Tomasz Adamowicz: Geometric Function and Mapping Theory - selected Topics and Open Problems


Geometric function and mapping theory (GF & MT for short) grows from the following two problems: (I) To find a counterpart of conformal mappings and holomorphic functions in higher real dimensions in $\mathbb{R}^n$ for $n \ge 3$ and in more general spaces, and (II) to investigate to what extend geometric properties of harmonic functions are preserved for solutions of more general elliptic equations and systems of equations and for (quasi)minima of variational integrals? Both questions have been the subject of intensive studies for more than a century and been successfully investigated in Euclidean domains, on manifolds and in general metric spaces. The presentation is designed to be a little showcase of GF&MT and we plan to discuss the following topics: 1. Quasiconformal mappings and their relations to p-harmonic and variable exponent $p(\cdot)$-harmonic equations. 2. Harnack type inequalities and boundary Harnack inequalities for $p$-harmonic and $p(\cdot)$-harmonic equations, the growth of solutions to variable exponent PDEs. 3. Systems of quasilinear elliptic equations and related geometric problems: three-spheres theorems, the Rado–Kneser–Choquet theorems, isoperimetric inequalities. 4. Analysis and PDEs on metric measure spaces (including the setting of Heisen-berg groups).