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

TL;DR

This paper constructs a smooth, axially symmetric solution to a classical one-phase free boundary problem in higher dimensions, featuring a catenoid-type free boundary, extending known two-dimensional solutions to higher dimensions.

Contribution

It provides the first explicit higher-dimensional example of a free boundary with catenoid shape, confirming a conjecture from prior work.

Findings

Existence of a smooth axially symmetric solution in $\\mathbb{R}^N$

Free boundary of catenoid type in higher dimensions

Extension of 2D solutions to N-dimensional setting

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}.