Hölder continuity of $ p(x)$-harmonic functions

The question on the Holder continuity of solutions of the -Laplace equation with measurable summability index bounded away from one and infinity is studied. In the case when the domain of definition , , of the equation is partitioned by a hyperplane into parts and such that has a logarithmic modulus of continuity at a point from either side it is proved that solutions of the equation are Holder-continuous at . The case when has a logarithmic modulus of continuity in and is considered separately. It is proved that smooth functions in are dense in the class of solutions.