363 SOBOLEV 



Sobolev has studied the dynamics of an elastic body. He 

 formulated the theory of plane waves in an elastic semi -space 

 with a boundary free from tension, and elucidated the general 

 concept of a surface wave. Together with V. I. Smirnov, he 

 worked out a new method of investigating the propagation and 

 reflection of elastic waves from rectilinear boundaries— a 

 method which is associated with functionally invariant solutions 

 of wave propagation on a plane. Sobolev also worked out a new 

 method of integrating linear and non-linear equations with 

 partial derivates of the hyperbolic type. He carried out re- 

 search on the boundary problem in an n-dimensional space for 

 poly-harmonic equation in the presence of a degenerate bounda- 

 ry; he established an almost-periodic solution of the boundary 

 problems of linear hyperbolic equations, investigated the de- 

 pendence of the solutions of hyperbolic equations on disturbing 

 forces, initial and final conditions, and solved new boundary 

 problems for these equations. In his investigation Sobolev 

 formulated a series of new concepts; generalized derivative, 

 generalized solution of equations with partial derivatives, 

 generalized differential operator. With the aid of these con- 

 cepts, he was able to formulate and solve some fundamental 

 problems in mathematical physics. Future development of 

 these ideas of Sobolev led to the establishment of the theory of 

 the so-called generalized functions. Sobolev also studied the 

 properties of functional space. 



As of 1961, Sobolev was a Member of the Presidium, Siberi- 

 an Branch U.S.S.R. Academy of Sciences, and Director of the 

 Institute of Mathematics and Computation Center, Siberian 

 Branch U.S.S.R. Academy of Sciences. 

 Bibliography: 



Some Uses of Functional Analysis in Mathematical Physics. 



Leningrad: 1950. 



Equations of Mathematical Physics, 3rd ed. Moscow: 1954. 



Formulae for mechanical curvatures in n-dimensional space. 



Doklady Akad. Nauk S.S.S.R. 137, #3, 527-30 (1961). 



The interpolation of functions of n-variables. Doklady Akad. 



Nauk S.S.S.R. 137, #4, 778-81 (1961). 



Cube formulae on a sphere, invariants in reformed finite 



groups of isolation. Doklady Akad. Nauk S.S.S.R. 146, #2, 



310-13 (1962). 



Number of formula branches on a sphere. Doklady Akad. 



Nauk S.S.S.R. 146, #4, 770-73 (1962). 

 Office: Moscow University 



Moscow, USSR 



