On a free boundary problem in electrostatics
Gérard A. Philippin · Mathematical Methods in the Applied Sciences · 1990
Abstract Let Ωi ⊂ ℝN, i = 0, 1, be two bounded separately star‐shaped domains such that \documentclass{article}\pagestyle{empty}\begin{document}$ \Omega _0 \supset \bar \Omega _1 $\end{document} . We consider the electrostatic potential u defined in \documentclass{article}\pagestyle{empty}\begin{document}$ \Omega : = \Omega _0 \backslash \bar \Omega _1 $\end{document} : The geometry of the two boundary components Γ0 and Γ1 is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions magnified image Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.