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Solutions of the dirichlet problem for elliptic systems in a disk

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Abstract

We study 2×2 second-order elliptic systems, which can be written as a single equation with complex coefficients. In an arbitrary bounded region with smooth boundary, we obtain necessary and sufficient conditions on the trace relation of a solution, which we apply in the case of a disk. We prove existence and uniqueness theorems for a solution in a Sobolevskii space of an equation which is not properly elliptic. In particular, we prove that the properties of the problem determine the angle between the bicharacteristics. If it is π-rational, then there is no uniqueness, but if it is π-irrational, then the smoothness of the solution of the Dirichlet problem depends on the order of its approximation by π-rational numbers; but if it is nonreal, then the problem has the usual properties for the elliptic case.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Academy of Sciences of the Ukraine, Donetsk

    V. P. Burskii

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  1. V. P. Burskii
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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1307–1313, October, 1992.

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Burskii, V.P. Solutions of the dirichlet problem for elliptic systems in a disk. Ukr Math J 44, 1197–1203 (1992). https://doi.org/10.1007/BF01057674

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  • DOI: https://doi.org/10.1007/BF01057674

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