an:00665425
Zbl 0808.31004
Hansen, Wolfhard; Nadirashvili, Nikolai
A converse to the mean value theorem for harmonic functions
EN
Acta Math. 171, No. 2, 139-163 (1993).
00022058
1993
j
31B05 31C35 60J45
harmonic functions; \(r\)-median function; converse to the mean value theorem; minimal fine topology; Martin compactification
Let \(U\) be a bounded domain in \(\mathbb{R}^ d\), \(d\geq 1\), \(\rho(x)= \text{dist} (x,U^ c)\), and \(r:U\to \mathbb{R}\) a function satisfying \(0<r\leq\rho\). Let \(B^ x\), \(x\in U\), denote the open ball centered at \(x\) with radius \(r(x)\). A Lebesgue measurable function \(f\) on \(U\) satisfying
\[
f(x)= {1\over {\lambda(B^ x)}} \int_{B^ x} f d\lambda
\]
for every \(x\in U\) (where \(\lambda\) denotes the Lebesgue measure), is said to be \(r\)-median. Results of the type under what conditions is an \(r\)-median function \(f\) actually harmonic, are usually known as a converse to the mean value theorem.
The main result of the present paper states that if \(f\) is \(r\)-median, continuous (on \(U\)) and \(h\)-bounded (i.e. \(| f|\leq h\) with \(h\) harmonic on \(U\)), then \(f\) is harmonic on \(U\). As a rather simple consequence of this result, the authors show that an \(r\)-median, \(h\)- bounded, Lebesgue measurable function \(f\) on \(U\) is harmonic provided that the function \(r\) is bounded away from zero on compact subsets of \(U\), thus improving the result of \textit{W. A. Veech} [Ann. Math., II. Ser. 97, 189-216 (1973; Zbl 0282.60048)], where \(U\) was assumed to be a Lipschitz domain.
The proof of the results is analytic, but with the strong probabilistic flavor in the background. It uses the minimal fine topology of the Martin compactification of \(U\), an appropriate (transfinite) sweeping of measures, and certain properties of the Schr??dinger equation \(\Delta u- \delta\rho^{-2} 1_ A=0\) on \(U\) (\(\delta>0\), \(A\) a suitable subset of \(U\)).
Z.Vondra??ek (Saarbr??cken)
Zbl 0282.60048