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SMALL VOLUME FRACTION LIMIT OF THE DIBLOCK COPOLYMER PROBLEM: I. SHARPINTERFACE FUNCTIONAL ∗
"... Abstract. We present the first of two articles on the small volume fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharpinterface version of the functional and consider a limit in which the volume ..."
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Abstract. We present the first of two articles on the small volume fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharpinterface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called particles) remainsO(1). Using the language of Γconvergence, we focus on two levels of this convergence and derive firstand secondorder effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into “point particles. ” At the highest level, the effective energy is entirely local and contains information about the structure of each particle but no information about their spatial distribution. At the next level we encounter a Coulomblike interaction between the particles, which is responsible for the pattern formation. We present the results here in both three and two dimensions.
Nonoriented solutions of the Eikonal equation
"... We study a new formulation for the eikonal equation ∇u  = 1 on a bounded subset of R 2. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1dimensional subspaces, with div P ∈ L 2. We prove existence and uniqueness for solutions of the equation P div P = 0. We give ..."
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We study a new formulation for the eikonal equation ∇u  = 1 on a bounded subset of R 2. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1dimensional subspaces, with div P ∈ L 2. We prove existence and uniqueness for solutions of the equation P div P = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. The idea of the proof is to apply a generalized method of characteristics introduced in [8] to a suitable vector field m satisfying P = m ⊗ m. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals. AMS Cl. 35L65, 35B65. 1
Stripe Patterns and the Eikonal Equation
"... In this note we describe the behaviour of a stripeforming system that arises in the modelling of block copolymers. Part of the analysis concerns a new formulation of the eikonal equation in terms of projections. For precise statements of the results, complete proofs, and references, we refer to [4] ..."
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In this note we describe the behaviour of a stripeforming system that arises in the modelling of block copolymers. Part of the analysis concerns a new formulation of the eikonal equation in terms of projections. For precise statements of the results, complete proofs, and references, we refer to [4] and [3].
Partial Differential Equations Nonoriented Solutions of the Eikonal Equation
"... Received *****; accepted after revision +++++ Presented by We study a new formulation for the eikonal equation ∇u  = 1 on a bounded subset of R 2. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1dimensional subspaces, with div P ∈ L 2. We prove that solutions of ..."
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Received *****; accepted after revision +++++ Presented by We study a new formulation for the eikonal equation ∇u  = 1 on a bounded subset of R 2. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1dimensional subspaces, with div P ∈ L 2. We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighbourhood of a regular closed curve. Résumé Nous étudions une nouvelle formulation de l’équation eikonale ∇u  = 1 sur un sousensemble borné de R 2.Au lieu d’un champ de vecteurs ∇u, nous considérons un champ P de projections orthogonales sur les sousespaces de dimension 1, avec div P ∈ L 2. Nous montrons que les solutions de cette équation propagent la direction comme dans l’équation eikonale classique. Nous montrons aussi que les solutions existent si et seulement si le domaine est un voisinage tubulaire d’une courbe régulière fermée. 1. Stripe patterns and the eikonal equation Many patternforming systems produce parallel stripes, both straight and curved. In this note we report on a new mathematical description of curved striped patterns. We recently studied the behaviour of a stripeforming energy, and investigated a limit process in which the stripe width tends to zero [4]. In that limit the stripes not only become thin, but also uniform in width, and the stripe pattern comes to resemble the level sets of a solution of the eikonal equation. The rigorous version of this statement, in the