Coupled second order singular perturbations for phase transitions

The Quarterly Journal of Mathematics, 2017

In this paper we study, via Γ-convergence techniques, the asymptotic behaviour of a family of coupled singular perturbations of a non-convex functional of the type as a variational model to address two-phase transition problems under the volume constraints and where the additional unknown ρ interplays with ∇u in the formation of interfaces.

We study the Γ−convergence as ε → 0 + of the family of degenerate functionals Q ε (u) = ε Ω ADu, Du dx + 1 ε Ω W (u) dx where A(x) is a symmetric, non negative n×n matrix on Ω (i.e. A(x)ξ, ξ ≥ 0 for all x ∈ Ω and ξ ∈ R n) with regular entries and W : R → [0, +∞) is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix A, we obtain a minimal interface criterion for the Γ−limit functional exploiting some tools of Analysis in Carnot-Carathéodory spaces. We extend some previous results obtained for the non degenerate perturbations Q ε in the classical gradient theory of phase transitions.

A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy, precisely,

Comm. Pure Appl. Math, 2002

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1989

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.

Journal de Mathématiques Pures et Appliquées, 2006

We study the following two phase elliptic singular perturbation problem:

Journal of Differential Equations, 2002

Transactions of the American Mathematical Society, 2008

We consider a class of nonconvex functionals of the gradient in one dimension, which we regularize with a second order derivative term. After a proper rescaling, suggested by the associated dynamical problems, we show that the sequence {F ν } of regularized functionals Γ-converges, as ν → 0 + , to a particular class of free-discontinuity functionals F , concentrated on SBV functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with F , using the minimizing movements method. We show that the minimizing movement starting from an initial datum with a finite number of discontinuities has jump positions fixed in space and whose number is nonincreasing with time. Moreover, there are a finite number of singular times at which there is a dropping of the number of discontinuities. In the interval between two subsequent singular times, the vector of the survived jumps is determined by the system of ODEs which expresses the L 2 -gradient of the Γ-limit. Furthermore the minimizing movement turns out to be continuous with respect to the initial datum. Some properties of a minimizing movement starting from a function with an infinite number of discontinuities are also derived.

Transactions of the American Mathematical Society, 1995

The paper deals with the boundary value problem ex+xx-x2 = 0, with jc(0) = A, x(T) = B for A, B, T > 0 and e > 0 close to zero. It is shown that for T sufficiently big, the problem has exactly three solutions, two of which reach negative values. Solutions reaching negative values occur for T > T(e) > 0 and we show that asymptotically for e-> 0, T(e) ~-ln(e), i.e. lime_o _ wj) = 1 ■ The main tools are transit time analysis in the Liénard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems.

2003

The behavior of the solution of the below problem (E†), (BC†), (TC†) is studied when the small parameter † tends to 0. ( ¡†u00(x) + fi(x)u0(x) + fl(x)u(x) = f(x); x 2 (a;b); ¡ ¡ "(x)v0(x) ¢0 + fi(x)v0(x) + fl(x)v(x) = g(x); x 2 (b;c); (E†)