This talk is about sets and mappings that appear in geometric measure theory on a non-Euclidean metric space, the first Heisenberg group. It is known that this group endowed with a left invariant sub-Riemannian distance does not contain any subset of positive k-dimensional Hausdorff measure which is the Lipschitz image of a set in R^k for k>1. Classically, k-rectifiable sets are defined as sets which can be essentially covered by a countable union of Lipschitz images of sets in R^k, but that does not yield anything interesting in the Heisenberg group for k>1. Instead, research has shown that so-called 'intrinsic' Lipschitz graphs (in the sense of B. Franchi, R. Serapioni and F. Serra Cassano) are the correct class of sets to consider in connection with 3-rectifiability in the first Heisenberg group. Our goal is to study whether the same holds true for quantitative concepts of rectifiability (in the spirit of the uniform rectifiability theory developed by G. David and S. Semmes in the Euclidean setting). We characterize sets with big pieces of intrinsic Lipschitz graphs as those sets which satisfy a weak geometric lemma for vertical beta-numbers and which have big vertical projections. The proof involves nonlinear PDEs, which do not play a role for the analogous problem in Euclidean space. This is joint work with V. Chousionis and T. Orponen.
The talk concerns my work with Anna Zatorska-Goldstein. We investigate the methods leading to an existence result for degenerate parabolic problems on a bounded domain in R^n. The key challenge is to manipulate with embeddings of classical and weighted Sobolev Spaces.
The studies of prime ends have long history involving various approaches, for example due to Caratheodory, Nakki, Vaisala and Zorich. In the talk we present some of those theories, including recent developments in metric measure spaces. Moreover, we discuss problems of continuous and homeomorphic extensions of mappings to the topological and prime end boundaries in the Euclidean setting and the setting of metric measure spaces for a class of mappings generalizing quasiconformal mappings.