Existence of quadrature surfaces for positive measures with finite support (Q1320304)

The main result is as follows. For any positive measure \(\mu\) with finite support in \(\mathbb{R}^ n\), \(n \geq 2\), there is a bounded open set \(\Omega \supset \text{supp} \mu\) such that \[ \int_{\partial \Omega} h dH^{n- 1} = \int_ \Omega h d \mu \] for all functions \(h\) harmonic in \(\overline \Omega\); here \(H^{n-1}\) denotes the \((n-1)\)-dimensional Hausdorff measure. The proof uses the free boundary problem technique of \textit{H. W. Alt} and \textit{L. A. Caffarelli} [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)].