M. Basheleishvili
abstract:
In the paper we consider the boundary value problem for homogeneous equations of
statics of the theory of elastic mixtures in a circular domain, and in an
infinite domain with a circular hole, when projections of the displacement
vector on the normal and of the stress vector on the tangent are prescribed on
the boundary of the domain. The arbitrary analytic vector j
appearing in the general representation of the displacement vector is sought as
a double layer potential whose density is a linear combination of the normal and
tangent unit vectors. Having chosen the displacement vector in a special form,
we define the projection of the density on the normal by the function given on
the boundary. To find the projection of the stress vector on the tangent, we
obtain a singular integral equation with the Hilbert kernel. Using the formula
of transposition of singular integrals with the Hilbert kernel, we obtain
expressions for the projection on the tangent of the above-mentioned density.
Assuming that the function is Hölder
continuous, the projection of the displacement vector on the normal and its
derivative are likewise Hölder continuous.
Under these conditions the obtained expressions for the displacement and stress
vectors are continuous up to the boundary. The theorem on the uniqueness of
solution is proved, when the boundary is a circumference. The projections of the
displacement vector on the normal and tangent are written explicitly. Using
these projections, the displacement vector is written in the form of the
integral
Poisson type formula.
Mathematics Subject Classification: 74B05
Key words and phrases: Potentials, singular equation with a Hilbert kernel, formula of interchange of of singular integrals, general representation of the displacement and stress vectors, analogues of the general Kolosov-Muskhelishvili's representations