Asymptotic analysis for some boundary value problems ‎in thin domains with friction laws

Abstract

This thesis focuses on the study of the asymptotic analysis of some boundary value problems in a ‎three-dimensional thin domain Ω^ε with nonlinear boundary conditions of friction type on a part of ‎the boundary. The main idea of this study is to show how to derive two-dimensional limit problems ‎when the thickness tends to zero for three types of bilateral contacts problems involving Tresca's or ‎Coulomb's friction law. We start first with an incompressible fluid governed by the Brinkman ‎equation. Then the second problem concerns a mathematical model describing the static process of ‎contact between a piezoelectric body and a foundation. Finally, the third work carried out is ‎devoted to the transmission problem for the linear elasticity equation with a nonlinear dissipative ‎term. Precisely, we have transformed the original problems posed in the domain Ω^ε into new ‎equivalent problems on a fixed domain Ω independent of a small parameter ε, and by using a new ‎scale and several inequalities we prove some estimates and convergence theorems. Then, we obtain ‎the limit problems with the weak generalized equation and its uniqueness.‎