Periodic Reporting for period 4 - QSHvar (Quantitative stochastic homogenization of variational problems)
Reporting period: 2024-02-01 to 2025-01-31
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 limited 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 the calculus of variations. The first one is to prove the 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 the 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 projects. Especially the successful modeling in the last part would possibly lead to a better understanding of the potential of geothermal energy.
Many of the actual goals of the proposal come from problems in the calculus of variations. The first one is to prove the 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 the 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 projects. Especially the successful modeling in the last part would possibly lead to a better understanding of the potential of geothermal energy.
So far, the main results stem from smooth convex variational problems. It is well-known, by the seminal works of De Giorgi and Nash, together with Schauder estimates, that minimizers are smooth. This is often called the Hilbert 19th problem. A conjecture, which we solved in two long papers, was that in the context of homogenization, one could prove the corresponding theory in the case of homogenization. In essence, this required higher-order regularity theory in the nonlinear setting. Moreover, we have obtained some of the first quantitative results in the context of homogenization and free boundary problems. In particular, we have shown, again in the context of homogenization, that the free boundaries of the obstacle problem have large-scale regularity in a quantified way. These types of problems have been open for a long time.
Finally, the most significant results in the project were obtained in the context of high-contrast homogenization. A quantitative theory has been open for a long time, and there were many conjecture by physicists. For example, before our work, superdiffusive scaling limits in random environments were understood only in an averaged (annealed) sense, and precise variance asymptotics with universal prefactors were conjectural. We have proved, in a quenched setting, that the variance grows as conjectured by physicists with an explicit constant. We further established a quenched invariance principle under the same scaling. Achieving such pathwise control was widely thought to be out of reach because incompressible drifts generate long-range, dynamically evolving correlations. Moreover, we introduced so-called coarse-grained ellipticity, a new observable that measures the effective ellipticity ratio of the operator after integrating out fluctuations below a given scale. By analyzing scale-by-scale behavior for this quantity, obtained through a direct analytic argument rather than the combinatorial or perturbative techniques typical in physics, we produced the first fully rigorous renormalization group framework.
Finally, the most significant results in the project were obtained in the context of high-contrast homogenization. A quantitative theory has been open for a long time, and there were many conjecture by physicists. For example, before our work, superdiffusive scaling limits in random environments were understood only in an averaged (annealed) sense, and precise variance asymptotics with universal prefactors were conjectural. We have proved, in a quenched setting, that the variance grows as conjectured by physicists with an explicit constant. We further established a quenched invariance principle under the same scaling. Achieving such pathwise control was widely thought to be out of reach because incompressible drifts generate long-range, dynamically evolving correlations. Moreover, we introduced so-called coarse-grained ellipticity, a new observable that measures the effective ellipticity ratio of the operator after integrating out fluctuations below a given scale. By analyzing scale-by-scale behavior for this quantity, obtained through a direct analytic argument rather than the combinatorial or perturbative techniques typical in physics, we produced the first fully rigorous renormalization group framework.
The most striking results during the project were obtained from the topic stemming from geothermal energy modeling. Actually, we pushed the theory well beyond this goal. The fluid flow in the rock can be modeled by the so-called Darcy's law, stating that the flow velocity is parallel to the change of pressure multiplied by a material parameter. The material parameter can be assumed to have a certain law, in accordance with measurements from several sites globally. This suggests a stochastic partial differential equation with extreme oscillations in the material parameter. It has been quite challenging to find the right concept to analyze the corresponding equation. In a series of new works, discussed above, we established a new theoretical framework to tackle such problems. These achievements supply new conceptual language (coarse-grained ellipticity) and deliver new techniques, which open intriguing new possibilities to analyze a wide range of physically relevant models.