Coupled second order singular perturbations for phase transitions

Abstract

Abstract. The asymptotic behavior of a family of singular perturbations of a non-convex second order functional of the typeis studied through Γ-convergence techniques as a variational model to address two-phase transition problems.

Singular perturbations for phase transitions

“…We conclude this survey by mentioning some recent results obtained by Baía, Barroso, Chermisi & Matias[3] where the authors addressed the asymptotic behaviour, as a small parameter ε tends to zero, of a sequence of functionals of the formE ε (u) = 1 ε Ω f (x, u(x), ε∇u(x), ε 2 ∇ 2 u(x)) dxobtained as a singular perturbation of a non-convex second order functional of the typeΩ f (x, u(x), ∇u(x), ∇ 2 u(x)) dx where f (·, u(·), ∇u(·), ∇ 2 u(·))represents the free energy of a mixture of d fluids (d ∈ N, d ≥ 2), occupying a fixed container Ω ⊂ R N (N ∈ N, N ≥ 2), and is a function of the density u = (u 1 , ..., u d ) and its first and second order derivatives. The bulk energy density f is assumed to be continuous, positive and such that for all x ∈ Ω the function f (x, ·, O, O) achieves its minimum value zero at exactly two vectors α, β ∈ R d + , α = β.…”

Domain Formation in Membranes Near the Onset of Instability

“…As we will see in the sequel (see (2.3)) the constant m d represents the surface energy density per unit area of the limit energy. The fact that m d is characterized by the cell problem (2.2) is to be expected in this type of singular perturbations problems (see, e.g., [2], [7], [8]). As it turns out, in the case in which only first order derivatives are considered in the energy functionals, m d reduces to a one-dimensional geodesic distance between the wells for an appropriate metric involving the double-well potential W (see [13]).…”

“…The Γ-convergence of similar energies has been considered before (e.g. [18,2]), however there are some differences with the functional (2.1) (for example when q = 0) and will be addressed in a separate paper.…”

A model for phase transitions with competing terms

“…Thus, by (H4), using the fact that c n → 1 and since u n and ε n λ n are uniformly bounded in L ∞ and λ n is bounded in L 1 , we have with x j ∈ Ω ν \ H ν and c j ∈ R + 0 . Following the procedure in [6], this result can be generalised for…”

“…u(x) dx = V f relies on the fact that g(u) = 0 ⇔ u ∈ {α, β} and can be achieved following an argument analogous to the one used in Lemma 4.3 in[6].…”