On the regularity of $p(x)$-harmonic functions

The talk is based on a joint work with Y.A.Altukhov (Vladimir State University). I will present a series of results on the regularity of solutions to divergent elliptic equations of the $p(x)$-Laplacian type. Most of well-known facts for solutions to equations of such type (as well as in the theory of the corresponding functional spaces) requires at least “logarithmic Hölder-continuity” of the nonlinearity characteristics $p(x)$. We deal with the cases when the condition is not valid. Firstly, we are interested in the case when $p(x)$ is discontinuous, but has a distinct geometric structure. Secondly, we study equations with $p(x)$ having modulus of continuity that is weaker than the logarithmic one. Special attention is devoted to the case when $p(x)$ is continuous only at a given point.