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Matthias Röger

Publications and source records attributed to Matthias Röger.

34 records · Page 2Linked to original sources
arXiv2015

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 $\mathbb{R}^3$ 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 Röger [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.

math.AP↗
arXiv2014

Variational analysis of a mesoscale model for bilayer membranes

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third author (Arch. Ration. Mech. Anal. 193, 2009). The energy is both non-local and non-convex. It combines a surface area and a Monge-Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. Here we extend key results of our previous analysis to the three dimensional case. First we prove a general lower estimate and formally identify a curvature energy in the zero-thickness limit. Secondly we construct a recovery sequence and prove a matching upper-bound estimate.

math.AP↗
arXiv2013

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 $C^2$-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.

math.AP↗
arXiv2013

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.

math.AP↗
arXiv2013

A phase field model for the optimization of the Willmore energy in the class of connected surfaces

We consider the problem of minimizing the Willmore energy connected surfaces with prescribed surface area which are confined to a finite container. To this end, we approximate the surface by a phase field function $u$ taking values close to +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. A diffuse interface approximation for the area functional, as well as for the Willmore energy are well known. We address the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit.

math.AP↗
arXiv2013

Variational analysis of a mean curvature flow action functional

We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of generalized hypersurfaces this functional consists of the sum of the squares of the mean curvature and of the velocity vector, integrated over time and space. For given initial and final conditions we investigate the corresponding action minimization problem. We give a generalized formulation and prove compactness and lower semicontinuity properties of the action functional. Furthermore we characterize the Euler-Lagrange equation for smooth stationary points and investigate conserved quantities. Finally we present an explicit example and consider concentric spheres as initial and final data and characterize in dependence of the given time span the properties of the minimal rotationally symmetric connection.

math.AP↗
arXiv2013

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 $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

math.AP↗
arXiv2012

Tightness for a stochastic Allen--Cahn equation

We study an Allen-Cahn equation perturbed by a multiplicative stochastic noise which is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive uniform energy bounds and prove tightness of of solutions in the sharp interface limit, and show convergence to phase-indicator functions.

math.AP↗
arXiv2011

Turing instabilities in a mathematical model for signaling networks

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction-diffusion system in the inner volume to a reaction-diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction-diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.

math.AP↗
arXiv2010

Confined elastic curves

We consider the problem of minimizing Euler's elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values +1 on the inside and -1 on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known. Implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces.

math.AP↗
arXiv2009

Convergence of perturbed Allen-Cahn equations to forced mean curvature flow

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally we discuss some applications.

math.AP↗
arXiv2008

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.

math.AP↗
arXiv2007

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.

math.AP↗
arXiv2007

Travelling-wave analysis of a model describing tissue degradation by bacteria

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the `degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

math.AP↗
arXiv2007

Sobolev regularity and an enhanced Jensen inequality

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ 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: $u\in W^{1,2}$ 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 Euler-Lagrange equation.

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