Abstract
Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.
Similar content being viewed by others
References
V. P. Burskii, “On a solution of the Dirichlet problem for elliptic systems in a disk,”Ukr. Mat. Zh.,44, No. 10, 1307–1313 (1992).
V. P. Burskii, “Boundary-value problems for a second-order hyperbolic equation in a disk,”Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 22–29 (1987).
N. N. Luzin,An Integral and a Trigonometric Series [in Russian], Gostekhteoretizdat, Moscow (1951).
V. P. Burskii,Theorems on Traces of a Solution of String Equation in a Disk [in Russian], Preprint No. 85.23, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985).
F. John, “The Dirichlet problem for a hyperbolic equation,”Am. J. Math.,63, No. 1, 141–154 (1941).
R. A. Aleksandryan, “Spectral properties of operators generated by systems of differential equations of Sobolev type,”Tr. Mosk. Mat. Obshch.,9, 455–505 (1960).
S. G. Ovsepyan, “On ergodicity of continuous automorphisms and uniqueness of a solution of the Dirichlet problem for the string equation. II,”Izv. Akad. Nauk Arm. SSR,2, No. 3, 195–209 (1967).
V. P. Burskii, “On the violation of uniqueness of a solution of the Dirichlet problem for elliptic systems in a disk,”Mat. Zametki,48, No. 3, 32–37 (1990).
L. Bers, F. John, and M. Shehter,Partial Differential Equations [Russian translation], Mir, Moscow (1974).
A. Ya. Khinchin,Continued Fractions [in Russian], Nauka, Moscow (1961).
B. I. Ptashnik,Incorrect Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.
Rights and permissions
About this article
Cite this article
Burskii, V.P. Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem. Ukr Math J 45, 1659–1668 (1993). https://doi.org/10.1007/BF01060856
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01060856