Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Anja Schlömerkemper
Universität Würzburg
Uniformly Γ-equivalent theories for nonconvex discrete systems
Within the context of the multiscale analysis I will focus on continuum limits of nonconvex discrete systems.
In particular I will present results for a one-dimensional chain of atoms which interact through nearest and
next-to-nearest neighbour interactions being nonconvex. The aim is to find a good approximation of the
energy functional of this system for a large number of atoms. The method is based on Γ-convergence and
moreover on a concept introduced by A. Braides and L. Truskinovsky called uniformly Γ-equivalent theory,
which I will recall. Then I will show how this can be applied to the above mentioned system.
Literatur
Braides, A., Truskinovsky, L. (2008). Asymptotic expansions by Γ-convergence. Continuum Mech. Thermodyn.,
20, 21–62.
Scardia, L., Schlömerkemper, A., Zanini, C (2011). Boundary Layer Energies for nonconvex discrete systems.
DOI: 10.1142/S0218202511005210. Forthcoming article in Math. Models Methods Appl. Sci.
Bernd Schmidt
Universität Augsburg
On discrete-to-continuum limits for brittle fracture
We study a two-dimensional discrete system of atoms which allows for brittle fracture in the continuum
limit. In particular, under suitable conditions it is shown that cracks occur along a special cleavage line.
As an application, we can e.g. rigorously justify the reduction to a one-dimensional chain model for the
investigation of brittle materials. (Joint work with M. Friedrich).
Matthias Schneider
Ruprecht Karls-Universität Heidelberg
Closed magnetic geodesics
We give existence results for closed curves with prescribed geodesic curvature in Riemannian surfaces.
Moreover, we will discuss higher dimensional versions, e.g. existence of closed surfaces with prescribed
curvature in three dimensional Riemannian manifolds.
Literatur
Rosenberg, H. and Schneider, M. (2011) Embedded constant curvature curves on convex surfaces. ar-
Xiv:1105.1609.
Rosenberg, H. and Smith, G. (2010) Degree Theory of Immersed Hypersurfaces. arXiv:1010.1879v1.
Schneider, M. (2010) Closed magnetic geodesics on closed hyperbolic Riemann surfaces. ar-
Xiv:1009.1723.
Schneider, M. (2011) Closed magnetic geodesics on S 2 . J. Differential Geometry 87 343-388.
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