KTH-Uppsala PDE days

  • Date:
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
  • Lecturer: Henrik Shahgholian (KTH), Tuomo Kuusi (Alto University)
  • Contact person: Kaj Nyström
  • Seminarium

10.15-11.00: Henrik Shahgholian, KTH: From fluid flow in cones to boundary Harnack for PDEs with RHS

Abstract: A simple home-made experiment shows interesting behaviour of fluid flow on a table close to corners of the table. The experiment reveals a new Boundary Harnack Principle for PDEs, with right hand sides. (Based on a recent work with Mark Allen.)

11.15-12.00: Tuomo Kuusi, Alto University: Homogenization, linearization and large-scale regularity for nonlinear elliptic equations

Abstract: I will consider nonlinear, uniformly elliptic equations with variational structure and random, highly oscillating coefficients satisfying a finite range of dependence, and discuss the corresponding homogenization theory. I will recall basic ideas how to get quantitative rates of homogenization for nonlinear uniformly convex problems. After this I will discuss our recent work proving that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). These results lead to a better understanding of differences of solutions to the nonlinear equation. As a consequence, we obtain a large-scale C^{0,1}-type estimate for differences of solutions and improve the regularity of the homogenized Lagrangian by showing that it has the same regularity as the original heterogeneous Lagrangian, up to C^{2,1}.