Mathematical study of the equation describing the evolution of the surface of a film

Abstract

We consider a crystal constituted by an elastic substrate and a film with a small thickness. This crystal being constrained, it appears morphological instabilities.

We are interested in the evolution of the free boundary of the film, which is parametrized by a function denoted by f. The three‐dimensional model here considered is detailed in [8]. This model consists in solving a coupled system of partial derivative equations. The first equations are the linearized elasticity equations posed in the solid, the boundary of which depends on the evolution surface. The second equation is the evolution equation, depending on the elastic displacement. This model is first classically simplified in order to obtain a two‐dimensional model by assuming that the crystal is infinite in one dimension. Besides, under some hypotheses, we derive a wide class of models which the unknown is the map of the film–vapor surface and solves a nonlinear partial derivatives equation, which is independent of the displacement of the solid. Some of those models might blow up in finite time as the physicists expect.

In this paper, we study the existence and uniqueness of solution of this model, by constructing approximated problems under very weak assumptions. To this end, we assume that the initial map of the free boundary is small enough in an appropriate space.