Geometric analysis seminar: Anders and Jana Björn (Linköping University)

Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces (Anders Björn)

Abstract: Newtonian Sobolev functions in $N^{1,p}(\mathcal{P})$ are better defined than a.e., since other ``representatives'' do not have upper gradients in $L^p$.

Here $\mathcal{P}=(\mathcal{P},d,\mu)$ is a metric measure space with $0 < \mu(B)<\infty$ for every ball $B$.

It has been an open problem since the late 1990s whether functions in $N^{1,p}(\mathcal{P})$ are always quasicontinuous.

The most general results is due to Eriksson-Bique and Poggi-Corradini (2024) who showed this when $\mathcal{P}$ is locally complete.

Quasicontinuity is also closely connected to whether the associated (Sobolev) capacity is an outer capacity.

In this talk I will take a look at these questions if we replace the $L^p$ norm by a more general norm $X$.

I will consider Banach function space norms and even more general Banach function lattice norms. A particular focus will be on $X=L^\infty$.

Perron method and boundary regularity for nonlinear nonlocal problems (Jana Björn)

Abstract: (joint with Anders Björn and Minhyun Kim)

Consider a nonlocal Dirichlet problem for nonlinear operators of $s$-fractional $p$-Laplace type, such as $(-\Delta_p)^s u = 0$, on a general bounded open set $\Omega$ in $\R^n$.

We define Perron solutions for arbitrary exterior Dirichlet data on the complement of $\Omega$ and study when the upper and the lower Perron solutions coincide.

Our definition of Perron solutions generalizes two earlier definitions and leads to some useful properties for the Perron solutions, including uniqueness results and perturbations on sets of zero capacity. Perron solutions appear also in a characterization of regular boundary points. Finally, regular boundary points and sets of zero capacity will be compared for different values of $p$ and $s$, including the local case $s=1$.