Main scientific results of Academician A.A. Samarskii in mathematical physics.
The candidate (Ph.D.) thesis of A.A. Samarskii was based on his research of perturbation of the discrete spectrum of Laplace operator during variation of the boundary that was carried out in 1947-1948. This problem was discussed by Erenthest, A. Vitt, S. Shubin in connection to some models of atom and, as A.A. Samarskii found out, not all their suggested ideas were correct.
In the statement of A.A. Samarskii, this problem is formulated as follows.
Let G - is a domain in two- or three-dimensional Euclid space with sufficiently smooth boundary Γ. Let E - is a sub-domain of G, Uε(E) - is an open ε-neighborhood of the set E. Consider two eigenvalue problems:
- are the eigenvalues for the quasifixation on the set E.
A.A. Samarskii proved that if the capacity of the set E is zero, then the eigenvalues of the problem (1) do not change during fixation, i.e.
and for the first eigenvalue, the equality of the set capacity to zero is a necessary and sufficient condition of that it does not change, i.e. that the following equality holds:
A.A. Samarskii obtained the following estimate for the variation of the first eigenvalue
where c(E,Γ) - is the capacity of the set E relative to the boundary Γ, u1(x) - is the first eigenfunction of the problem (1). A.A. Samarskii showed that this estimate is precise.
I.G. Petrovskii was one of the official reviewers of this dissertation. He called the work of A.A. Samarskii "An outstanding candidate dissertation that is worth publishing completely".
It is worth to note that after the work of A.A. Samarskii, the usage of a notion of capacity in the spectral theory of the Laplace operator became widespread.
In the same years of post-graduate studying, A.A. Samarskii co-authored with A. Tikhonov a series of papers on electrodynamics, on the excitation of electromagnetic waves in radio waveguides. In this series of papers they formulated the general problem of the radiation of waves in unbounded domains. For the hollow radio waveguides of arbitrary cross section, they found the existence of solutions of the problem of excitation of arbitrary extraneous current. It was found that the radiation problem can be represented as a superposition of normal waves and it was rigorously proven that the system of normal waves forms a basis.
In 1948 A.A. Samarskii and A.N. Tikhonov formulated the principle of the limiting amplitude to obtain a unique solution of the Helmholtz equation
in an infinite domain. This principle implies that the function υ(M), - the amplitude of the steady state, is naturally treated as the limit
where the function u(M,t) is the solution of the problem
It was shown in the paper "On the principle of radiation" that for Ω=R3 the limit (3) actually exists and satisfies the Sommerfeld radiation conditions.
The ideas put forward in this paper in many respects anticipated many modern studies on the relationship between the steady and unsteady formalism in the scattering theory, and the work was, in fact, the first study on this topic.
The very principle of the limiting amplitude became the subject of research of many mathematicians.
In 1958, A.A. Samarskii studied the Cauchy problem for the heat conductivity equation with piece-wise smooth coefficients. He obtained the following result. Let the curves
do not intercross pairwisely, are piece-differentiable and the derivatives ηi(t) satisfy the Gelder condition with the factor γ>1/2; let the coefficients a(x,t) and b(x,t) are differentiable with respect to x and satisfy the Gelder condition with respect to t in the domains
Then the equation
has a unique classical solution in the domain
This result is a fundamental contribution to the theory of partial differential equations and the research in this field continues in the preset time as well.
As L.I. Kamynin showed, the smoothness conditions for the curves ηi(t) in the considered class of functions cannot be improved (the factor γ cannot be replaced by 1/2).
The paper "On discontinuous solutions of quasilinear equation of the first order" studies the generalized solution of a quasilinear equation of the form
This equation is interpreted as a consequence of the integral conservation law, the solution is defined as a function satisfying this conservation law. The existence and uniqueness of such solution are proven.