A homogenization result in the gradient theory of phase transitions

Abstract

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In particular, the case where the scale ε of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as ε → 0, to an anisotropic interfacial energy.

Sharp interface limit of a multi-phase transitions model under nonisothermal conditions

“…Here the periodic structure of the material is modeled by the periodicity of the function W in the first variable. The sharp interface model was derived by Braides and Zeppieri in [17] for the one dimensional case N = 1 and by Fonseca, Hagerty, Popovici, and the first author in [22] (see also [20]) for N > 1. When the homogenization takes place at the level of the singular perturbation we refer to [7,8,27,28].…”

“…In view of (H.2), we can assume m large enough so that the sets z i (Q r m ) are pairwise disjoint. Let us remark that the advantage to work with (22) instead of ( 21) is that the latter is a purely geometric problem, i.e., the functional that we aim at minimizing does not depend on the specific choice of the parametrization. We are able to prove an explicit upper bound on the Euclidean length of certain solutions to the minimization problem in (22) (see Proposition 3.2).…”

“…Let us remark that the advantage to work with (22) instead of ( 21) is that the latter is a purely geometric problem, i.e., the functional that we aim at minimizing does not depend on the specific choice of the parametrization. We are able to prove an explicit upper bound on the Euclidean length of certain solutions to the minimization problem in (22) (see Proposition 3.2). Furthermore, since the only property of Z that is needed in the proof is that it is the union of the images of a compact convex set through the z i 's, the argument also works for the case where Z = {z 1 (x), .…”

“…if z ∈ R M is sufficiently close to Z , where d i (z) denotes the distance between z and z i (Q ρ m ). Given p, q ∈ R M we want to show that solutions to (22) have uniformly finite Euclidean length. We only discuss the case where p and q belong to a neighborhood of z i (Q ρ m ) for some i ∈ {1, .…”

“…, let us denote by q one projection of q on z i (Q ρ m ). Then the curves obtained by first connecting q and q with a segment and then q and p with a curve in z i (Q ρ m ) is a solution to the minimization problem in (22). The proof of this uses the co-area formula and that each curve connecting p and q must traverse every level set of d i lower than d i (q).…”