Calculus of Variations and Geometric Measure Theory
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H. Aikawa - A. Bjorn - J. Bjorn - N. Shanmugalingam

DICHOTOMY OF GLOBAL CAPACITY DENSITY IN METRIC MEASURE SPACES

created by shanmugal on 08 Jan 2017
modified on 19 Jun 2018

[BibTeX]

Accepted Paper

Inserted: 8 jan 2017
Last Updated: 19 jun 2018

Journal: Advances in Calc. Var.
Year: 2017

Abstract:

The variational capacity $\text{cap}_p$ in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every $E\subset\mathbb{R}^n$, \[ \inf_{x\in\mathbb{R}^n}\frac{\text{cap}_p(E\cap B(x,r),B(x,2r))}{\text{cap}_p(B(x,r),B(x,2r))} \] is either zero or tends to $1$ as $r \to \infty$. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a \p-Poincar\'e inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in $\mathbb{R}^n$.

It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

Keywords: metric space, capacitarily stable collection, Capacitary potential, capacity density


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