Asymptotic analysis of a Signorini-type problem

Resumo:  Signorini type boundary conditions appear in many problems in applied mathematics deriving from engineering and physics. We can think, for instance, to lubrication and filtration processes, hydrodynamics, plasticity, crack theory, optimal control problems etc..The weak formulations of such kind of problems involve variational inequalities corresponding to non linear free boundary-value problems (see [1], [2] and [3]). In this talk it is investigated the effect of a Signorini type interface condition on the asymptotic behaviour, as ε tends to zero, of a problem posed in a ε-periodic domain with inclusions. The Signorini type condition is expressed in terms of a set of equalities ad inequalities involving the jump of the solution on the interface and its conormal derivative via a parameter γ. Different limit problems are obtained according to different values of γ. The most interesting cases are γ = −1 ad γ = 1. Indeed in the first case, at the limit, we obtain a non linear homogenized problem, where the effective diffusion term is defined through the solution of a variational inequality with a non negative jump on the reference cell interface. In the case γ = 1, at the limit, we get an obstacle type problem in the whole domain.
References
[1] H. Brezis, Problemes unilateraux, J. Math. Pures Appl. 51 (1972) 1–168.
[2] G. Duvaut and J.L. Lions, Les in ́equations en m ́ecanique et en physique
(Dunod, Paris, 1972).
[3] D. Kinderleherer and G. Stampacchia, An Introduction to Variational In-
equalities and Their Applications, Classics in Applied Mathematics (Aca-
demic Press, New York, 2000).