39 research outputs found
Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp Interface Functional
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 particles) remains O(1). Using the
language of Gamma-convergence, we focus on two levels of this convergence, and
derive first and second order 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
Coulomb-like interaction between the particles, which is responsible for the
pattern formation. We present the results here in both three and two
dimensions.Comment: 37 pages, 1 figur
Small Volume Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional
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 (arXiv:0907.2224), we consider here
the full diffuse-interface functional and address the limit in which epsilon
and the volume fraction tend to zero but the number of minority phases (called
particles) remains O(1). Using the language of Gamma-convergence, we focus on
two levels of this convergence, and derive first- and second-order 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 size of each particle but no information about their spatial
distribution. At the next level we encounter a Coulomb-like interaction between
the particles, which is responsible for the pattern formation. We present the
results in three dimensions and comment on their two-dimensional analogues
Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere
On the two dimensional sphere, we consider axisymmetric critical points of an
isoperimetric problem perturbed by a long-range interaction term. When the
parameter controlling the nonlocal term is sufficiently large, we prove the
existence of a local minimizer with arbitrary many interfaces in the
axisymmetric class of admissible functions. These local minimizers in this
restricted class are shown to be critical points in the broader sense (i.e.,
with respect to all perturbations). We then explore the rigidity, due to
curvature effects, in the criticality condition via several quantitative
results regarding the axisymmetric critical points.Comment: 26 pages, 6 figures. This version is to appear in ESAIM: Control,
Optimisation and Calculus of Variation
Global minimizers for axisymmetric multiphase membranes
We consider a Canham-Helfrich-type variational problem defined over closed
surfaces enclosing a fixed volume and having fixed surface area. The problem
models the shape of multiphase biomembranes. It consists of minimizing the sum
of the Canham-Helfrich energy, in which the bending rigidities and spontaneous
curvatures are now phase-dependent, and a line tension penalization for the
phase interfaces. By restricting attention to axisymmetric surfaces and phase
distributions, we extend our previous results for a single phase
(arXiv:1202.1979) and prove existence of a global minimizer.Comment: 20 pages, 3 figure
Anisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes
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 upon the existence of a
certain associated vector field, we find necessary and sufficient conditions
for a function to be a minimizer. We apply these results to 2D bar codes to
find explicit regimes ---in terms of the fidelity parameter and smallest length
scale of the bar codes--- for which a perfect bar code is recoverable via
minimization of the functionals. Via a discretization reformulated as a linear
program, we perform numerical experiments for all functionals demonstrating
their denoising and deblurring capabilities.Comment: 34 pages, 6 figures (with a total of 30 subfigures); errors corrected
in Version 3, see Errata 1.1, 4.4, and 6.6 (v3 numbering) for more
informatio