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Key words: hyperbolic equations, Lamb problem, wave propagation,
Chidhood and youth
Sergei L'vovich Sobolev (1908 - 1989) was born in St. Petersburg (Leningrad) in the family of Lev Aleksandrovich Sobolev, a barrister. S.L.Sobolev's grandfather on his father's side came from a family of Siberian Cossacks. Sergej L'vovich was bereaved of his father in early childhood and was raised by his mother, Natal'ya Georgievna, who was a very educated woman, a teacher of literature and history. Natal'ja Georgievna also had the second speciality: she graduated from a medical institute and worked as a tutor at the First Leningrad Medical Institute. She cultivated in S.L.Sobolev such qualities as adherence to principles, honesty, and purposefulness which characterized him as a scholar and personality. Sergei L'vovich studied the course of secondary school himself, taking the greatest interest in mathematics. During the civil war he and his mother lived in Kharkov. When living there, he studied at preparatory courses of a workers' night technical secondary school for one semester. At the age of 15 he knew the whole course of mathematics, physics, chemistry, and other sciences to the extent of secondary school, has read many classics of the Russian and world literature, books on philosophy, medicine and biology. Having moved from Kharkov to Petersburg in 1923, Sergei L'vovich entered the graduate class of School No. 190 and graduated from it with honours in 1924. After graduation he could not enter the university because he was under age (he was 15 years old). He began to study at the First State Art School in the piano class.
In 1925 S.L.Sobolev entered the Faculty of Physics and Mathematics of Leningrad State University (LSU) continuing at the same time to study at the Art School. In LSU, S.L.Sobolev attended lectures of Professors N.M.Gyunter, V.I.Smirnov, G.M.Fikhtengol'ts, and others. He made his diploma on analytic solutions of a system of differential equations with two independent variables under the supervision of N.M.Gyunter. At that time LSU was a large mathematical research center which maintained remarkable traditions of the Petersburg mathematical school famous for its profound discoveries and connected with the names of P.L.Chebychev, A.M.Lyapunov, and A.A.Markov.
Having graduated from the Leningrad State University in 1929, Sergei L'vovich began to work at the Theoretical Department of the Leningrad Seismological Institute under the guidance of V.I.Smirnov. At that period he solved, in close co-operation with V.I.Smirnov, a few of fundamental mathematical problems of wave propagation theory. In 1930 Sergei L'vovich published in the Proceedings of the Seismological Institute an article on the wave equation in an inhomogeneous medium. This work and his subsequent publications on the same subject are important from a mathematical viewpoint because they originated the well-known Sobolev method for solving of the Cauchy problem for hyperbolic equations of 2nd order. Many important solutions of the wave equation, e.g., homogeneous solutions of the zero degree of homogeneity, are functionally invariant. Reflecting functionally invariant solutions from a plane boundary under classical boundary conditions, we again obtain functionally invariant solutions. With the aid of this new method S.L.Sobolev, together with V.I.Smirnov, explicity solved the famous Lamb problem of finding the displacement of an elastic half-plane under the action of a concentrated impulse. With the aid of the superposition principle the three-dimennnsional axia symmetric case of the Lamb problem was also solved. It is easy to see that when a plane step wave is incident on a corner (zero before the wave front, unity behind it) the solution should be a homogeneous function of the zero degree of homogeneity. The technique of homogeneous functionally invariant solutions turned out to be very convenient here.
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