New results in stochastic homogenization of variational models

这个项目的目的是在变分中所选泛函的几乎必然伽玛收敛的意义上的随机齐化。在这样做的时候，要么现有结果中的假设被显著削弱，要么我们研究的模型到目前为止还没有在随机齐化意义上考虑。这项工作可以分为四个独立的子项目：1)具有简并的平稳相互作用系数的铁磁Ising系统，2)随机晶格上的铁磁XY模型，3)流形上的随机齐次模型，4)(静态)非线性弹性的随机齐次化。在1)中，改进了已有的结果，即两个磁性粒子之间的相互作用权重不再需要从上面有界，而只需要满足最小随机可积性假设。在方案2)中，研究了随机排列的自旋粒子的类涡旋拓扑奇性。对于给定的XY模型，在齐化(既不是周期的，也不是平稳的，遍历的)背景下，还没有关于Gamma收敛的结果。项目3)致力于定义流形上适当的平稳性概念，它允许考虑聚合物网络的相关模型及其在曲面上的齐次化。在最后一个方案4)中，我们试图证明非线性弹性理论中能量随机齐化的第一个部分结果。特别地，我们证明了来自标准齐化理论的多胞公式提供了Gamma极限的一个上界。每个项目都有许多可能性，可以推广到关于积分电流的能量泛函的非常抽象的考虑，或者到导致非线性弹性的完全伽马收敛。

The aim of the project is the stochastic homogenization in the sense of almost sure gamma convergence of selected functionals in the calculus of variations. In doing so, either assumptions in existing results are significantly weakened or we investigate models for which there have been no considerations in the sense of stochastic homogenization so far. The work can be divided into four independent subprojects: 1) Ferromagnetic Ising systems with degenerate stationary interaction coefficients, 2) The ferromagnetic XY model on stochastic lattices, 3) Models with stochastic homogeneity on manifolds, 4) Stochastic homogenization in (static) nonlinear elasticity. In 1), existing results are improved in the sense that the interaction weights between two magnetic particles no longer need to be bounded from above, but only need to satisfy minimal stochastic integrability assumptions. In project 2), vortex-like topological singularities for randomly arranged spin particles are studied. For the given XY-model, no result on gamma-convergence in the context of homogenization (neither periodic nor stationary, ergodic) exists yet. Project 3) is devoted to the definition of a suitable stationarity notion on manifolds, which allows to consider relevant models for polymer networks and their homogenization also on curved surfaces. In the last project 4) we try to prove first partial results for the stochastic homogenization of energies in nonlinear elasticity theory. In particular, we show that the multi-cell formula from standard homogenization theory provides an upper bound for the Gamma-limit. Each of the projects has numerous possibilities for generalization up to very abstract considerations of energy functionals on integral currents or to a full gamma convergence result in nonlinear elasticity.