$H^2$ regularity for the $p(x)-$Laplacian in two-dimensional convex domains

In this paper we study the $H^2$ global regularity for solutions of the $p(x)-$Laplacian in two dimensional convex domains with Dirichlet boundary conditions. Here $p:\mathrm{\Omega }\to [p_1,\mathrm{\infty })$ with $p\in Lip(\bar{\mathrm{\Omega }})$ and $p_1>1$.