## 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 $$\text{supp} \mu$$.

### MSC:

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