The high school which he entered was called the 190th School of Leningrad at the time although previously it had been called the Lentovskii High School. In  (see also ) it is explained that this school :-
... was founded during the First Russian Revolution by the foremost St Petersburg teachers for pupils who had been excluded from the State Schools and Technical Colleges because of their participation in the revolutionary movement.After graduating from high school in 1925, Sobolev entered the Physics and Mathematics Faculty of Leningrad State University where his talents were quickly spotted by Smirnov who had returned to Leningrad three years earlier. Sobolev became interested in differential equations, a topic which would dominate his research throughout his life, and even at this stage in his career he produced new results which he published.
By 1929 Sobolev had completed his university education and he began to teach in a number of different educational establishments. For example his first appointment was in 1929 at the Theoretical Department of the Seismological Institute of the USSR Academy of Sciences. However, in addition, he taught at the Leningrad Electrotechnic Institute in 1930-31.
In 1932 the Steklov Institute of Physics and Mathematics was divided into separate Departments of Mathematics and of Physics. Vinogradov headed the Mathematics Department and invited Sobolev to join the Department. By this time, however, Sobolev had already ( and ):-
... published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations.Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation. These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium. The methods also led them to a complete theory of Rayleigh surface waves and Sobolev went on to solve problems on diffraction. Sobolev was honoured for this outstanding work by election as Corresponding Member of the USSR Academy of Sciences in 1933.
On 28 April 1934, at the general meeting of the Division of Mathematical and Natural Sciences of the USSR Academy of Sciences, a decision was taken to split the Departments of the Steklov Institute of Physics and Mathematics, which had been created two years earlier, into two independent Institutes, the Steklov Mathematical Institute and the Lebedev Physical Institute. In the same year the Steklov Mathematical Institute was moved from Leningrad to Moscow and Sobolev went with the new Institute to Moscow. By 1935 Sobolev was head of the Department of the Theory of Differential Equations at the Institute.
During the 1930s Sobolev introduced notions which were fundamental in the development of several different areas of mathematics :-
The study of Sobolev function spaces, which he introduced in the 1930s, immediately became a whole area of functional analysis. Sobolev's notion of generalised function (distribution) turned out to be especially important; with further developments by Schwartz and Gelfand, it became one of the central notions of mathematics.While working in Moscow, Sobolev built on the standard variational method for solving elliptic boundary value problems by introducing these Sobolev function spaces. He gave inequalities on the norms on these spaces which were important in the theory of embedding function spaces. He applied his methods to solve difficult problems in mathematical physics.
In 1939 Sobolev was elected a full member of the USSR Academy of Sciences. He was only 31 years of age at the time of his election which was a remarkable achievement. It made him the youngest full member of the Academy of Sciences and in fact he remained the youngest member for quite a few years.
At the beginning of World War II, the Steklov Mathematical Institute was moved from Moscow to Kazan. In the October of that year Sobolev was appointed as Director of the Institute and in the spring of 1943 he supervised the move of the Institute back to Moscow. Sobolev became one of the first recipients of a Stalin prize (later called a State prize) in the first presentation of these prizes in 1941. His period as Director of the Institute ended in February 1944.
A new area of his research involved the study of the motion of a fluid in a rotating vessel. He was led to study a number of new problems which ( and ):-
... led him to lay the foundations of the theory of operators in a space with an indefinite metric, and to introduce new ideas in the spectral theory of operators. These ideas in the main concern generalised solutions of non-classical boundary value problems.In 1950 he published his famous text Applications of functional analysis in mathematical physics (in Russian). An English translation was published by the American Mathematical Society in 1963. Schwartz's book on the the theory of distributions appeared in the same year.
In the early 1950s Sobolev's work turned towards computational mathematics and in 1952 he became head of the first department of computational mathematics in the Soviet Union when he organised the first such department at Moscow State University. However in 1956 he joined with a number of colleagues in proposing ways in which the large areas of Russia in the east could be opened up with educational initiatives. The scheme was to set up a number of Institutes for Scientific research to balance the large number of high quality educational establishments in the east of the Soviet Union.
After the plan was approved, Sobolev spent some time in Moscow recruiting staff and organising the establishment of an Institute in Novosibirsk. In  and  his contribution to the Institute in Novosibirsk is stressed:-
During the difficult formative years of the Institute Sobolev, by his excellent example, infused his young colleagues with the best habits for scientific work. In ten years, under the leadership of Sobolev, the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences has become one of the greatest centres for the mathematical sciences of international status.In 1958 Sobolev was part of the Soviet delegation to the International Mathematical Union, the delegation being led by Vinogradov, and Sobolev attended the International Congress at Edinburgh that year and gave an invited address on partial differential equations. From 1960 until 1978 Sobolev, in addition to his work at the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences, was a professor at Novosibirsk University.
During the 1960s much of Sobolev's research was directed towards numerical methods, in particular to interpolation. Although interpolation for functions of a single variable was well worked out, the problem of interpolation in many dimensions was largely unsolved. Sobolev applied his theories of generalised functions and of embeddings of function spaces to cubature formulae, the multi-dimensional analogues of quadrature formulae for functions of one variable. Again a major text by Sobolev Introduction to the theory of cubature formulae has been extremely influential in this area.
Sobolev received many honours for his fundamental contributions to mathematics. He was elected to many scientific societies, including the USSR Academy of Sciences, the Académie des Sciences de France, and the Accademia Nazionale dei Lincei. He was awarded many prizes, including three State Prizes and the 1988 M V Lomonosov Gold Medal from the USSR Academy of Sciences.
Article by: J J O'Connor and E F Robertson
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