On one phase free boundary problem in ${\mathbb{R}}^N$

Abstract

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in ${\mathbb{R}}^N$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}$ (\cite{Pacard}). The existence of such solution is conjectured in \cite [Remark 2.4]{Pacard}.

42 pages

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