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.