Gradient Theory Of Phase Transitions Research Articles

Erratum to: A homogenization result in the gradient theory of phase transitions

We would like to thankWill Feldman and Peter S. Morfe for pointing out the typo in the cell formula of Definition 1.3 in [1]. Here we present the correct formula for the energy density of the limiting functional, together with the minor modifications needed to adjust accordingly the proofs of the results presented in the paper.

A homogenization result in the gradient theory of phase transitions

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 \varepsilon 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 \varepsilon \to 0 , to an anisotropic interfacial energy.

The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).

Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions

We study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible connection with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino, focusing close to the helimagnet-ferromagnet transition point corresponding to the critical value of the frustration parameter $\alpha=4$. We reformulate this problem for any $\alpha\geq0$ in the framework of surface energies in nonconvex discrete systems with nearest neighbors ferromagnetic and next-to-nearest neighbors antiferromagnetic interactions and we link it to the gradient theory of phase transitions, by showing a uniform equivalence by $\Gamma$-convergence on $[0,4]$ with Modica-Mortola type functionals.

A weighted gradient theory of phase transitions with a possibly singular and degenerate spatial inhomogeneity

This paper studies the asymptotic behavior of a perturbed variational problem for the Cahn–Hilliard theory of phase transitions in a fluid, with spatial inhomogeneities in the internal free energy term. The inhomogeneous term can vanish or become infinite and it can also behave as an appropriate power of the distance from the boundary. The standard minimal interface criterion will be recovered even in spite of such severe degeneracies and/or singularities.

Unique minimizer for a random functional with double-well potential in dimension 1 and 2

We add a random bulk term, modelling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. We show that in d � 2 there exists, for almost all the realizations of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast to the case when the random bulk term is absent. In the latter case there are two minimizers which are (in law) invariant under translations in space.

The Gibbs–Thomson relation for non homogeneous anisotropic phase transitions

In this paper we prove the Gibbs–Thomson relation between the coarse grained chemical potential and the non homogeneous and anisotropic mean curvature of a phase interface within the gradient theory of phase transitions thus proving a generalization of a conjecture stated by Gurtin and proved by Luckhaus and Modica in the homogeneous and isotropic case.

Sharp-Interface Limit of a Ginzburg–Landau Functional with a Random External Field

We add a random bulk term, modeling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double-well potential. For the resulting functional we study the asymptotic properties of minimizers and minimal energy under a rescaling in space, i.e., on the macroscopic scale. By bounding the energy from below by a coarse-grained, discrete functional, we show that for a suitable strength of the random field the random energy functional has two types of random global minimizers, corresponding to two phases. Then we derive the macroscopic cost of low energy “excited” states that correspond to a bubble of one phase surrounded by the opposite phase.

Gradient theory of phase transitions with a rapidly oscillating forcing term

We consider the Gamma-limit of a family of functionals which model the interaction of a material that undergoes phase transition with a rapidly oscillating conservative vector field. These functionals consist of a gradient term, a double-well potential and a vector field. The scaling is such that all three terms scale in the same way and the frequency of the vector field is equal to the interface thickness. Difficulties arise from the fact that the two global minimizers of the functionals are nonconstant and converge only in the weak L-2-topology.

Gradient theory of phase transitions in composite media

We study the behaviour of non-convex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes, depending on the frequency of the oscillations. In the two extreme cases, a separation of scales effect is described.

From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists uε, a critical point of the Allen-Cahn energy Eε(u) = ε2 ∫ |∇u|2 + ∫(1 − u2)2, whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function uε being a critical point of Eε under some volume constraint.

A compactness result in the gradient theory of phase transitions

We examine the singularly perturbed variational problem in the plane. As ε → 0, this functional favours |∇ψ| = 1 and penalizes singularities where |∇∇ψ| concentrates. Our main result is a compactness theorem: if {Eε(ψε)}ε↓0 is uniformly bounded, then {∇ψε}ε↓0 is compact in L2. Thus, in the limit ε → 0, ψ solves the eikonal equation |∇ψ| = 1 almost everywhere. Our analysis uses ‘entropy relations’ and the ‘div-curl lemma,’ adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.

Slow motion in the gradient theory of phase transitions via energy and spectrum

We describe some aspects of the Cahn-Hilliard and related equations. In particular we consider the dynamics of almost spherical interfaces and establish that almost spherical interfaces either persist forever or until they reach the boundary, a phenomenon which happens superslowly.

Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions

We study radially symmetric stationary points of the functionalwhere u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and ε accounts for surface energy. Under the further assumption of small energy, that isfor small ε, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation. Each of these solutions is monotone in the radial direction and converges as ε→0 to one of two possible radially symmetric single interface minimizers of E0. Our main tool is the method of matched asymptotic expansions from which we construct exact solutions.

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ɛ2¦▽ϱ¦2+ W(ϱ), where ϱ is the density. The main result is that λɛ is asymptotically equal to ɛEλ/d+o(ɛ), with E the interfacial energy, per unit surface area, of the interface between phases, λ the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.

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

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.

Gradient theory of phase transitions with boundary contact energy

We study the asymptotic behavior as ε → 0+ of solutions of the variational problems for the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid. We assume that the internal free energy, per unit volume, is given by ε2|∇ρ|2 + W(ρ) and the contact energy with the container walls, per unit surface area, is given by εσ(ρ), where ρ is the density. The result is that such solutions approximate a two-phases configuration satisfying a variational principle related to the equilibrium configuration of liquid drops.