathematical Institute.
Sergei Lvovich Sobolev died on January 3, 1989. He was buried at the Novodevichy cemetery in Moscow.
Memorial plaque on the building of the Institute of Mathematics. S.L. Sobolev SB RAS