Homogenization of a semilinear variational inequality in a thick multi-level junction

Taras A Mel’Nyk,Iulia A Nakvasiuk

doi:10.1186/s13660-016-1051-y

Taras A Mel’Nyk, Iulia A Nakvasiuk

Open Access

https://doi.org/10.1186/s13660-016-1051-y

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Abstract

We consider a semilinear variation inequality in a thick multi-level junction $\Omega_{\varepsilon}$ , which is the union of a domain $\Omega_{0}$ (the junction’s body) and a large number of thin cylinders. The thin cylinders are divided into m classes depending on the geometrical characteristics and the semilinear perturbed boundary conditions of the Signorini type given on their lateral surfaces. In addition, the thin cylinders from each class are ε-periodically alternated along some manifold on the boundary of the junction’s body. The purpose is to study the asymptotic behavior of the solution $u_{\varepsilon}$ of this variation inequality as $\varepsilon\to0$ , i.e. when the number of the thin cylinders from each class infinitely increases and their thickness tends to zero. The passage to the limit is accompanied by special intensity factors $\{ \varepsilon^{\alpha_{k}}\}_{k=1}^{m}$ in the boundary conditions. We establish two qualitatively different cases in the asymptotic behavior of the solution depending on the value of parameters $\{{\alpha_{k}}\}_{k=1}^{m}$ . For each case we prove a convergence theorem. As a consequence, we see that $u_{\varepsilon}$ converges (as $\varepsilon\to0$ ) to the solution of the corresponding nonstandard homogenized problem and show that the semilinear boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders from each class.

Highlights

Introduction and statement of the problemIt is known that some properties of materials are controlled by their geometrical structure
The present paper is devoted to further development of the asymptotic method proposed in [ ], where we studied the linear Signorini boundary-value problem in a thick junction
In the Signorini boundary conditions there are two alternative sets of boundary conditions the solution must satisfy and it is not a priori known what of the two sets is satisfied for each point

Summary

Define also the subset

). By the same way as in Section we can give equivalent definitions of a weak solution to the problem For this purpose define the multi-valued function. Au , φ – g ≥ F, φ ∀φ ∈ K , where the linear functional F ∈ H ∗ is defined as follows: F, v := fv dx ∀v ∈ H. A function u ∈ K is called a weak solution to problem

Due to the inequality m

Kε and

It is known that x χk ε w

Dk x ε