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Almost periodicity of solutions of the first and the second boundary value problems for the wave equation in cylindrical domains was proved in the works: Muckenhoupt, C. F.: J. Math. Phys. Massachusetts Inst. of Technology, 8, 163–199 (1929); Sobolev, S. L.: Dokl. Akad. Nauk SSSR, 48, 570–573 (1945); 48, 646–648 (1945); 49, 12–15 (1945). — Ed.
See, for instance, the works: Aleksandryan, R. A.: Tr. Moskov. Mat. Obshch., 9, 455–505 (1960); Denchev, R. T.: Dokl. Akad. Nauk SSSR, 126, 259–262 (1959). Ed.
See references in: Berezanskii, Yu. M.: Expansions in Eigenfunctions of Selfadjoint Operators. Naukova Dumka, Kiev (1965). — Ed.
The notion of the eigenfunctional or the generalized eigenfunction was introduced by R. A. Aleksandryan in his Ph. D. Thesis (Moskovsk. Gosudarstv. Univ., Moscow (1949)). — Ed.
The main results of R. A. Aleksandryan’s thesis (Moskovsk. Gosudarstv. Univ., 1949) are published in the works: Dokl. Akad. Nauk SSSR, 73, 631–634 (1950); 73, 869–872 (1950); Izv. Akad. Nauk Arm. SSR, Ser. Fiz.-Mat. Nauk, 10, 69–83 (1957); Tr. Moskov. Mat. Obshch., 9, 455–505 (1960). — Ed.
Zelenyak, T. I.: Differ. Uravn., 2, 47–64 (1966) — Ed.
The continuous dependence of solutions of mixed problems for the Sobolev equation for n = 2; 3 were studied also in: Zelenyak, T. I.: Dokl. Akad. Nauk SSSR, 164, 1225–1228 (1965). — Ed.
Gagliardo, E.: Rend. Sem. Mat. Univ. Padova, 27, 284–305 (1957) — Ed.
See the following: John, F.: Amer. J. Math., 63, 141–154 (1941); Vakhaniya, N. N.: Dokl. Akad. Nauk SSSR, 116, 906–909 (1957); Arnold, V. I.: Izv. Akad. Nauk SSSR, Ser. Mat., 25, 21–86 (1961); Finzi, A.: Ann. Sci. Ecole Norm. Sup., 69, 371–430 (1952). — Ed.
For the review and references see the paper by Aleksandryan R. A., Berezanskii Yu. M., Il’in V. A., Kostyuchenko A. G. in the book: Partial Differential Equations (Proceedings of the Symposium Dedicated to the 60th Anniversary of Academician S. L. Sobolev) Nauka, Moscow (1970), pp. 3–35. — Ed.
Vishik, M. I.: Mat. Sb., 39, 51–148 (1956); Ladyzhenskaya, O. A.: The Mathematical Problems of Dynamics of Viscous Incompressible Fluid. Nauka, Moscow (1961); English edition: Gordon and Breach Science Publishers, New York — London (1963). — Ed.
The review and references can be found in the book: Zelenyak, T. I.: Selected Questions of Qualitative Theory of Partial Differential Equations. Novosibirsk. Gosudarstv. Univ., Novosibirsk (1970). — Ed.
S. L. Sobolev’s investigations of the problem on small oscillations of a rotating fluid originated the most intense interest in equations not solvable with respect to the highest-order derivative \( \mathcal{A}_0 D_t^l u + \sum\limits_{k = 0}^{l - 1} {\mathcal{A}_{l - k} } D_t^k u = f \) where A 0, A 1;...;A l are linear differential operators in x = (x 1;...; x n). At present, equations of such form are often called equations of Sobolev type in the literature. Bibliographical comments and extensive references devoted to the theory of boundary value problems for equations of Sobolev type can be found in the book: Demidenko, G. V., Uspenskii, S. V.: Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative. Marcel Dekker, New York, Basel (2003). — Ed.
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Sobolev, S.L. (2006). On a Class of Problems of Mathematical Physics. In: Demidenko, G.V., Vaskevich, V.L. (eds) Selected Works of S.L. Sobolev. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-34149-1_11
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