Densities with the mean value property for harmonic functions in a Lipschitz domain
HTML articles powered by AMS MathViewer
- by Hiroaki Aikawa PDF
- Proc. Amer. Math. Soc. 125 (1997), 229-234 Request permission
Abstract:
Let $D$ be a bounded domain in $\mathbb {R}^{n}$, $n\ge 2$, and let $x_{0}\in D$. We consider positive functions $w$ on $D$ such that $h( x_{0}) = (\int _{D}w dx)^{-1} \int _{D} h w dx$ for all bounded harmonic functions $h$ on $D$. We determine Lipschitz domains $D$ having such $w$ with $\inf _{D} w>0$.References
- Hiroaki Aikawa, Integrability of superharmonic functions and subharmonic functions, Proc. Amer. Math. Soc. 120 (1994), no. 1, 109–117. MR 1169019, DOI 10.1090/S0002-9939-1994-1169019-7
- Alano Ancona, On strong barriers and an inequality of Hardy for domains in $\textbf {R}^n$, J. London Math. Soc. (2) 34 (1986), no. 2, 274–290. MR 856511, DOI 10.1112/jlms/s2-34.2.274
- W. Hansen and I. Netuka, Volume densities with the mean value property for harmonic functions, Proc. Amer. Math. Soc. 123 (1995), no. 1, 135–140. MR 1213859, DOI 10.1090/S0002-9939-1995-1213859-3
- Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
- Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37 (1968). MR 239264, DOI 10.7146/math.scand.a-10841
Additional Information
- Hiroaki Aikawa
- Affiliation: Department of Mathematics, Shimane University, Matsue 690, Japan
- Email: haikawa@riko.shimane-u.ac.jp
- Received by editor(s): July 13, 1995
- Received by editor(s) in revised form: August 1, 1995
- Communicated by: J. Marshall Ash
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 229-234
- MSC (1991): Primary 31A05, 31B05
- DOI: https://doi.org/10.1090/S0002-9939-97-03649-6
- MathSciNet review: 1363444