The Dirichlet problem and boundary regularity for local and nonlocal nonlinear elliptic equations in metric spaces

In this project we will study the Dirichlet problem and boundary regularity for elliptic equations in R^n and metric spaces in three different settings. I. For nonlinear 2nd order divergence type operators such as the p-Laplacian. Here the main focus will be on unbounded domains and boundary regularity at infinity.

In particular we want to obtain the Wiener criterion at infinity for unweighted and weighted R^n.II. For the p-Laplacian on finely open sets.

Here there are very few studies of the Dirichlet problem even on unweighted R^n, and boundary regularity is completely novel.Even Perron solutions in this context have so far only been considered in one of our recent preprints.III. For nonlocal nonlinear fractional elliptic equations.

Here there are a few results available for unweighted R^n, but even in that case many fundamental questions remain open. We want to take a more general perspective and consider these problems in metric spaces.

This goes completely beyond the current state-of-the-art and will include regional operators, weighted R^n, manifolds and fractals as special cases.As a fourth objective we want to develop the tools of sphericalization and inversions toour needs for studying the questions above.These are fundamental problems with wide use in applications.

Here the focus is on the mathematics, but a good theoretical understanding is vital for making relevant and fruitful mathematical modelling and simulations in future applications.