The gradient theory of phase transitions for systems with two potential wells

Abstract

SynopsisIn this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbationsof the nonconvex functionalwhere W:RN→R supports two phases and N ≧1. We obtain the Γ(L1(Ω))-limit of the sequenceMoreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of Eε(·) converging in L1(Ω) to a minimiser of E0(·) with minimal interfacial area.

On a Volume-Constrained Variational Problem

“…Similar results have been obtained for phasetransitions problems in which the formation of phases is driven by a double-well potential (see [13], [5], [8]); here the creation of interfaces is due to the volume constraints. In Section 5 we characterize fully the solutions of (M) when W (ξ) = |ξ| 2 , Ω is an interval and d > 1 (see Subsection 5.1).…”

Second order asymptotic development for the anisotropic Cahn–Hilliard functional

“…If the boundary condition u = 1 on ∂Ω is omitted and P Φ (E) is replaced by by P Φ (E, Ω) := Ω∩∂ * E Φ (ν E ) dH n−1 , this result has been established in [20], [21], [26] for the isotropic scalar-valued case, in [14] for the isotropic vectorvalued case, in [6], [23] for the anistropic, scalar-valued case, and in [4] for the anisotropic, vector-valued case (see also [7]). In the proof below we show how to take into account the boundary condition.…”

“…In this paper we study the second order term in the asymptotic development by Γ-convergence for the anisotropic Cahn-Hilliard functional (see, e.g., [20], [15], [21], [26], [14], [6], [4])…”

A result in asymptotic analysis for the functional of Ginzburg-Landau type with externally imposed multiple small scales in one dimension

“…The literature on the subject is extensive. Here we only mention [1,2] and [6][7][8][9][10][11][12][13][14][15]. Further list of references can be found in [1].…”