Wiener criterion for Cheeger \(p\)-harmonic functions on metric spaces. (English) Zbl 1120.31003

Björn, Jana

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 103-115 (2006).

The main theorem of the paper is the following result. Let \(X\) be a complete metric measure space with a doubling measure \(\mu\) supporting a \(p\)-Poincaré inequality. Let \(\Omega\subset X\) be open and bounded. Then the point \(x\in\partial\Omega\) is Cheeger \(p\)-regular if and only if for some \(\delta>0,\) \[ \int_0^\delta \left(\frac{\mathrm{Cap}_p(B(x,t)\setminus\Omega,B(x,2t))}{t^{-p}\mu(B(x,t))}\right)^{1/(p-1)}\frac{dt}{t}=\infty. \]
For the entire collection see [Zbl 1102.31001].

Reviewer: Roman Urban (Wrocław)

MSC:

31C45	Other generalizations (nonlinear potential theory, etc.)
31C15	Potentials and capacities on other spaces
35J65	Nonlinear boundary value problems for linear elliptic equations
35J60	Nonlinear elliptic equations

Keywords:

boundary regularity; capacity; metric measure space; Poincaré inequality