Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018
REGULARITY OF A BOUNDARY POINT FOR
Yu . A . Alkhu t ov
A. G. and N. G. Stoletov Vladimir State University
87, Gor’kogo St., Vladimir 600000, Russia
M. D. Surnachev
Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences
4, Miusskaya sq., Moscow 125047, Russia
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:24titles.
Dedicated to the memory of Vasilii Vasil’evich Zhikov
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
∇u) = 0 (1.1)
at a boundary point, where the exponent p is measurable and such that
1 <α p(x) β for almost all x ∈ D. (1.2)
Equations of the form (1.1) ﬁrst arose in works of Zhikov [1, 2] in connection with homoge-
nization of integrands of the form |∇u|
. Such equation also arise in mathematical modeling of
ﬂuids whose properties change under the action of electromagnetic ﬁeld or temperature –.
To deﬁne a solution to Equation (1.1), we introduce the class of functions
u ∈ W
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 92, 2018, pp. 5-25.
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