Geodesic
Continuity at boundary points of solutions of quasilinear elliptic equations with a~non-standard growth condition
Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1063-1117

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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 $1$. 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. 
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     author = {Yu. A. Alkhutov and O. V. Krasheninnikova},
     title = {Continuity at boundary points of solutions of quasilinear elliptic equations with a~non-standard growth condition},
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Yu. A. Alkhutov; O. V. Krasheninnikova. Continuity at boundary points of solutions of quasilinear elliptic equations with a~non-standard growth condition. Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1063-1117. http://geodesic.mathdoc.fr/item/IM2_2004_68_6_a0/