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The Dirichlet problem for \(p\)-harmonic functions on the topologist’s comb

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Abstract

In this paper we study the Perron method for solving the \(p\)-harmonic Dirichlet problem on the topologist’s comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.

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Acknowledgments

The author was supported by the Swedish Research Council. Part of this research was done while the author was a Fulbright scholar (supported by the Swedish Fulbright Commission) visiting the University of Cincinnati in 2010. He would like to thank Tomasz Adamowicz, Jana Björn and Nageswari Shanmugalingam for helpful discussions related to this research.

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Correspondence to Anders Björn.

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Björn, A. The Dirichlet problem for \(p\)-harmonic functions on the topologist’s comb. Math. Z. 279, 389–405 (2015). https://doi.org/10.1007/s00209-014-1373-8

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  • DOI: https://doi.org/10.1007/s00209-014-1373-8

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