KTH-Uppsala PDE days

March 5 at KTH and May 14 at Uppsala University

Supported by the Swedish Research Council


May 14, room 64119, Ångström, Uppsala

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}.


March 5, F11, KTH

10.15-11.00: Lorenzo Brasco, Ferrara: The Faber-Krahn inequality: old & new

Abstract: The celebrated Faber-Krahn inequality asserts that, among planar domains with given area, disks (uniquely) minimize the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary conditions. We review this classical result and discuss some of its applications. We also discuss the question of the stability of such an inequality. Then we present some recent results, by generalizing the discussion to the case of the fractional Laplacian. Some of the results presented are contained in a paper with Eleonora Cinti (Bologna) and Stefano Vita (Bologna).

11.15-12.00: Andreas Minne, KTH: Obstacle-Type Problems in the Subelliptic Setting

Abstract: The obstacle problem is one of the most famous problems in the field of Free Boundary Problems. In this talk we briefly present some classical results of the optimal regularity of solutions, and then show our recent theorem; namely that the optimal regularity can be achieved also in the subelliptic setting for related problems, but without any sign assumptions. Key ideas are the use of BMO-estimates, and a decay estimate for the so-called coincidence set. These results have been obtained in collaboration with Valentino Magnani (Universita di Pisa).

12-: Lunch

Title: Abstract:

Organizers: John Andersson, Erik Lindgren, Kaj Nyström and Henrik Shahgholian