Continuity at boundary points of solutions of quasilinear elliptic equations with a non-standard growth condition

We study the behaviour at boundary points of a solution of the Dirichlet problem with continuous boundary function for the Euler equation generated by the Lagrangian $|\nabla u|^{p(x)}/p(x)$ with variable$p=p(x)$ that has logarithmic modulus of continuity and satisfies the condition $1\lt p_1\leqslant p(x)\leqslant p_2\lt \infty$. We obtain a regularity criterion for a boundary point of Wiener type, an estimate for the modulus of continuity of the solution near a regular boundary point, and geometric conditions for regularity.