Electron. J. Diff. Equ., Vol. 2011 (2011), No. 67, pp. 1-15.

Vanishing p-capacity of singular sets for p-harmonic functions

Tomohiko Sato, Takashi Suzuki, Futoshi Takahashi

In this article, we study a counterpart of the removable singularity property of $p$-harmonic functions. It is shown that p-capacity of the singular set of any $p$-harmonic function vanishes, and such function is always weakly $N(p-1)/(N-p)$-integrable. Several related results are also shown.

Submitted April 4, 2011. Published May 18, 2011.
Math Subject Classifications: 35B05, 35B45, 35J15, 35J70.
Key Words: p-harmonic function; capacity; singular set; removable singularity; weak Sobolev space.

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Tomohiko Sato
Department of Mathematics, Faculty of Science
Gakushuin University
1-5-1 Mejiro, Toshima-ku, Tokyo, 171-8588, Japan
email: tomohiko.sato@gakushuin.ac.jp
Takashi Suzuki
Division of Mathematical Science, Department of System Innovation
Graduate School of Engineering Science, Osaka University
Machikaneyamacho 1-3, Toyonakashi, 560-8531, Japan
email: suzuki@sigmath.es.osaka-u.ac.jp
Futoshi Takahashi
Department of Mathematics, Graduate School of Science
Osaka City University
Sugimoto 3-3-138, Sumiyoshiku, Osakashi, 535-8585, Japan
email: futoshi@sci.osaka-cu.ac.jp

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