Approximation of the Willmore energy by a discrete geometry model

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $\Gamma$-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.Comment: 29 pages, 9 figures. v2: Minor revisions, references adde

On a stochastic particle model of the Keller-Segel equation and its macroscopic limit

The aim of this thesis is to derive the two-dimensional Keller-Segel equation for chemo- taxis from a stochastic system of N interacting particles in the situation in which bounded solutions are guaranteed to exist globally in time, that is in the case of subcritical chemo- sensitivityZiel dieser Arbeit ist die Herleitung der zwei-dimensionalen Keller-Segel Gleichung für Chemotaxis aus einem wechselwirkenden, stochastischen N-Teilchen System, wenn die Existenz von beschränkten, für alle Zeiten definierten Lösungen vorgegeben ist. Dies entspricht dem unterkritischen Fal

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Gamma-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies