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Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary. (English) Zbl 1403.35035

In this paper, the authors study the homogenization of a nonlinear monotone problem with Signorini boundary conditions in a brush domain \(\Omega_\epsilon\subset \mathbb{R}^N\), \(N\geq 2\), which consists of vertical cylinders with fixed height and diameter of order \(\epsilon\) distributed on a fixed base \(\epsilon\)-periodically. The authors study the asymptotic behavior of the problem as \(\epsilon\) tends to zero. To this end, they first establish a uniform (in \(\epsilon\)) \(H^1\)-estimate for the solutions. Then they identify the limit problem by using the method of oscillating test functions introduced by L. Tartar and by exploring the monotone relation for the monotone operators considered.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J60 Nonlinear elliptic equations
35R35 Free boundary problems for PDEs
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
47H05 Monotone operators and generalizations
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