An overview of some works of S.L. Sobolev

Abstract

This paper is a short essay about Sergei L'vovich Sobolev (1908–1989), an outstanding mathematician of the 20th century, who made fundamental contributions to the development of the modern theory of partial differential equations, equations of mathematical physics, functional analysis, function theory, and computational mathematics.

Keywords:

• Diffraction problem
• generalized derivatives
• Sobolev spaces
• embedding theorems
• generalized solutions
• cubature formulas

AMS Subject Classifications:

• 35-XX
• 46-XX

View publisher note:

Publisher's note

Sergei L'vovich Sobolev (1908–1989) is one of the most outstanding mathematicians of the 20th century. His results have had a huge impact on the development of modern theory of partial differential equations, equations of mathematical physics, functional analysis, function theory, and computational mathematics.

S.L. Sobolev's interest in mathematics had emerged in high school already, a program which he mastered individually. Later, as a student at Leningrad State University, he attended lectures by prominent professors, in particular, N.M. Gunther, V.I.  Smirnov, G.M. Fichtenholz. Under supervision of N.M. Gunther, he wrote his first scientific paper on partial differential equations of first order. The obtained results were unexpected even for Professor N.M. Gunther, since in the work of student Sergei Sobolev, counterexamples for a number of theorems of a well-known mathematician were constructed and new theorems were proved.

In 1929, after graduation from Leningrad State University, S.L. Sobolev was offered a position in the Theoretical Department, headed by V.I. Smirnov, of the Seismological Institute of the USSR Academy of Sciences. The Department focused on solving problems in the theory of elasticity. The tasks of studying the propagation of elastic waves in different media were given to the young specialist Sergei Sobolev.

In his first work written at the Institute, S.L. Sobolev [1] studied the problem of wave propagation in an inhomogeneous medium. Mathematically, this problem is reduced to the consideration of the Cauchy problem for the wave equation with variable coefficient (1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(x,y,z)(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}),\end{array}$(1) (2) $\begin{array}{rl}& u{|}_{t=0}={\phi }_1(x,y,z),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}={\phi }_2(x,y,z).\end{array}$(2) Earlier the Cauchy problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(x,y,z)(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}),\end{array}$(1) ), (2(2) $\begin{array}{rl}& u{|}_{t=0}={\phi }_1(x,y,z),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}={\phi }_2(x,y,z).\end{array}$(2) ) was studied by the outstanding French mathematician J. Hadamard. Its solution is contained in the monograph [2], published in 1923. In his work [1], S.L. Sobolev proposed a very elegant new method for constructing a solution to the Cauchy problem. He transformed it into an integral equation of Volterra type (3) $u(M_0,t)=\frac{1}{4\pi }{\iiint }_{D_t}[u]\mathrm{\Delta }\sigma (M,M_0)\mathrm{d}v+\frac{1}{4\pi }{\iint }_{S_t}(\sigma \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }n}-\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }n}{\phi }_1+\sigma \frac{\mathrm{\partial }\tau }{\mathrm{\partial }n}{\phi }_2)\mathrm{d}s,$(3) where $M=(x,y,z),M_0=(x_0,y_0,z_0),$ $\tau (M,M_0)$ is a solution to the eikonal equation ${\tau }_x^2+{\tau }_y^2+{\tau }_z^2=c^{-2}(M),$ $\sigma (M,M_0)$ is a solution to the equation $2\mathrm{\nabla }\sigma \mathrm{\nabla }\tau +\sigma \mathrm{\Delta }\tau =0,$ $S_t$ is a quasi-sphere with centre $M_0$ of radius t, whose equation has the form $\tau (M,M_0)=t$, $D_t$ is the domain bounded by the quasi-sphere $S_t$, and $[u]=u(M,t-\tau (M,M_0)).$ In the case of a constant coefficient $c(x,y,z)\equiv c_0$, the identities hold $\tau (M,M_0)\equiv \frac{|M-M_0|}{c_0},\sigma (M,M_0)\equiv \frac{1}{|M-M_0|}.$ Hence, the volume integral in (3(3) $u(M_0,t)=\frac{1}{4\pi }{\iiint }_{D_t}[u]\mathrm{\Delta }\sigma (M,M_0)\mathrm{d}v+\frac{1}{4\pi }{\iint }_{S_t}(\sigma \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }n}-\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }n}{\phi }_1+\sigma \frac{\mathrm{\partial }\tau }{\mathrm{\partial }n}{\phi }_2)\mathrm{d}s,$(3) ) vanishes and the integral over the sphere $S_t$ coincides with the Kirchhoff integral, which gives a representation of the solution to the Cauchy problem at the point $(M_0,t)$.

In the case of a variable, sufficiently smooth coefficient $c(x,y,z)>0$ (inhomogeneous medium), for sufficiently small t, equation (3(3) $u(M_0,t)=\frac{1}{4\pi }{\iiint }_{D_t}[u]\mathrm{\Delta }\sigma (M,M_0)\mathrm{d}v+\frac{1}{4\pi }{\iint }_{S_t}(\sigma \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }n}-\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }n}{\phi }_1+\sigma \frac{\mathrm{\partial }\tau }{\mathrm{\partial }n}{\phi }_2)\mathrm{d}s,$(3) ) has a unique solution that can be obtained using the usual method of successive approximations. This solution is a solution to the Cauchy problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(x,y,z)(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}),\end{array}$(1) ), (2(2) $\begin{array}{rl}& u{|}_{t=0}={\phi }_1(x,y,z),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}={\phi }_2(x,y,z).\end{array}$(2) ).

The obtained formula for the solution to the Cauchy problem is called Sobolev formula in the literature [3]. It is often used in solving applied problems in geophysics.

In 1930, the First All-Union Mathematical Congress was held in the Soviet Union (Kharkiv). S.L. Sobolev took part in the Congress and presented the talk ‘The Wave Equation in an Inhomogeneous Medium’, in which he spoke about his method for solving the Cauchy problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(x,y,z)(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}),\end{array}$(1) ), (2(2) $\begin{array}{rl}& u{|}_{t=0}={\phi }_1(x,y,z),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}={\phi }_2(x,y,z).\end{array}$(2) ). At this Congress J. Hadamard gave the plenary lecture ‘Partial Differential Equations and the Theory of Functions of a Real Variable’. J. Hadamard listened to S.L. Sobolev's talk and, after the report, approached S.L. Sobolev and said: ‘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’. (Ariadna Dmitrievna Soboleva, the wife of Sergei L'vovich Sobolev, present at this conversation, later had reported this episode.)

Afterwards, friendly relations between S.L. Sobolev and J. Hadamard were established. It is interesting to note that S.L. Sobolev wrote some of his works, published in the 30s in the Soviet Union, in French. Unfortunately, some important works of S.L. Sobolev were published in the Transactions of the Seismological Institute of the USSR Academy of Sciences, which are available only for a few mathematicians.

While working at the Seismological Institute, S.L. Sobolev carried out a number of deep investigations of elastic wave propagation problems. In particular, in 1932 he solved the famous Lamb problem on the propagation of perturbations in an elastic half-space under the action of a concentrated impulse on the interface of the medium. This is one of the problems set by H. Lamb in his work from 1904, which was the continuation of investigations of Poisson, Stokes, and Rayleigh. In this work, H. Lamb has obtained integral formulas that can be used to determine the components of displacements at any point of observation on the surface. In [4], S.L. Sobolev obtained formulas describing the components of displacements for longitudinal and transverse elastic waves at any point within the medium. To solve this problem, he first thoroughly studied the plane Lamb problem, proposing a new method to describe elastic waves moving at various angles towards the interface. In particular, he indicated a way to describe the known Rayleigh waves. Then, using Lamb's formulas and applying the method of superposition of plane waves, S.L. Sobolev obtained integral formulas of the components of displacements for elastic waves at any internal point of the medium. Based on the obtained result, he proposed for the first time a solution to the spatial Lamb problem, specifying: ‘A solution to equations of elasticity theory for a half-space consists of longitudinal and transverse waves reflected at various angles (sometimes greater than the limit angle), and, in addition, of Rayleigh wave’. Note that analyzing the obtained formulas, S.L. Sobolev identified special cases when discontinuities occur.

The solution to the described Lamb problem was the beginning of the study of a number of other problems related to elastic vibrations under the action of various forces assigned by H. Lamb. The obtained results were important not only for seismology-practitioners, but also became a source of new ideas for solving partial differential equations.

Later, in a series of papers written by S.L. Sobolev jointly with his adviser V.I. Smirnov [5,6], a method of functionally invariant solutions was developed to the wave equation with constant coefficient (4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) To construct functionally invariant solutions to Equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ), the authors proposed a very simple scheme based on the consideration of the linear equation (5) $l(v)t+m(v)x+n(v)y-k(v)=0$(5) with analytic coefficients that satisfy the condition $(l(v){)}^2=c^2((m(v){)}^2+(n(v){)}^2).$ It turns out that if one can find a solution $v=v(x,y,t)$ to Equation (5(5) $l(v)t+m(v)x+n(v)y-k(v)=0$(5) ), then it will be a functionally invariant solution to the wave equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ). Afterwards, S.L. Sobolev showed [7] that all smooth functionally invariant solutions to Equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ) can be obtained by solving an equation of the form (5(5) $l(v)t+m(v)x+n(v)y-k(v)=0$(5) ).

In [5,6], the method of functionally invariant solutions was used to construct solutions to a series of problems on the propagation of elastic waves on the plane and in space. Here the construction of solutions has a clear geometric character and does not use rather cumbersome techniques, related to Fourier transform and often leading to divergent integrals, as in H. Lamb's works. In particular, V.I. Smirnov and S.L. Sobolev solved the generalized Lamb problem on vibrations of an elastic half-space under the action of a force source located inside the half-space. They obtained formulas describing the components of displacements of elastic waves at any point within the medium. Based on the obtained formulas, they gave an interpretation of some physical phenomena, in particular, they concluded that at infinity elastic vibrations generate a wave of finite amplitude propagating at the speed of Rayleigh waves. They called this wave the generalized Rayleigh wave.

Later, using the method of functionally invariant solutions, S.L. Sobolev solved a number of fundamental problems in the theory of elasticity. In particular, he studied questions of the reflection and refraction of elastic waves from a plane boundary for free and forced vibrations, solved a problem of vibrations of a half-plane and an elastic layer under arbitrary initial conditions, a problem of Rayleigh wave propagation, and also a number of problems in the theory of wave diffraction (see, for example, [8,9]).

The results obtained on dynamical problems in the theory of elasticity were described in detail by S.L. Sobolev 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 Ph. Frank and R. Mises (1937). These results are used in modern mathematical methods of mineral exploration, in inverse problems of seismology, and in the study of cracks in an elastic medium.

In the second half of the last century, the idea of the method of functionally invariant solutions became widespread in solving various problems of the theory of partial differential equations.

Applied problems, that were studied in the Theoretical Department of the Seismological Institute, often required new approaches to the study of partial differential equations. In particular, one of the most important problems arising in the study of elastic waves was the problem of propagation of discontinuities. From a mathematical point of view, the description of elastic waves was reduced to the consideration of partial differential equations. However, the classical concept of solutions to differential equations did not allow to describe discontinuous processes mathematically strictly. Therefore, considering only classical solutions to differential equations, a researcher had incomplete compliance between the physical content of the problem and its mathematical reflection.

S.L. Sobolev repeatedly had to deal with this problem when studying the propagation of elastic waves, for example, when studying problems of diffraction of plane elastic waves in homogeneous media [10]. As it is known, for such media, equations of elasticity theory can be reduced to two wave equations of the form (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ) (with different coefficients $c_1^2>c_2^2>0$). Therefore, in problems of wave diffraction, it was naturally necessary to introduce a mathematically correct concept of a discontinuous or generalized solution.

For the first time, the problem of generalized solutions to partial differential equations had arisen in the middle of the 18th century in a famous dispute between the outstanding mathematicians L. Euler and J. d'Alembert about what to call a solution to the string vibration equation (6) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}.$(6) J. d'Alembert had thought that solutions to this equation must be continuously differentiable up to second order, showing that they have the form (7) $u(x,t)=f(x-ct)+g(x+ct).$(7) He had defended this concept of a solution in all his works (see, for example, the unpublished volume 9 of ‘Opuscules Mathématiques’). However, L. Euler had believed that the definition of a solution to Equation (6(6) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}.$(6) ) should not be so rigid and it should be adequate to the physical understanding of the problem. As arguments, L. Euler had indicated the geometric pattern of wave propagation depending on the initial profile of the string. In other words, L. Euler for the first time posed the problem of a generalized solution to the string vibration equation.

It is interesting to note that taking into account the initial profile of the string and its initial velocity $u{|}_{t=0}=\phi (x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=\psi (x),$ L. Euler had specified the form of functions f and g in (7(7) $u(x,t)=f(x-ct)+g(x+ct).$(7) ), obtaining the formula $u(x,t)=\frac{\phi (x+ct)+\phi (x-ct)}{2}+\frac{1}{2c}{\int }_{x-ct}^{x+ct}\psi (\xi )\mathrm{d}\xi ,$ which is called the d'Alembert formula in the literature, despite the fact that J. d'Alembert had considered it as incorrect.

Later, D. Bernoulli and J. Lagrange had intervened in the dispute between L. Euler and J. d'Alembert. But, of course, at that level of development of mathematics, this problem could not be solved in the 18th century. At that time, there was not even a generally accepted concept of a function!

In the 19th century, many famous mathematicians took part in discussions about the concept of generalized solutions to partial differential equations. At the beginning of the 20th century, some approaches to the concept of a generalized solution were discussed in the works of J. Hadamard, N. Wiener, N.M. Gunther, J. Leray, K.O. Friedrichs, and others. The idea of a new concept of a generalized solution seemed to be in the air! But it was in the works of S.L. Sobolev in the mid-30s that the definition of generalized solutions was given, which became generally accepted and ideologically fundamental for the development of the theory of partial differential equations.

As was already said, since the beginning of the 30s, S.L. Sobolev was actively engaged in the study of elastic wave propagation and, in particular, at the formal level, considered discontinuous solutions to the wave equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ). In an effort to overcome this formalism, in 1934 S.L. Sobolev introduced the concept of the generalized solution to Equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ) and studied its properties. As the generalized solution to the wave equation in a domain G, he proposed to call a summable function $u(x,y,t)$, for which there was a sequence of classical solutions to this equation $\{u_m(x,y,t)\}$ converging to $u(x,y,t)$ in a Lebesgue space, i.e. $\Vert u_m(x,y,t)-u(x,y,t),L_1(G)\Vert \to 0,m\to \mathrm{\infty }.$ Studying the properties of generalized solutions, S.L. Sobolev established that for the function $u(x,y,t)$ to be a generalized solution to Equation (4(4) $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}).$(4) ) in the domain G it is necessary and sufficient to satisfy the integral relation ${\iiint }_{G}u(x,y,t)(\frac{{\mathrm{\partial }}^2\phi (x,y,t)}{\mathrm{\partial }{t}^2}-c^2(\frac{{\mathrm{\partial }}^2\phi (x,y,t)}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\phi (x,y,t)}{\mathrm{\partial }{y}^2}))\mathrm{d}x\mathrm{d}y\mathrm{d}t=0$ for any smooth and in G compactly supported function $\phi (x,y,t)$. Note that when proving this criterion, S.L. Sobolev applied the averaging operator introduced by him using his famous Sobolev function of ‘cap-shaped’ type.

Based on these results, S.L. Sobolev solved the general diffraction problem for the two-dimensional wave equation.

He presented the obtained results at the Second All-Union Mathematical Congress (Leningrad, June 1934). The results on wave diffraction were described in the section ‘Equations of Mathematical Physics’ in the talk ‘The Problem of Diffraction on Riemann Surfaces’ [11, p. 87], while those on generalized solutions were presented in the section ‘Differential and Integral Equations’ in the report ‘Generalized Solutions to the Wave Equation’ [11, p. 88].

This is how the strict definition of the generalized solution to the wave equation first appeared. Subsequently, it was generalized for various classes of partial differential equations and used in numerous works of mathematicians around the world. In the present time, this direction is still very actively developing.

An interesting historical fact should be pointed out. As it has already been mentioned, the problem of the generalized solution to the string vibration equation has been posed by L. Euler in the middle of the 18th century. At this time, he was living and working in Russia in St. Petersburg. And almost 200 years later, in a more general formulation, the problem of generalized solutions was solved by the young Soviet mathematician S.L. Sobolev in the same city, but under a different name – Leningrad.

While working on diffraction problems and investigating the problem of generalized solutions, S.L. Sobolev continued to study wave propagation in inhomogeneous media. First, he generalized his result [1] on the Cauchy problem (1(1) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=c^2(x,y,z)(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}),\end{array}$(1) ), (2(2) $\begin{array}{rl}& u{|}_{t=0}={\phi }_1(x,y,z),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}={\phi }_2(x,y,z).\end{array}$(2) ) to the case when the initial conditions are given on an arbitrary spatially oriented surface $\{t=T(x,y,z)\}$ [12]. Then he proceeded to consider the Cauchy problem for second-order hyperbolic equations with sufficiently smooth variable coefficients (8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) (9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) Developing his method [1,12], S.L. Sobolev reduced the construction of a solution to the problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ) to the consideration of an integral equation of Volterra type [13] generalizing Equation (3(3) $u(M_0,t)=\frac{1}{4\pi }{\iiint }_{D_t}[u]\mathrm{\Delta }\sigma (M,M_0)\mathrm{d}v+\frac{1}{4\pi }{\iint }_{S_t}(\sigma \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }n}-\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }n}{\phi }_1+\sigma \frac{\mathrm{\partial }\tau }{\mathrm{\partial }n}{\phi }_2)\mathrm{d}s,$(3) ). Assuming that there was a sufficiently smooth solution to the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ), he wrote the solution to this integral equation as a convergent series (10) $u(x_0,t_0)=\sum _{l=0}^{\mathrm{\infty }}{J}_l(x_0,t_0),$(10) where $J_l(x_0,t_0)$ are some integrals defined recursively; moreover, the integral $J_0(x_0,t_0)$ is determined by the initial conditions (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ) and depends on the coefficients of Equation (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ). Therefore, knowing that there is a classical solution to the Cauchy problem and using formula (10(10) $u(x_0,t_0)=\sum _{l=0}^{\mathrm{\infty }}{J}_l(x_0,t_0),$(10) ), an algorithm can be specified for constructing an approximate solution. This result was also presented by S.L. Sobolev at the Second All-Union Mathematical Congress [11, p. 87–88] (three reports at such a representative forum!).

In 1934 the ‘Moscow period’ of the S.L. Sobolev's activity began. He moved to Moscow and was appointed head of the Department at the Steklov Mathematical Institute. In his first works in Moscow, S.L. Sobolev developed the results that he had reported about at the Mathematical Congress in Leningrad [11]. In 1935, in the Proceedings of the Steklov Institute of Mathematics, he published the paper [14], in which a deployed presentation of the concept of a generalized solution to the wave equation and a detailed solution to the problem of wave diffraction were given. In the same year, in Doklady of the USSR Academy of Sciences, his paper [15] was published with the presentation of the idea of proving the solvability of the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ). Namely, S.L. Sobolev proposed first to study the generalized Cauchy problem, i.e. the Cauchy problem in the space of functionals, and then to find out the conditions under which the generalized solution would be classical.

At the formal level, the generalized problem is to find a functional U as a solution to the hyperbolic equation (11) $L(D_t,D_x)U=F$(11) with initial conditions (12) $U{|}_{t<0}=0,$(12) where the functional on the right-hand side in (11(11) $L(D_t,D_x)U=F$(11) ) is defined by the formula $(F,\phi )={\iint }_{R_{+}^{2k+2}}f\phi \mathrm{d}x\mathrm{d}t+{\int }_{R^{2k+1}}(u_0\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}+u_1\phi ){|}_{t=0}\mathrm{d}x$ for any smooth compactly supported functions $\phi \in \mathrm{\Phi }$.

At that time, S.L. Sobolev's idea was unexpected, moreover, the level of development of functional analysis did not allow to give a mathematically strict definition of a generalized solution to the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ). Therefore, S.L. Sobolev needed to develop a mathematical tool that could be used to implement his idea. To do this, in [15], he introduced the space of ‘test functions’ Φ, indicated the convergence of a sequence in it, defined the space of continuous functionals on Φ, and indicated their main properties and the possibility of regularization. In the space of functionals, he defined the simplest linear operations, in particular, differentiation and multiplication by a function, and then, a linear differential operator. All this made it possible to give meaning to (11(11) $L(D_t,D_x)U=F$(11) ), (12(12) $U{|}_{t<0}=0,$(12) ) and thereby introduce the concept of a generalized solution to the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ). In the same paper, S.L. Sobolev showed how the existence and uniqueness of the generalized solution to the Cauchy problem can be proved.

The work [15] is a brief outline of the basics of the theory of generalized functions proposed by S.L. Sobolev. A detailed presentation of the results [15], as well as the proof of the existence of a classical solution to the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ) is contained in his famous work [16], published in Matematicheskii Sbornik in 1936 (in French). The foundation of the modern theory of generalized functions was laid in these two works; moreover, the main ideas and constructions from the papers [15,16] entered the theory of generalized functions almost unchanged. The widespread use of the theory of generalized functions (or the theory of distributions) and its intensive development began in the early 50s after the publication of the monograph of L. Schwartz [17], which contained further development of the theory.

The emergence of the theory of generalized functions is one of the main events in mathematical analysis in the 20th century. It had a huge impact on the development of the theory of partial differential equations. A number of fundamental results, obtained since the beginning of the century in the theory of differential equations, gradually brought the appearance of such a theory closer. Some investigations carried out by prominent mathematicians at that time could have contributed to the emergence of ideas for the theory of generalized functions. They had been very close in subject matter to the works of S.L. Sobolev. Such studies are undoubtedly the works of J. Hadamard on the Cauchy problem for hyperbolic equations, the works of N. Wiener on the telegraph equation, the works of N.M. Gunther devoted to equations of potential theory, and finally, the recent studies of S.L. Sobolev on generalized solutions to the wave equation and the diffraction problem [11,14]. It is interesting that in 1967 the outstanding French mathematician J. Leray [18] wrote about the theory of generalized functions in his review of the works of S.L. Sobolev:

“The applications of distribution theory in all areas of mathematics, theoretical physics, and numerical analysis remind of the dense forest hiding the tree whose seeds it has grown from. However, we know that if Sobolev had fail to make his discovery about 1935 in Russia, it would be committed in France by 1950 and somewhat later in Poland; the USA also flatters itself that they would make this discovery in the same years”.

It should be emphasized that the theory of generalized functions appeared in the mid-30s in the works of S.L. Sobolev as a mathematical tool that allowed solving one of the most important problems of the theory of partial differential equations. S.L. Sobolev came to study this problem by solving important applied problems from the dynamical theory of elasticity. In this regard, we remind that S.L. Sobolev was a representative of the St. Petersburg – Leningrad mathematical scientific school, founded by the outstanding Russian mathematician and mechanic P.L. Chebyshev. A characteristic feature of the activity of this school was the close connection between the considered mathematical problems and important applied problems of science and technology.

In the following years, S.L. Sobolev developed the theory of generalized functions in a new direction. Based on the concept of a generalized derivative, he introduced and studied new function spaces, which later became known in the literature as Sobolev spaces $W_p^l$. The space of functions $W_p^l(G),l\in N,p\ge 1,G\subseteq R^n,$ is characterized by the fact that its elements f have all generalized derivatives $D_x^{\alpha }f$ in the domain G up to the lth order being summable with the pth power on G. For these spaces, S.L. Sobolev considered various ways of normalization and established some connections between spaces $W_p^l$ and classical spaces $C^k$ and $L_q$, proving the first embedding theorems [19,20].

Studying properties of the introduced spaces and using embedding theorems, he applied them to the investigation of boundary value problems in multidimensional domains [19] for the polyharmonic equation ${\mathrm{\Delta }}^{m}u=0,x\in G,$ and solved the Cauchy problem for the second-order quasilinear hyperbolic equations $u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t,u,\mathrm{\nabla }u)u_{x_i{x}_j}=f(x,t,u,\mathrm{\nabla }u),x\in R^{2k+1},t>0,$ $u{|}_{t=0}=u_0(x),u_t{|}_{t=0}=u_1(x),$ where the functions $a_{i,j}(x,t,y_0,y),f(x,t,y_0,y)$ are sufficiently smooth and satisfying some summability [21].

At present, spaces of functions with generalized derivatives and embedding theorems for them have become a classic tool in the study of a wide range of problems in many areas of mathematics. S.L. Sobolev's ideas and methods 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 modern mathematical analysis, in which a new direction of research has been formed, called ‘the theory of Sobolev spaces’.

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 A.D. Soboleva recalled: ‘S.L. Sobolev often said that he was in debt to the USSR Academy of Sciences and he would try to justify his status as 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 his subsequent works, S.L. Sobolev no longer returned to the discussion of the theorem on the solvability of the Cauchy problem for linear hyperbolic equations with variable coefficients [16]. However, if one gets acquainted with his works [22–24], then a hypothesis unvoluntarily occurs that S.L. Sobolev thought over another version of the proof of the solvability of the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ). The first two works published in 1937 and 1938 were devoted to one class of multidimensional integro-differential equations. Such equations arise when solving problems for hyperbolic equations. In the third work written in 1941, a new problem was studied for hyperbolic equations with boundary conditions on a characteristic conoid. It is quite possible that these three works were preparatory to creating a multidimensional analogue of the Riemann method and were aimed at the development of a constructive way to prove the existence theorem for the Cauchy problem (8(8) $\begin{array}{rl}& u_{tt}-\sum _{i,j=1}^{2k+1}{a}_{i,j}(x,t)u_{x_i{x}_j}+\sum _{i=1}^{2k+1}{b}_i(x,t)u_{x_i}+c(x,t)u=f(x,t),x\in R^{2k+1},t>0,\end{array}$(8) ), (9(9) $\begin{array}{rl}& u{|}_{t=0}=u_0(x),\frac{\mathrm{\partial }u}{\mathrm{\partial }t}{|}_{t=0}=u_1(x).\end{array}$(9) ).

However, fate decreed otherwise. In 1941, at the very beginning of the Great Patriotic War, S.L. Sobolev was appointed Director of the Steklov Mathematical Institute. In 1941–1943, he did a lot for the organization of applied research related to military subjects at the Institute. In 1944, S.L. Sobolev was invited to join a group of prominent scientists of the USSR to work in the Soviet atomic project. He left the post of Director of the Steklov Mathematical Institute and went to work in the laboratory headed by I.V. Kurchatov. Soon he was appointed Deputy of I.V. Kurchatov and the Head of the Scientific Council of the laboratory. Since that time, for more than 10 years, the work in the atomic project for S.L. Sobolev was most important.

Sergei L'vovich Sobolev was an ardent patriot of his Motherland. Working in the Soviet atomic project, he was directly involved in solving important applied problems of defense significance. In 1951, S.L. Sobolev received the highest award in the Soviet Union: ‘for exceptional services to the country during the execution of a special task of the Government’, he was awarded the title of the Hero of the Socialist Labor.

Despite his enormous occupation in the atomic project, S.L. Sobolev implemented a number of fundamental works in mathematics during these years. Here are some of them.

In 1945, in a series of works [25–27], S.L. Sobolev studied almost-periodicity of classical solutions to boundary value problems in cylindrical domains for the second-order hyperbolic equations with variable coefficients $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}-\sum _{i,j=1}^n\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(a_{i,j}(x)\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_j})+c(x)u=0,x\in \mathrm{\Omega },t>0,$ and a generalized solution to the first boundary value problem for the wave equation $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}-\sum _{i=1}^3\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}_i^2}=0,x\in \mathrm{\Omega },t>0.$ For obtaining the results, a mathematical tool developed by him when studying generalized solutions and spaces of functions with generalized derivatives was significantly used. The works [25–27] initiated the direction in the theory of partial differential equations connected with the investigation of the behavior of solutions to boundary value problems for nonstationary equations at infinity.

In 1950, his famous book ‘Some Applications of Functional Analysis in Mathematical Physics’ [28] was published. This book contains some results on analysis and differential equations obtained in the works of S.L. Sobolev in the 30s, as well as a number of new ones. In particular, the theory of Lebesgue and Sobolev function spaces, embedding theorems, a variational method for solving boundary value problems for elliptic equations, and the theory of the Cauchy problem for linear and quasilinear hyperbolic equations of second order are presented. This book had and still has a huge impact on the development of modern theory of differential equations, function theory, and computational mathematics. It is a handbook for many beginning researchers.

In 1954, S.L. Sobolev's paper [29] was published, which has already become a classic, devoted to the study of small vibrations of a rotating ideal fluid. The mathematical formulation was reduced to solving boundary value problems for a system of differential equations of the form (13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) where $v=v(x,y,z,t)$ is the vector of velocity, $p=p(x,y,z,t)$ is the pressure, and ω is the vector of angular velocity. In the case f = 0, g = 0, $\omega =(0,0,1)$, each component of the solution to System (13(13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) ) is a solution to the equation (14) $\mathrm{\Delta }{u}_{tt}+u_{zz}=0.$(14) System (13(13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) ) and Equation (14(14) $\mathrm{\Delta }{u}_{tt}+u_{zz}=0.$(14) ) belong to the classes of equations not solvable with respect to the highest-order time derivative. Apparently, such equations were first considered in 1885 in the famous work of the outstanding French mathematician A. Poincaré [30]. However, there were no results on the solvability of any boundary value problem for these equations in the literature. In his work [29], S.L. Sobolev proved the solvability of the Cauchy problem, the first and second boundary value problems for System (13(13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) ) and Equation (14(14) $\mathrm{\Delta }{u}_{tt}+u_{zz}=0.$(14) ), established the continuous dependence of solutions on the initial data, and obtained an explicit formula for the solution to the Cauchy problem. In the work [31], which was closely related to [29], a question of stability of motion of a symmetric spinning top filled with fluid was investigated. In these works, a number of new problems of mathematical physics were formulated; it was the first in-depth study of equations not solvable with respect to the highest-order time derivative. Therefore, in the literature, System (13(13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) ) is called the Sobolev system and Equation (14(14) $\mathrm{\Delta }{u}_{tt}+u_{zz}=0.$(14) ) is called the Sobolev equation. Note that in both works, S.L. Sobolev widely applied methods of functional analysis and, in particular, the new mathematical tool developed in the study of Sobolev spaces.

S.L. Sobolev's investigations of Equations (13(13) $v_t-[v,\omega ]+\mathrm{\nabla }p=f,\mathrm{div}v=g,$(13) ), (14(14) $\mathrm{\Delta }{u}_{tt}+u_{zz}=0.$(14) ) were continued by his disciples: R.A. Aleksandryan, R.T. Denchev, V.I. Lebedev, V.N. Maslennikova, S.G. Ovsepyan, N.N. Vakhania, G.V. Virabyan, and T.I. Zelenyak. It is also well known that after the appearance of the works of S.L. Sobolev ‘…I.G. Petrovskii pointed out the need to study general differential equations and systems not solvable with respect to the highest-order time derivative (systems that do not belong to the Kovalevskaya type systems)’ (see [32, p. 27]).

S.L. Sobolev's works [29,31] initiated a new direction in the study of various classes of equations and systems not solvable with respect to the highest-order derivative (15) $\mathcal{L}(D_x)D_t^{m}U+\sum _{k=0}^{m-1}{\mathcal{L}}_{m-k}(D_x)D_t^{k}U=F(t,x).$(15) In the literature, such equations are often called Sobolev type equations. At present, this direction is developing intensively. Great interest in equations of the form (15(15) $\mathcal{L}(D_x)D_t^{m}U+\sum _{k=0}^{m-1}{\mathcal{L}}_{m-k}(D_x)D_t^{k}U=F(t,x).$(15) ) is caused, on the one hand, by the fact that such equations arise in many important applied problems [33,34], on the other hand, by the natural desire of mathematicians to study new objects.

During many years of work in the Soviet atomic project, S.L. Sobolev and his group had to solve many specific problems for equations of mathematical physics. It has always been said about S.L. Sobolev that he had a colossal penetrating power in solving problems of mathematical physics. But in order for theoretical solutions of applied problems to be used in practice, it was necessary to organize numerical calculations. At the same time, the calculations had to be carried out with necessary accuracy and within a specified time limit. At that time, in addition to arithmometers and digital tables, scientists and engineers had nothing else at their disposal, and many calculations were conducted manually! S.L. Sobolev led the main calculations, so in these years he paid much attention to the issues of computational mathematics. In particular, he developed a concept of closure of a computational algorithm, investigated discrete problems arising in approximation of differential and integral equations of mathematical physics. He was one of the first to understand the need to create computer facilities and to train highly qualified personnel capable of working with them and maintaining them. He was one of the initiators of the creation of large computers in the USSR and headed the first Chair of Computational Mathematics in the Soviet Union at Moscow State University. S.L. Sobolev was the first mathematician in Europe who personally wrote programs for the first computers and numerically solved multidimensional problems of mathematical physics. Solving such problems was often reduced to the study of multidimensional integral equations, and for them, apparently, S.L. Sobolev even then invented his first cubature formulas. Later, S.L. Sobolev recalled: ‘Working at the Institute of Atomic Energy, I acquired a taste for computational mathematics and realized its exceptional potential’.

Awareness of the need for the development of computational mathematics determined S.L. Sobolev's scientific interests for many years to come. The change of scientific thematics fell on his ‘Siberian period’.

In 1956 Academicians M.A. Lavrent'ev, S.L. Sobolev, and S.A. Khristianovich appealed to the Government of the USSR with a proposal to create a new scientific centre in the East of the Soviet Union. In 1957 the Government decided to organize the Siberian Branch of the USSR Academy of Sciences consisting of several research institutes, including the Institute of Mathematics. Academician S.L. Sobolev was appointed Director of the Institute. In 1958 the ‘Siberian period’ of S.L. Sobolev's activity began. After spending a year to fill the positions in several departments of the future Institute of Mathematics, he moved with his colleagues to Siberia to Novosibirsk, three thousand kilometers away from Moscow.

‘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 scientific 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’.

The main topic was also new: it was cubature formulas, i.e. formulas for approximate calculation of integrals over multidimensional domains $\mathrm{\Omega }\subseteq R^n$ (16) ${\int }_{\mathrm{\Omega }}\phi (x)\mathrm{d}x\approx \sum _{k=1}^N{c}_k\phi (x^k),$(16) where points $x^1,\dots ,x^N$ and parameters $c_1,\dots ,c_N$ are called nodes and coefficients of the cubature formula, respectively. S.L. Sobolev had often said: ‘After moving from Moscow to Novosibirsk, my thoughts are occupied by cubature formulas’.

The problem of approximate integration of functions is one of the main problems of the theory of computation. For functions of one real variable, the appearance of approximate integration formulas dates back to the 18th century (trapezoidal rule, Simpson's rule). More precise formulas appeared in the 19th century (Gregory's rule, Gaussian rule, Chebyshev's rule). But formulas for the approximate calculation of multidimensional integrals appeared only in the 20th century, and unlike one-dimensional formulas, they are extremely computationally time-consuming.

For 25 years of work in Siberia, S.L. Sobolev created his own theory of cubature formulas. When creating the theory, he proposed the functional approach, which was so characteristic of the entire scientific work of S.L. Sobolev. In this approach, it is assumed that the set of integrable functions φ belongs to some Banach space B and the cubature formula (16(16) ${\int }_{\mathrm{\Omega }}\phi (x)\mathrm{d}x\approx \sum _{k=1}^N{c}_k\phi (x^k),$(16) ) is juxtaposed with the error functional (17) $l(x)={\chi }_{\mathrm{\Omega }}(x)-\sum _{k=1}^N{c}_k\delta (x-x^k),$(17) where ${\chi }_{\mathrm{\Omega }}(x)$ is the characteristic function of the domain Ω and $\delta (x)$ is the Dirac delta function. The functional (17(17) $l(x)={\chi }_{\mathrm{\Omega }}(x)-\sum _{k=1}^N{c}_k\delta (x-x^k),$(17) ) is linear and continuous on B, i.e. it is an element of $B^{\ast }$.

In the theory of cubature formulas, the algebraic and functional approaches are separated. They define different quality criteria for cubature formulas. In the algebraic approach, the best formula is considered to be the one that is accurate on larger sets of polynomials. In the functional approach, the quality is based on the norm of the error functional, the optimal cubature formula is such that its error functional, for given number of nodes N, has the smallest norm in $B^{\ast }$. Knowledge of the norm of the error functional allows to obtain guaranteed estimates of the accuracy of the cubature formula on functions from the space B. This is the essential advantage of the functional approach over the algebraic one.

In S.L. Sobolev's theory of cubature formulas, four main directions can be distinguished.

The first direction, which apparently began in the 50s, is related to three-dimensional cubature formulas, which have high algebraic orders of accuracy. As it is known, the construction of the cubature formula with given nodes, which is exact on polynomials of degree l, is reduced to solving a system of linear equations, the dimension of which depends on l. The higher l is, the greater is the dimension of the system. But for domains of integration with some symmetry, S.L. Sobolev proposed algorithms for constructing invariant cubature formulas, for which the dimensions of the corresponding systems of linear equations could be significantly reduced. Finding nodes and calculating the coefficients of such formulas can be very efficient and allows to achieve high accuracy for given amount of calculations.

The second direction is devoted to asymptotically optimal lattice cubature formulas in the spaces $L_2^m$. The norm in the space $L_2^m$ is defined by the expression $\Vert \phi ,L_2^m(\mathrm{\Omega }){\Vert }^2={\int }_{\mathrm{\Omega }}\sum _{|\alpha |=m}\frac{m!}{\alpha !}|D_x^{\alpha }\phi (x){|}^2\mathrm{d}x$ and the error functional of the lattice cubature formula is a generalized function of the form (18) $l(x)={\chi }_{\mathrm{\Omega }}(x)-\sum _{hH\gamma \in \mathrm{\Omega }}{h}^{n}c[\gamma ]\delta (x-hH\gamma ),$(18) where H is an orthogonal matrix of dimension $n\times n$, $detH=1$, the parameter h>0 is the step of the lattice, and γ is a column vector of integers. It is assumed that the domain Ω lies strictly inside the fundamental parallelepiped ${\mathrm{\Omega }}_0=\{x\in R^n:x=Hy,0\le y_j<1,j=1,\dots ,n\}.$ S.L. Sobolev proved that for the norm of any error functional of the form (18(18) $l(x)={\chi }_{\mathrm{\Omega }}(x)-\sum _{hH\gamma \in \mathrm{\Omega }}{h}^{n}c[\gamma ]\delta (x-hH\gamma ),$(18) ) with a small step h, the estimate (19) $\Vert l,L^{m\ast }\Vert \ge B_{n,m}|\mathrm{\Omega }{|}^{1/2}{h}^m+O(h^{m+1})$(19) holds, where $B_{n,m}=(2\pi {)}^{-m}\sqrt{\zeta (H,2m)}$ is defined by the Epstein function $\zeta (H,2m)$. Cubature formulas, the norms of the error functionals for which differ from the lower bound (19(19) $\Vert l,L^{m\ast }\Vert \ge B_{n,m}|\mathrm{\Omega }{|}^{1/2}{h}^m+O(h^{m+1})$(19) ) by a small value of higher order, are called asymptotically optimal cubature formulas. Studying such formulas, S.L. Sobolev, in particular, specified a constructive way of creating a class of cubature formulas, the error functionals for which have the form (20) $l(x)=\sum _{\gamma \in B_L^{(1)}}{l}_0(\frac{x}{h}-H\gamma )+\sum _{\gamma \in B_L^{(2)}}{l}_{\gamma }(\frac{x}{h}-H\gamma ),$(20) where $B_L^{(1)}$ is a set of points from Ω that are more than Lh away from the boundary $\mathrm{\partial }\mathrm{\Omega }$, $B_L^{(2)}$ is a set of points that are at most Lh away from the boundary $\mathrm{\partial }\mathrm{\Omega }$, $supp{l}_{\gamma }\subset \{x:|x|\le L\}$. He called such functionals the error functionals with regular boundary layer and proved an upper bound for their norm (21) $\Vert l,L^{m\ast }\Vert \le B_{n,m}|\mathrm{\Omega }{|}^{1/2}{h}^m+K{h}^{m+1}$(21) with a constant K independent of h. It follows from estimates (19(19) $\Vert l,L^{m\ast }\Vert \ge B_{n,m}|\mathrm{\Omega }{|}^{1/2}{h}^m+O(h^{m+1})$(19) ), (21(21) $\Vert l,L^{m\ast }\Vert \le B_{n,m}|\mathrm{\Omega }{|}^{1/2}{h}^m+K{h}^{m+1}$(21) ) that for a fixed matrix H and a lattice step h tending to zero, the corresponding cubature formulas are asymptotically optimal. Herewith the constant $B_{n,m}$ depends on the matrix H. Therefore, to minimize the norm of the error functional, this matrix should be chosen so that for a given m, the corresponding Epstein zeta function takes the minimal value. This leads to the study of non-trivial problems in number theory.

The third direction is S.L. Sobolev's research on cubature formulas in classes of infinitely differentiable functions. He considered spaces of periodic functions of many variables with given behavior of $L_2^m$-norms as $m\to \mathrm{\infty }$. Studying lattice cubature formulas in these spaces, S.L. Sobolev obtained an asymptotic representation of the logarithm of the norm of the error functional, and an interesting fact was established. Namely, if the norm of the error functional of the cubature formula for classes of functions of finite smoothness decreases in a power-law manner as $h\to 0$, then for infinitely differentiable functions it decreases exponentially.

The fourth direction relates to S.L. Sobolev's research on optimal lattice cubature formulas in spaces $L_2^m$. One of the key results in this direction is an analytical algorithm for finding the coefficients of optimal formulas with a given lattice of nodes. To do this, S.L. Sobolev introduced and studied new spaces of functions of many discrete variables that were analogous to the Sobolev spaces $W_p^m$, $L_p^m$. These spaces were used in an algorithm for finding the coefficients of optimal formulas, which was based on solving a discrete analogue to the Wiener–Hopf problem. To solve this problem, S.L. Sobolev defined and investigated a special discrete operator $D_{hH}^m[\gamma ]$ analogic to the polyharmonic operator ${\mathrm{\Delta }}^m$. Its action on a function of discrete argument was represented by a convolution with a special kernel. Using the properties of this operator, S.L. Sobolev obtained an analytical formula for the desired coefficients.

In 1974, S.L. Sobolev's fundamental monograph ‘Introduction to the Theory of Cubature Formulas’ [35] was published, in which all four directions of the theory of cubature formulas were described and a number of important unsolved problems was indicated.

In 1983 the ‘Siberian period’ of the S.L. Sobolev's activity was over, and in 1984 he returned to Moscow to continue his work at the Steklov Mathematical Institute at the department headed by Academician S.M. Nikol'skii. In 1988, S.L. Sobolev prepared for publication the monograph [36], in which he outlined the theory of the most well-known functional spaces, in particular, spaces of generalized functions. He described in detail the main operations in the spaces, various normalizations, and integral representations. For weighted spaces $W_p^l$, embedding theorems were proved, the issues of density of compactly supported functions were investigated, and the stabilization at infinity was studied. S.L. Sobolev planned to write a continuation of the book on the theory of cubature formulas [35]. However, it was published only in 1996 jointly with his disciple V.L. Vaskevich [37].

The outstanding scientist and public leader S.L. Sobolev was an excellent teacher who stimulated a variety of talented disciples and followers. Under his leadership, candidate dissertations (PhD theses) were defended by Kh.L. Smolitskii (1940), V.I. Kondrashov (1942), I.A. Yakovlev (1948), R.A. Aleksandryan (1949), O.A. Ladyzhenskaya (1949), G.V. Mukhina (1950), V.N. Maslennikova (1954), N.P. Trifonov (1954), A.A. Dezin (1957), V.I. Lebedev (1957), M.M. Lavrent'ev (1957), N.S. Bakhvalov (1958), N.N. Vakhania (1958), A.L. Krylov (1958), R.T. Denchev (1960), Yu.I. Gil'derman (1962), T.I. Zelenyak (1962), E.G. D'yakonov (1962), A.Kh. Gudiev (1963), I.G. Globenko (1963), G.N. Salikhov (1964), G.V. Virabyan (1965), V.R. Portnov (1967), Ts.B. Shoinzhurov (1967), V.I. Polovinkin (1968), L.V. Voitishek (1971), V.Ya. Ivrii (1972), Z.Zh. Zhamalov (1975), P.E. Berkhin (1976), S.Yu. Prishchepionok (1976), N.I. Blinov (1978), M.M. Zarubin (1981), Kh.M. Shadimetov (1982), V.L. Vaskevich (1983).

S.L. Sobolev taught at Leningrad State University, Leningrad Electrotechnical Institute, Military Transport Academy of the Red Army, Moscow State University, and Moscow Institute of Physics and Technology. He was one of the founders of Novosibirsk State University, where he gave the very first lecture at the opening of the university in 1959, he founded the Chair of Differential Equations and for many years taught courses on ordinary differential equations and equations of mathematical physics, special courses on the theory of generalized functions and cubature formulas.

The brilliant scientific and public activity of S.L. Sobolev, which gave him a great reputation in the USSR, received proper international recognition. He was a foreign member of the French Academy of Sciences, the Accademia Nazionale dei Lincei in Rome, the Academy of Sciences in Berlin, an honorary member of the Edinburgh Royal Society, the Moscow Mathematical Society and the American Mathematical Society. He received honorary degrees from various universities all over the world. The contributions of S.L. Sobolev had earned him many state awards. In 1988 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 was buried at the Novodevichy Cemetery.

Toward the centenary of S.L. Sobolev, two volumes of his selected works [38,39] were published at the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. The volumes contain his fundamental papers on partial differential equations, equations of mathematical physics, functional analysis, function theory, computational mathematics and the theory of cubature formulas.

The biography of S.L. Sobolev, his interesting scientific-popular and publicistic articles, a number of rare documents, as well as the memoirs of contemporaries, colleagues, friends, and his wife are contained in the book [40]. The full list of S.L. Sobolev's publications can be found in [41]. The website of the Sobolev Institute of Mathematics contains the webpage [42] dedicated to S.L. Sobolev with links to resources about him, his papers and activities, as well as the webpage [43] about conferences in his honor.

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