On Smooth Solutions to One Phase-Free Boundary Problem in $\mathbb{R}^{n}$
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Abstract We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{n}$, $n\geq 3.$ Its free boundary is of “catenoid” type. This is a higher dimensional analogy of the Hauswirth–Helein–Pacard solution [18] in $\mathbb{R}^{2}$. The existence of such solution is conjectured in [18, Remark 2.4]. This is the 1st nontrivial smooth solution to the one phase-free boundary problem in higher dimensions.
2015 ◽
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