A harmonic quadrature formula characterizing bi-infinite cylinders
Let \(m\) and \(n\) be positive integers, let \(m\geq 2\), let \(K_r= \{x\in \mathbb{R}^m\): \(|x|< r\}\) and let \(\lambda_m\) denote \(m\)- dimensional Lebesgue measure. It can be shown that, if \(h\) is harmonic and integrable on \(K_r \times \mathbb{R}^n\), then \[ \int h d\lambda_{m+n}= \lambda_m (K_r) \int h(0, \xi) d\lambda_n (\xi). \tag{1} \] The main result of this paper is an inverse result as follows. Let \(D\) be an open subset of \(\mathbb{R}^{m+n}\) such that \((\overline {D})^0 =D\) and \(\{0 \}^m\times \mathbb{R}^n \subset D\), and such that \(D\) is contained in some (arbitrarily large) cylinder. If (1) holds for every integrable harmonic function \(h\) on \(D\), then \(D= K_r \times \mathbb{R}^n\). The extreme cases where \(n=0\) (that is, \(D\) is a ball) and where \(m=1\) (that is, \(D\) is a strip) had been solved previously.
scientific article; zbMATH DE number 784591