p -harmonic measure is not additive on null sets

José G. Llorente[1]; Juan J. Manfredi[2]; Jang-Mei Wu[3]

  • [1] Department de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Spain
  • [2] Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260, USA
  • [3] Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, USA

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 2, page 357-373
  • ISSN: 0391-173X

Abstract

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When 1 < p < ∞ and p ≠ 2 the p -harmonic measure on the boundary of the half plane ℝ + 2 is not additive on null sets. In fact, there are finitely many sets E 1 , E 2 ,..., E κ in ℝ , of p -harmonic measure zero, such that E 1 ∪ E 2 ∪ . . . ∪ E κ = ℝ .

How to cite

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Llorente, José G., Manfredi, Juan J., and Wu, Jang-Mei. "$p$-harmonic measure is not additive on null sets." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.2 (2005): 357-373. <http://eudml.org/doc/84563>.

@article{Llorente2005,
abstract = {When $1&lt;p&lt;\infty $ and $p\ne 2$ the $p$-harmonic measure on the boundary of the half plane $\mathbb \{R\}^2_+$ is not additive on null sets. In fact, there are finitely many sets $E_1$, $E_2$,...,$E_\{\kappa \}$ in $\mathbb \{R\}$, of $p$-harmonic measure zero, such that $ E_1 \cup E_2 \cup ... \cup E_\{\kappa \}\!=\!\mathbb \{R\}$.},
affiliation = {Department de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Spain; Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260, USA; Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, USA},
author = {Llorente, José G., Manfredi, Juan J., Wu, Jang-Mei},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {-Laplacian; -harmonic function; -superharmonic functions; Choquet capacity; -harmonic measure},
language = {eng},
number = {2},
pages = {357-373},
publisher = {Scuola Normale Superiore, Pisa},
title = {$p$-harmonic measure is not additive on null sets},
url = {http://eudml.org/doc/84563},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Llorente, José G.
AU - Manfredi, Juan J.
AU - Wu, Jang-Mei
TI - $p$-harmonic measure is not additive on null sets
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 2
SP - 357
EP - 373
AB - When $1&lt;p&lt;\infty $ and $p\ne 2$ the $p$-harmonic measure on the boundary of the half plane $\mathbb {R}^2_+$ is not additive on null sets. In fact, there are finitely many sets $E_1$, $E_2$,...,$E_{\kappa }$ in $\mathbb {R}$, of $p$-harmonic measure zero, such that $ E_1 \cup E_2 \cup ... \cup E_{\kappa }\!=\!\mathbb {R}$.
LA - eng
KW - -Laplacian; -harmonic function; -superharmonic functions; Choquet capacity; -harmonic measure
UR - http://eudml.org/doc/84563
ER -

References

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