Project acronym QSHvar
Project Quantitative stochastic homogenization of variational problems
Researcher (PI) Tuomo Kuusi
Host Institution (HI) HELSINGIN YLIOPISTO
Call Details Consolidator Grant (CoG), PE1, ERC-2018-COG
Summary 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.
Summary
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.
Max ERC Funding
1 312 500 €
Duration
Start date: 2019-08-01, End date: 2024-07-31
Project acronym QUAMAP
Project Quasiconformal Methods in Analysis and Applications
Researcher (PI) Kari ASTALA
Host Institution (HI) AALTO KORKEAKOULUSAATIO SR
Call Details Advanced Grant (AdG), PE1, ERC-2018-ADG
Summary The use of delicate quasiconformal methods, in conjunction with convex integration and/or nonlinear Fourier analysis, will be the common theme of the proposal. A number of important outstanding problems are susceptible to attack via these methods. First and foremost, Morrey's fundamental question in two dimensional vectorial calculus of variations will be considered as well as the related conjecture of Iwaniec regarding the sharp $L^p$ bounds for the Beurling transform. Understanding the geometry of conformally invariant random structures will be one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calder\'on's inverse conductivity problem will also be considered, as well as the applications to imaging. Further goals are to be found in fluid mechanics and scattering, as well as the fundamental properties of quasiconformal mappings, interesting in their own right, such as the outstanding deformation problem for chord-arc curves.
Summary
The use of delicate quasiconformal methods, in conjunction with convex integration and/or nonlinear Fourier analysis, will be the common theme of the proposal. A number of important outstanding problems are susceptible to attack via these methods. First and foremost, Morrey's fundamental question in two dimensional vectorial calculus of variations will be considered as well as the related conjecture of Iwaniec regarding the sharp $L^p$ bounds for the Beurling transform. Understanding the geometry of conformally invariant random structures will be one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calder\'on's inverse conductivity problem will also be considered, as well as the applications to imaging. Further goals are to be found in fluid mechanics and scattering, as well as the fundamental properties of quasiconformal mappings, interesting in their own right, such as the outstanding deformation problem for chord-arc curves.
Max ERC Funding
2 280 350 €
Duration
Start date: 2019-09-01, End date: 2024-08-31
