The basic mixed plane boundary value problem of equations of statics of the elastic mixture theory is considered in a simply connected domain when the displacement vector is given on one part of the boundary and the stress vector on the remaining part. The problem is investigated using the general displacement vector and stress vector representations obtained in [Basheleishvili, Georgian Math. J. 4: 223–242, 1997]. These representations enable us to reduce the considered problem to a system of singular integral equations with discontinuous coefficients of special kind. The solvability of this system in a certain class is proved, which implies that the basic plane boundary value problem has a solution and this solution is unique.

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The problem of a stressed state in a nonhomogeneous infinite plane consisting of two different anisotropic half-planes and having slits of the finite number on the interface line is investigated. It is assumed that a difference between the displacement and stress vector values is given on interface line segments; on the edges of slits we have the following data: boundary values of the stress vector (problem of stress) or displacement vector values on the one side of slits, and stress vector values on the other side (mixed problem). Solutions are constructed in quadratures.