Evolution of elastic thin films with curvature regularization via minimizing movements

Abstract

Abstract. The evolution equation, with curvature regularization, that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate is considered. The film is strained due to the mismatch between the crystalline lattices of the two materials. Here, short time existence, uniqueness and regularity of the solution are established using De Giorgi's minimizing movements to exploit the L 2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporationcondensation case in the presence of elasticity.

Derivation of a heteroepitaxial thin-film model

“…In [14] the existence and the shape of island profiles, which enforces the presence of nonzero contact angles, has been analyzed in the constraint of faceted profiles. In [18] a mathematical justification of island nucleation was provided by deriving scaling laws for the minimal energy in terms of e 0 and the film volume, and then extended in [2] to the situation of unbounded domains, in the two regimes of small-and large-slope approximations for the profile function h. Finally, the evolutionary problem for thin-film profiles has been studied in dimension two in [11] for the evolution driven by surface diffusion, and in [22] for the growth in the evaporation-condensation case (see also [7,13] for a related model describing vicinal surfaces in epitaxial growth). Recently the analysis of [11] has been extended to three dimensions in [12].…”

Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization

“…It is interesting to observe that the gradient flow of the free-energy functional G with respect to an L 2 -Riemannian structure, (instead of H −1 ) leads to a fourth order evolution equation, which describes motion by evaporation-condensation (see [14,28] and [32], where the two-dimensional case was studied analytically).…”

“…This can be viewed as the evolutionary counterpart of the static theory developed in [11,23,25,22,9,15] in the two-dimensional case and in [10] in three dimensions. The two dimensional formulation of the same evolution problem has been addressed in [24] (see also [32] for the case of motion by evaporation-condensation).…”

Scaling Law and Reduced Models for Epitaxially Strained Crystalline Films

“…We deal only with the stationary setting, but we refer to [20,39] for recent results on the time evolution problem. Finally, we point out that it might be interesting to investigate the model without the hypothesis that the domain of the film is a subgraph (see [10]).…”