The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

ABSTRACT

In this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.

KEYWORDS:

• Boundary regularity
• capacity
• Dirichlet problem
• doubling measure
• Kellogg property
• metric space
• p-harmonic function
• Poincaré inequality
• regular point
• resolutive-regular point
• weak Kellogg property

AMS SUBJECT CLASSIFICATIONS:

• Primary: 31C45
• Secondary: 31E05
• 35J66
• 35J92
• 49Q20

Acknowledgements

The idea to study resolutive-regularity is due to Tomas Sjödin (private communication).

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the Swedish Research Council, [grant number 621-2007-6187], [grant number 621-2011-3139], [grant number 2016-03424].