Phasenübergänge in Kristallen

我们考虑了自由能密度依赖于化学物质局部浓度的晶体固体的弹性，根据这些物质的浓度，晶体的不同几何构型(相)是优选的。化学物质的局部浓度通过扩散而改变。扩散和弹性松弛的时间尺度是不同的。在模型中，我们考虑了分离不同相区域的超曲面的表面能。我们计划通过研究离散模型的连续极限来加深对表面能和弹性能相互作用的理解。为了达到这些目的，我们计划采用和扩展来自分析(隐式时间离散化、弱收敛、G-极限、有界变差函数)和几何测度论的方法(根据Brakke/Soner的变折的几何测度论革命)。

We consider the elasticity of cristalline solids whose free energy density depends on the local concentration of chemical substances in such a way that according to the concentration of these substances different geometrical configurations (phases) of the crystal are preferred. The local concentration of the chemical substances is changed by diffusion. The time scales for the diffusion and the elastic relaxation are different. In the models we consider the surface energy of the hypersurfaces separating regions of different phase is taken into account. We plan to enhance the understanding of the interplay of surface energy and elastic energy by investigating the continuum limit of discrete models. For these aims, we plan to adapt and extent methods from analysis (implicit time discretization, weak convergence, G-Limit, functions of bounded variation) and from geometric measure theory Revolution of varifolds according to Brakke/Soner).