836 research outputs found
The Allen-Cahn Action functional in higher dimensions
The Allen-Cahn action functional is related to the probability of rare events
in the stochastically perturbed Allen-Cahn equation. Formal calculations
suggest a reduced action functional in the sharp interface limit. We prove in
two and three space dimensions the corresponding lower bound. One difficulty is
that diffuse interfaces may collapse in the limit. We therefore consider the
limit of diffuse surface area measures and introduce a generalized velocity and
generalized reduced action functional in a class of evolving measures. As a
corollary we obtain the Gamma convergence of the action functional in a class
of regularly evolving hypersurfaces.Comment: 33 pages, 4 figures; minor changes and addition
Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids
We introduce a new sharp interface model for the flow of two immiscible,
viscous, incompressible fluids. In contrast to classical models for two-phase
flows we prescribe an evolution law for the interfaces that takes diffusional
effects into account. This leads to a coupled system of Navier-Stokes and
Mullins-Sekerka type parts that coincides with the asymptotic limit of a
diffuse interface model. We prove the long-time existence of weak solutions,
which is an open problem for the classical two-phase model. We show that the
phase interfaces have in almost all points a generalized mean curvature.Comment: 26 page
Gamma convergence of a family of surface--director bending energies with small tilt
We prove a Gamma-convergence result for a family of bending energies defined
on smooth surfaces in equipped with a director field. The
energies strongly penalize the deviation of the director from the surface unit
normal and control the derivatives of the director. Such type of energies for
example arise in a model for bilayer membranes introduced by Peletier and
R\"oger [Arch. Ration. Mech. Anal. 193 (2009)]. Here we prove in three space
dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to
a specific curvature energy. In order to obtain appropriate compactness and
lower semi-continuity properties we use tools from geometric measure theory, in
particular the concept of generalized Gauss graphs and curvature varifolds.Comment: 29 page
Control of the isoperimetric deficit by the Willmore deficit
In the class of smoothly embedded surfaces of sphere type we prove that the
isoperimetric deficit can be controlled by the Willmore deficit
Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks
Signaling molecules play an important role for many cellular functions. We
investigate here a general system of two membrane reaction-diffusion equations
coupled to a diffusion equation inside the cell by a Robin-type boundary
condition and a flux term in the membrane equations. A specific model of this
form was recently proposed by the authors for the GTPase cycle in cells. We
investigate here a putative role of diffusive instabilities in cell
polarization. By a linearized stability analysis we identify two different
mechanisms. The first resembles a classical Turing instability for the membrane
subsystem and requires (unrealistically) large differences in the lateral
diffusion of activator and substrate. The second possibility on the other hand
is induced by the difference in cytosolic and lateral diffusion and appears
much more realistic. We complement our theoretical analysis by numerical
simulations that confirm the new stability mechanism and allow to investigate
the evolution beyond the regime where the linearization applies.Comment: 21 pages, 6 figure
Confined structures of least bending energy
In this paper we study a constrained minimization problem for the Willmore
functional. For prescribed surface area we consider smooth embeddings of the
sphere into the unit ball. We evaluate the dependence of the the minimal
Willmore energy of such surfaces on the prescribed surface area and prove
corresponding upper and lower bounds. Interesting features arise when the
prescribed surface area just exceeds the surface area of the unit sphere. We
show that (almost) minimizing surfaces cannot be a -small perturbation of
the sphere. Indeed they have to be nonconvex and there is a sharp increase in
Willmore energy with a square root rate with respect to the increase in surface
area.Comment: 27 pages, 3 figure
Convergence of phase-field approximations to the Gibbs-Thomson law
We prove the convergence of phase-field approximations of the Gibbs-Thomson
law. This establishes a relation between the first variation of the
Van-der-Waals-Cahn-Hilliard energy and the first variation of the area
functional. We allow for folding of diffuse interfaces in the limit and the
occurrence of higher-multiplicities of the limit energy measures. We show that
the multiplicity does not affect the Gibbs-Thomson law and that the mean
curvature vanishes where diffuse interfaces have collided.
We apply our results to prove the convergence of stationary points of the
Cahn-Hilliard equation to constant mean curvature surfaces and the convergence
of stationary points of an energy functional that was proposed by Ohta-Kawasaki
as a model for micro-phase separation in block-copolymers.Comment: 25 page
Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow
This paper is concerned with diffuse-interface approximations of the Willmore
flow. We first present numerical results of standard diffuse-interface models
for colliding one dimensional interfaces. In such a scenario evolutions towards
interfaces with corners can occur that do not necessarily describe the adequate
sharp-interface dynamics.
We therefore propose and investigate alternative diffuse-interface
approximations that lead to a different and more regular behavior if interfaces
collide. These dynamics are derived from approximate energies that converge to
the -lower-semicontinuous envelope of the Willmore energy, which is in
general not true for the more standard Willmore approximation
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