A mathematical resolution of some equations describing the evolution of a surface of a constraint material

We consider a cristal structure, constituted by an elastic substrate and a film with a small thickness. The lattice parameters between the film and the substrate are nor the same; consequently, a strain appears in the structure. This strain generates morphologies (see [1,2]). The difficulty consists in finding the profile of the film-vapor surface at any rime, which depends on the elastic displacement of the structure. To this end a physical model, detailed in [2], consists in solving a coupled system of partial derivative equations. The unknowns are the elastic displacement of the structure and the profile of the evolution surface. The elastic displacement solves the linearized elasticity equations posed over the domain occupied by the structure. The boundary of this domain depends on the evolution surface. The second equation is the evolution equation. depending on the elastic displacement by a term of the surface energy. This model is greatly simplified in order to obtain a decoupled two-dimensional model: the map of the film-vapor surface solves a non-linear partial derivatives equation, which is independent of the displacement of the structure. In this Note. we give some results of the existence and uniqueness of a solution for this model under some assumptions about the first derivative of the map. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.