Linear elliptic equations in composite media with
anisotropic fibres are concerned. The media consist of a periodic set of anisotropic
fibres with low conductivity, included in a connected matrix with high conductivity.
Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with
that in the transverse directions.
The coefficients of the elliptic equations depend on the conductivity.
This work is to derive the
H\"older and the gradient $L^p$ estimates (uniformly in the period size
of the set of anisotropic fibres as well as in the conductivity ratio of
the fibres in the transverse directions to the connected matrix)
for the solutions of the elliptic equations.
Furthermore, it is shown that, inside the fibres, the solutions have higher regularity
along the fibres than in the transverse directions.