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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 sharp-interface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called(More)
We present the second of two articles on the small volume-fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional [SIAM we consider here the full diffuse-interface functional and address the limit in which ε(More)
We address the branching of magnetic domains in a uniaxial ferromagnet. Our thesis is that branching is required for a domain pattern to have nearly{minimal energy. To show this, we consider the nonlocal, nonconvex variational problem of micromagnetics. We identify the scaling law of the minimum energy, by proving a rigorous lower bound which matches the(More)
The intermediate state of a type-I superconductor is a classical example of energy-driven pattern-formation, first studied by Landau in 1937. Three of us recently derived five different rigorous upper bounds for the ground state energy, corresponding to different microstructural patterns, but only one of them was complemented by a lower bound This paper(More)
The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving(More)
This paper explores the evolution of a sharp interface model for phase separation of copolymers in the limit of low volume fraction. Particles both exchange material as in usual Ostwald ripening, and migrate because of an effectively repulsive nonlocal energetic term. Coarsening via mass diffusion only occurs while particle radii are small, and they(More)
We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the L 1 distance to the signal, both with and without the presence of a deconvolution operator. Based(More)
We consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers. Our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal Cahn–Hilliard functional. Based upon two parameters which characterize the phase diagram, we give a preliminary analysis of the(More)