Abstract
In a domain \(G \subset \mathbb{R}^{n + 1} \) with cylindrical ends at infinity, we consider a general elliptic dissipative boundary value problem. The coefficients of the imaginary part of the operator of the problem vanish as \(x \to \infty ,x \in G\) The asymptotic behavior of the solutions is expressed in terms of incoming and outgoing “waves” (the amplitudes of such waves can grow at infinity). We introduce an (augmented) scattering matrix and, in terms of this matrix, we compute the number of linearly independent solutions to the homogeneous problem vanishing at infinity with a given rate. We discuss the statement of a problem with the so-called radiation conditions. The natural radiation conditions (only outgoing “waves” occur in asymptotic formulas for solutions) can be applied in any case. Other admissible radiation conditions for the problem under consideration are connected with the natural ones via scattering matrices. Bibliography: 12 titles.
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Kalvine, V.O., Neittaanmäki, P. Dissipative Elliptic Problems in Domains with Cylindrical Ends, Scattering Matrices, and Radiation Conditions. Journal of Mathematical Sciences 120, 1093–1108 (2004). https://doi.org/10.1023/B:JOTH.0000014838.09839.48
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DOI: https://doi.org/10.1023/B:JOTH.0000014838.09839.48