Existence and geometric properties of solutions of a free boundary problem in potential theory. (English) Zbl 0846.31005

Let \(0 \leq g\), \(h \in L^\infty (\mathbb{R}^N)\) \((N \geq 2)\) be two given density functions, at least one of which bounded away from zero outside a compact set and \(g\) (Hölder) continuous. We prove that for any compactly supported positive measure \(\mu\) which is sufficiently concentrated (e.g. has sufficiently high \((N - 1)\)-dimensional density) there exists a bounded open set \(\Omega \subset \mathbb{R}^N\) such that the Newtonian potential of the measure \(h{\mathcal L}^N \lfloor \Omega + g {\mathcal H}^{N - 1} \lfloor \partial \Omega\) agrees with that of \(\mu\) outside \(\Omega\). Some regularity of \(\partial \Omega\) is obtained, as well as several results on the geometry of \(\Omega\). Example: if \(h\) and \(g\) are constant then, for any \(x \in \partial \Omega\), the inward normal ray of \(\partial \Omega\) at \(x\) (if it exists) intersects the closed convex hull of \(\text{supp} \mu\).


31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35R35 Free boundary problems for PDEs
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