Results 1  10
of
10
Partial localization, lipid bilayers, and the elastica functional. in prep
, 2006
"... Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that thi ..."
Abstract

Cited by 17 (11 self)
 Add to MetaCart
(Show Context)
Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of twodimensional spatial mass densities, exhibits partial localization and displays the ‘solidlike ’ behavior of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, i.e. resemble thin structures. Deviation from a specific uniform thickness, creation of ‘ends’, and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gammaconvergence result. We prove that a rescaled version of the energy functional converges in the zerothickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and nonfracture; if these conditions are met, then the limit value is given by the classical Elastica (bending) energy of the curve.
Stability of monolayers and bilayers in a copolymerhomopolymer blend model
, 2009
"... We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface OhtaKawasaki energy for straight mono and bilay ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
(Show Context)
We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface OhtaKawasaki energy for straight mono and bilayers and determining the latter’s sign. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Both for the monolayer and for the bilayer there exist parameter regions where these structures are unstable. For strong repulsive interaction between the monomer types in the diblock copolymer the bilayer is always stable with respect to interface perturbations, irrespective of the domain size. The monolayer is only stable for small domain size. In the course of our computations we also give a Green’s function for the Laplacian on a twodimensional periodic strip.
Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality
"... Abstract. We derive a new criterion for a realvalued function u to be in the Sobolev space W 1,2 (Rn). This criterion consists of comparing the value of a functional ∫ f (u) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values con ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We derive a new criterion for a realvalued function u to be in the Sobolev space W 1,2 (Rn). This criterion consists of comparing the value of a functional ∫ f (u) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values converges to zero as the convolutions approach u, and we prove that the rate of convergence to zero is connected to regularity: u ∈ W 1,2 if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the EulerLagrange equation. Mathematics Subject Classification (2000): missing. 1.
Hydrophobic Effect in a Continuum Model of the
, 2009
"... Abstract. We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules. This paper extends previous work of Blom and Peletier [European J. Appl. Math., 15 (2004), pp. 487508] in the following ways. (a) It formulates a more general ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules. This paper extends previous work of Blom and Peletier [European J. Appl. Math., 15 (2004), pp. 487508] in the following ways. (a) It formulates a more general model of the hydrophobic effect to facilitate connections with microscale simulations and firstprinciples analysis. (b) It clarifies the meaning and role of the model parameters. (c) It outlines a method for determining parameter values so that physicallyrealistic bilayer density profiles can be obtained, for example for use in macroscale simulations. Points (a)(c) suggest that the model has potential to robustly connect some micro and macroscale levels of multiscale blood flow simulations. The mathematical modelling in point (a) is based upon a consideration of the underlying physics of intermolecular forces. The governing equations thus obtained are minimized by gradient flows via a novel numerical approach; this enables point (b). The numerical results are shown to behave physically in terms of the effect of background concentration, in contrast to the earlier model which is shown here to not display the expected behaviour. A “shorttail ” approximation of
unknown title
, 2008
"... Hydrophobic effect in a continuum model of the lipid bilayer ..."
(Show Context)
Arch. Rational Mech. Anal. (2006) Digital Object Identifier (DOI) 10.1007/s002050060011y SelfContact for Rods on Cylinders
"... We study selfcontact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a secondorder functional of a single scalar variable, and the ..."
Abstract
 Add to MetaCart
(Show Context)
We study selfcontact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a secondorder functional of a single scalar variable, and the selfcontact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cutandpaste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of selfcontact points is continuous, a result that contrasts with known examples from contact problems in free rods. 1.
Stability of monolayers and bilayers in a copolymerhomopolymer blend model
, 2007
"... We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface Ohta–Kawasaki energy for straight mono and bilay ..."
Abstract
 Add to MetaCart
(Show Context)
We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface Ohta–Kawasaki energy for straight mono and bilayers and determining the sign of the latter. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Both for the monolayer and for the bilayer there exist parameter regions where these structures are unstable. For strong repulsive interaction between the monomer types in the diblock copolymer the bilayer is always stable with respect to interface perturbations, irrespective of the domain size. The monolayer is only stable for small domain size. In the course of our computations we also give a Green’s function for the Laplacian on a twodimensional periodic strip.
Stability of monolayers and bilayers in a
"... copolymerhomopolymer blend model ..."
(Show Context)
SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY
, 2007
"... Abstract. We derive a new criterion for a realvalued function u to be in the Sobolev space W 1,2 (R n). This criterion consists of comparing the value of a functional R f(u) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values con ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We derive a new criterion for a realvalued function u to be in the Sobolev space W 1,2 (R n). This criterion consists of comparing the value of a functional R f(u) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values converges to zero as the convolutions approach u, and we prove that the rate of convergence to zero is connected to regularity: u ∈ W 1,2 if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the EulerLagrange equation. 1.