Capacities, Poincaré inequalities and gluing metric spaces.

Abstract

This thesis consists of an introduction, and one research paper with results related to potential theory both in the classical Euclidean setting, as well as in quite general metric spaces. The introduction contains a theoretical and historical background of some basic concepts, and their more modern generalisations to metric spaces developed in the last 30 years. By using upper gradients it is possible to define such notions as first order Sobolev spaces, p -harmonic functions and capacity on metric spaces. When generalising classical results to metric spaces, one often needs to impose some structure on the space by making additional assumptions, such as a doubling condition and a Poincar´e inequality. In the included research paper, we study a certain type of metric spaces called bow-ties , which consist of two metric spaces glued together at a single designated point. For a doubling measure µ , we characterise when µ supports a Poincar´e inequality on the bow-tie, in terms of Poincar´e inequalities on the separate parts together with a variational p -capacity condition and a quasiconvexity-type condition. The variational p -capacity condition is then characterised by a sharp measure decay condition at the designated point. We also study the special case when the bow-tie consists of the positive and negative hyperquadrants in R n , equipped with a radial doubling measure. In this setting, we characterise the validity of the p -Poincar´e inequality in various ways, and then provide a formula for the variational p -capacity of annuli centred at the origin.