# AG Biomathematik

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Item New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow(2023) Knüttel, Sascha; Röger, Matthias; Schweizer, BenShow more In this thesis we derive a higher order diffuse approximation of the Willmore energy from contributions by Karali and Katsoulakis [KK07], who studied a diffuse approximation of mean curvature flow. We prove Γ–convergence in smooth limit points for the sum of diffuse perimeter and the higher order diffuse Willmore energy in dimensions 2 and 3. Moreover, we prove the convergence on arbitrary time intervals towards weak solutions of mean curvature flow. We also consider a gradient-free diffuse approximation of the Willmore energy in the sense of Γ–convergence which we derive from a gradient-free diffuse approximation of the perimeter by Amstutz and van Goethem [AVG12]. We prove the lim sup–property for the Γ–convergence towards a multiple of the Willmore energy. In addition, we consider L2-type gradient flows of both diffuse Willmore energies, and give an asymptotic convergence result. Formally these constitute diffuse approximations of mean curvature flow and Willmore flow. In a restricted class of diffuse phase-field evolutions, we prove that these gradient flows convergence towards rescaled mean curvature flow and rescaled Willmore flow, respectively. References [AVG12] S. Amstutz and N. Van Goethem. Topology optimization methods with gradient-free perimeter approximation. Interfaces Free Bound., 14(3):401–430, 2012. [KK07] G. Karali and M. A. Katsoulakis. The role of multiple microscopic mechanisms in cluster interface evolution. J. Differential Equations, 235(2):418–438, 2007.Show more Item Relaxation analysis in a data driven problem with a single outlier(2020-06-24) Röger, Matthias; Schweizer, BenShow more We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.Show more Item Stochastic mean curvature flow(2020) Dabrock, Nils; Röger, Matthias; Hofmanova, MartinaShow more In this thesis we study a stochastically perturbed mean curvature flow (SMCF). In case of graphs, existence of weak solutions has already been established for one-dimensional and two-dimensional periodic surfaces with a spatially homogeneous perturbation. We extend this result by proving existence of solutions in arbitrary dimensions perturbed by noise which is white in time and colored in space. In addition, we work with a stronger notion of solution which corresponds to strong solutions in the PDE sense. For this, we give a new interpretation of graphical SMCF as a degenerate variational stochastic partial differential equation (SPDE) with compact embedding. In order to infer existence of an approximating sequence, we extend the theory of variational SPDEs such that we can treat SMCF within this framework. In order to pass to the limit with the approximating sequence, we prove new a-priori bounds for graphical SMCF. With this a-priori bounds, we can characterize the large-time behavior of solutions in case of spatially homogeneous noise. In particular, we will prove that solutions become asymptotically constant in space and behave like the driving noise in time. This strengthens a previously established one-dimensional large-time result by extending it to higher dimensions and proving stronger convergence. Furthermore, we propose a numerical scheme for graphical SMCF which employs the variational interpretation we have analyzed before. Using this scheme, we present Monte-Carlo simulations visualizing the energy estimates we have used in the analytic part of this thesis. Moreover, we discuss the regularity and uniqueness of solutions and give conditional results for both. We complement the previous results, especially the existence result, by investigating how they extend to SMCF with respect to an anisotropic notion of curvature.Show more Item The diffuse interface approximation of the Willmore functional in configurations with interacting phase boundaries(2018) Zwilling, Carsten; Röger, Matthias; Schweizer, BenShow more In this thesis we study a diffuse interface approximation of the sum of the area and Willmore functional for which Gamma-convergence has already been established in the case of small space dimensions and smoothly bounded sets. We extend this result to a larger class of configurations with nonsmooth phase boundaries and explicitly allow intersecting boundary curves. We also analyze the interaction of parallel planar phase phields and discuss their slow motion under the L2-gradient flow of the diffuse Willmore functional. Moreover, we prove the existence of a new class of periodic entire solutions to the stationary Allen-Cahn equation in two dimensions.Show more Item Mathematical analysis of a spatially coupled reaction-diffusion system for signaling networks in biological cells(2016) Hausberg, Stephan Erik; Röger, Matthias; Schweizer, BenShow more The main purpose of this thesis was the mathematical analysis of certain spatially coupled Reaction-Diffusion Systems arising in the description of Signaling Networks in biological cells. The spatial coupling between a diffusion in the cytosolic bulk and reaction and diffusion processes on the boundary surface is given by a Robin-type boundary condition that introduces a source term in the boundary equations. This thesis provides existence and well-posedness results for prototypes of such models. The main results were the following: Considering regular data and classical diffusion operators (in case of the surface equations expressed by the Laplace-Beltrami operator) we find existence of classical solutions and well-posedness based on an operator splitting approach that decouples bulk and surface equations. Complementary to the classical setting we have also considered possibly nonsmooth diffusion operators that for example can model heterogeneous domains with specific properties on the membrane. The existence and well-posedness result was based on an implicit discretization in time which reduces the given nonlinear parabolic system to a sequence of nonlinear recursive elliptic problems. This was then solved by an application of the theory of monotone perturbed operators. Furthermore, the results were applied to rigorously justify an asymptotic model reduction to a nonlocal two-variable system on the membrane.Show more