Quantitative stochastic homogenization of variational problems

ended

The proposal addresses various multiscale problems which lie at the intersection of probability theory and the analysis of partial differential equations and calculus of variations. Most of the proposed problems fit under the framework of stochastic homogenization, that is, the study of large-scale statistical properties of solutions to equations with random coefficients. In the last ten years, there has been significant progress made in developing a quantitative theory of stochastic homogenization, meaning that one can now go beyond limit theorems and prove rates of convergence and error estimates, and in some cases even characterize the fluctuations of the error. These new quantitative methods give us new tools to attack more difficult multi-scale problems that have until now resisted previous approaches, and consequently to solve open problems in the field. Many of the actual goals of the proposal come from problems in calculus of variations. Apart from qualitative results, many fundamental questions in quantitative theory are completely open, and our recent results suggest a way to tackle these problems. The first one is to prove regularity properties of homogenized Lagrangian under rather general assumptions on functionals, and to solve a counterpart for Hilbert's 19th problem in the context of homogenization. The second project is to attack so-called Faber-Krahn inequality in the heterogeneous case. This is a very involved problem, but again recent development in the theory of homogenization makes the attempt plausible. The final part of the proposal involves new mathematical approaches and subsequent computational research supporting the geothermal power plant project being built by St1 Deep Heat Ltd in Espoo, Finland.