New results in stochastic homogenization of variational models

• Applicant: Dr. Matthias Ruf
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530813503

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.

• DFG Programme: Research Grants