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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A problem of Baernstein on the equality of the $p\mspace{1mu}$-harmonic measure of a set and its closure


Authors: Anders Björn, Jana Björn and Nageswari Shanmugalingam
Journal: Proc. Amer. Math. Soc. 134 (2006), 509-519
MSC (2000): Primary 31C45; Secondary 30C85, 31A25, 31B20, 31C15, 46E35
Published electronically: August 12, 2005
MathSciNet review: 2176020
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Abstract: A. Baernstein II (Comparison of $p\mspace{1mu}$-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If $G$ is a union of $m$open arcs on the boundary of the unit disc $\mathbf{D}$, then is $\omega_{a,p}(G)=\omega_{a,p}(\overline{G})$, where $\omega_{a,p}$ denotes the $p\mspace{1mu}$-harmonic measure? (Strictly speaking he stated this question for the case $m=2$.) For $p=2$ the positive answer to this question is well known. Recall that for $p \ne 2$ the $p\mspace{1mu}$-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.

The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when $1<p<2$. In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function $\chi_G$ is the restriction to $\partial \mathbf{D}$ of a Sobolev function from $W^{1,p}(\mathbf{C})$.

For $p \ge 2$ it is no longer true that $\chi_G$ belongs to the trace class. Nevertheless, we are able to show equality for the case $m=1$ of one arc for all $1<p<\infty$, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.

Finally we show that in a certain sense the equality holds for almost all relatively open sets.


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Additional Information

Anders Björn
Affiliation: Department of Mathematics, Linköpings Universitet, SE-581 83 Linköping, Sweden
Email: anbjo@mai.liu.se

Jana Björn
Affiliation: Department of Mathematics, Linköpings Universitet, SE-581 83 Linköping, Sweden
Email: jabjo@mai.liu.se

Nageswari Shanmugalingam
Affiliation: Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025
Email: nages@math.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08187-6
PII: S 0002-9939(05)08187-6
Keywords: Ahlfors regular, Dirichlet problem, $d$-set, Lipschitz domain, Minkowski dimension, $p\mspace{1mu}$-harmonic function, $p\mspace{1mu}$-harmonic measure, Sobolev function, starshaped, trace, unit disc
Received by editor(s): September 27, 2004
Published electronically: August 12, 2005
Additional Notes: We thank Juha Heinonen for drawing our attention to the question of Baernstein
The first two authors were supported by the Swedish Research Council and Gustaf Sigurd Magnuson’s fund of the Royal Swedish Academy of Sciences. The second author did this research while she was at Lund University
The third author was partly supported by NSF grant DMS 0243355.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.