Geodesic
Second-order elliptic systems in the half-plane
Izvestiya. Mathematics , Tome 70 (2006) no. 6, pp. 1233-1264.

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We consider boundary-value problems in the upper half-plane for second-order elliptic systems with constant higher coefficients. Using the Bitsadze transformation, we reduce these problems to equivalent problems for analytic functions. This approach enables us to obtain explicit formulae for the solutions of basic boundary-value problems and to study the Fredholm solubility of these problems. (In particular, we obtain an analytic expression for the index.) We work in weighted Hölder and Hardy spaces. 
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A. P. Soldatov. Second-order elliptic systems in the half-plane. Izvestiya. Mathematics , Tome 70 (2006) no. 6, pp. 1233-1264. http://geodesic.mathdoc.fr/item/IM2_2006_70_6_a6/

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