The gradient theory of phase transitions and the minimal interface criterion

@article{Modica1987TheGT,
  title={The gradient theory of phase transitions and the minimal interface criterion},
  author={L. Modica},
  journal={Archive for Rational Mechanics and Analysis},
  year={1987},
  volume={98},
  pages={123-142}
}
  • L. Modica
  • Published 1987
  • Physics
  • Archive for Rational Mechanics and Analysis
In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed function Wo of the density distribution u. The classical problem (cf. GURTIN [16]) of determining the stable configurations of the fluid is to minimize the total energy of the fluid, E(u) = f Wo(u(x)) dx, t2 

Figures from this paper

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 phaseExpand
Gradient theory of phase transitions with boundary contact energy
Abstract We study the asymptotic behavior as e → 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 internalExpand
Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids
Abstract In this paper we extend the Van der Waals-Cahn-Hilliard theory of phase transitions to the case of a mixture of n non-interacting fluids. By describing the state of the mixture as given by aExpand
Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory
Abstract. We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions.Expand
Stable phase interfaces in the van der Waals–Cahn–Hilliard theory
Abstract We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals–Cahn–Hilliard energyExpand
On the singular limit in a phase field model of phase transitions
Abstract The limits of families of stable solutions for the equation e2 Δue − f(ue) + T(|x|) = 0 over radially symmetric domains with no-flux boundary conditions are discribed. Particular emphasis isExpand
Higher-order phase transitions with line-tension effect
The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of theExpand
Monotonicity of the Energy for Entire Solutions of Semilinear Elliptic Equations
In the mathematical theory of phase transitions in Van der Waals fluids (see, for instance, Alikakos & Bates [AB], Caginalp [CA], Gurtin [GU], Hagau & Serrin [HS], Modica [M03]) the following problemExpand
The sharp interface limit of the van der Waals-Cahn-Hilliard phase model for fixed and time dependent domains
We first study the thermodynamic consistency of phase field models which include gradient terms of the density ρ in the free energy function, such as the van der Waals–Cahn–Hilliard model. It isExpand
Interactions between homogenization and phase-transition processes
The main topic of this note will be the application of homogenization theory to the study of phase transitions problems in composite media. The starting point of the research (see Section 2) is aExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 22 REFERENCES
Structured phase transitions on a finite interval
Abstract : VAN DER WAALS, in his classic paper, gave arguments in support of a compressible fluid whose free energy at constant temperature depends not only on the density, but also on the densityExpand
Nonlinear aspects of the Cahn-Hilliard equation
Abstract This paper treats phase separation within the context of the phenomenological Cahn-Hilliard equation, c t = ∇ · [ M ( c )∇( ∂f / ∂c - K ∇ 2 c )], where c is the concentration, M ( c ) is theExpand
On Phase Transitions with Bulk, Interfacial, and Boundary Energy
Consider a fluid which has free energy1 ψ(u) a prescribed function of density u, and which occupies a fixed container Ω, with Ω a bounded, open region in ℝ N .
Some Results and Conjectures in the Gradient Theory of Phase Transitions
In the van der Waals-Cahn-Hilliard theory of phase transitions the energy depends not only on the density, but also on the density gradient, a dependence introduced to account for the interfaceExpand
On the singular limit for a class of problems modelling phase transitions
The total variation is a measure of the complexity of a given solution to ${{(\varepsilon )} \mathord{\left/ {\vphantom {{(\varepsilon )} {\varepsilon ^2 u''_\varepsilon }}} \right.Expand
Minimal surfaces and functions of bounded variation
I: Parametric Minimal Surfaces.- 1. Functions of Bounded Variation and Caccioppoli Sets.- 2. Traces of BV Functions.- 3. The Reduced Boundary.- 4. Regularity of the Reduced Boundary.- 5. SomeExpand
Elliptic Partial Differential Equa-tions of Second Order
Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's RepresentationExpand
Minimal Surfaces of Codimension One
Introduction. 1. Differential Properties of Surfaces. 2. Sets of Finite Perimeter and Minimal Boundaries. 3. The Dirichlet Problem for the Minimal Surface Equation. 4. Unbounded Solutions.
Geometric Measure Theory
Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 SimpleExpand
Funzioni $BV$ e tracce
L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisationExpand
...
1
2
3
...