DOI 10.1007/s10958-018-3870-5 Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 REGULARITY OF A BOUNDARY POINT FOR THE p(x)-LAPLACIAN Yu. A. Alkhutov A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia firstname.lastname@example.org M. D. Surnachev Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences 4, Miusskaya sq., Moscow 125047, Russia email@example.com UDC 517.9 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. Bibliography:24 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov 1 Introduction Let D be a bounded domain in R , n 2. This paper is devoted to the behavior of the solutions to the Dirichlet problem in D for the equation p(x)−2 Lu =div(|∇u| ∇u) = 0 (1.1) at a boundary point, where the exponent p is measurable and such that 1 <α p(x) β for almost
Journal of Mathematical Sciences – Springer Journals
Published: Jun 2, 2018
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