We study the behavior of solutions to the Dirichlet problem for the p(x)-Laplacian with a continuous boundary function. We prove the existence of a weak solution under the assumption that p is separated from 1 and ∞. We present a necessary and sufficient Wiener type condition for regularity of a boundary point provided that the exponent p has the logarithmic modulus of continuity at this point.
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Dedicated to the memory of Vasilii Vasil’evich Zhikov
Translated from Problemy Matematicheskogo Analiza 92, 2018, pp. 5-25.
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Alkhutov, Y.A., Surnachev, M.D. Regularity of a Boundary Point for the p(x)-Laplacian. J Math Sci 232, 206–231 (2018). https://doi.org/10.1007/s10958-018-3870-5
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DOI: https://doi.org/10.1007/s10958-018-3870-5