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arXiv · 1705.07345

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

Yong Liu·Kelei Wang·Juncheng Wei
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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 $\mathbb{R}% ^{2}$ (\cite{Pacard}). The existence of such solution is conjectured in \cite [Remark 2.4]{Pacard}.

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Yong Liu, Kelei Wang, Juncheng Wei. 2017-06-06. On one phase free boundary problem in $\mathbb{R}^{N}$. https://arxiv.org/abs/1705.07345

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