A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure
2006 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, Vol. 134, no 3, 509-519 p.Article in journal (Refereed) Published
A. Baernstein II (Comparison of p-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 D, then is w a,p(G)=w a,p(G), where w a,p denotes the p-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≠2 the p-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 1G is the restriction to ∂D of a Sobolev function from W 1,p(C).
For p≥2 it is no longer true that XG belongs to the trace class. Nevertheless, we are able to show equality for the case m=1 of one arc for all 1, 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.
Place, publisher, year, edition, pages
2006. Vol. 134, no 3, 509-519 p.
Ahlfors regular, Dirichlet problem, Lipschitz domain, Minkowski dimension, Sobolev function, starshaped, trace, unit disc
IdentifiersURN: urn:nbn:se:liu:diva-18233DOI: 10.1090/S0002-9939-05-08187-6OAI: oai:DiVA.org:liu-18233DiVA: diva2:217010
First Published in Proceedings of the American Mathematical Society in 2006:
Anders Björn, Jana Björn and Nageswari Shanmugalingam , A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure, 2006, Proceedings of the American Mathematical Society, (134), 3, 509-519.
Copyright: American Mathematical Society