We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω ε which is the union of a domain Ω 0 and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.