Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem. (English) Zbl 1217.35019

The authors consider the elliptic problem \(-\Delta _{x}u(x)=f(x)\) in \(\Omega _{\varepsilon }\), with the homogeneous Dirichlet boundary condition \( u_{\varepsilon }(x)=0\) on \(\Gamma _{\varepsilon }\), with the Signorini-type conditions \(u_{\varepsilon }(x)\leq g(x)\), \(\partial _{\nu }u_{\varepsilon }(x)\leq \varepsilon d(x)\), \((u_{\varepsilon }(x)-g(x))(\partial _{\nu }u_{\varepsilon }(x)-\varepsilon d(x))=0\) on \(S_{\varepsilon }\), and with the Neumann boundary conditions \(\partial _{\nu }u_{\varepsilon }(x)=0\) on \( \partial \Omega _{\varepsilon }\backslash (S_{\varepsilon }\cup \Gamma _{\varepsilon })\). Here \(\Omega _{\varepsilon }\) is a thick junction of type 3:2:1 consisting of the body \(\Omega _{0}\) and of a large number of thin curvilinear and vertical cylinders posed on the top surface of \(\Omega _{0}\), contained in the plane \(\{x_{3}=0\}\). \(S_{\varepsilon }\) is the union of the lateral surfaces of the thin cylinders. \(\Gamma _{\varepsilon }\) is the union of the upper surfaces of the thin cylinders which are contained in the plane \(\{x_{3}=h\}\). \(f\), \(g\) and \(d\) are given functions and \(g\) belongs to some Sobolev space.
The authors propose a variational formulation of this problem introducing the convex set \(K_{\varepsilon }=\{\varphi \in H^{1}(\Omega _{\varepsilon };\Gamma _{\varepsilon }):\varphi |_{S\varepsilon }\leq g|_{S\varepsilon }\) a.e. on \(S_{\varepsilon }\}\). This leads to the definition of weak solutions of this problem. The main result of the paper describes the asymptotic behaviour of the weak solution of this problem. The authors first prove uniform estimates on this weak solution. They then define the notion of weak solution of the homogenized problem and prove the existence of a weak solution of this homogenized problem. Finally they introduce auxiliary functions in order to prove the convergence of the solution of the original problem to that of the limit one. The authors end their paper with the derivation of further properties of the homogenized solution first in the general case, then assuming some additional hypotheses on the data of the problem. In the general case, the authors prove that the equations in \(\Omega _{0}\) can also be treated by means of boundary integral equations.


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
47J05 Equations involving nonlinear operators (general)
74K30 Junctions
35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
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