On Smooth Solutions to One Phase-Free Boundary Problem in Rn

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in R-n, n >= 3. Its free boundary is of catenoid type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution [18] in 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.

Automated summary

In this paper, a smooth axially symmetric solution to the classical one phase free boundary problem in higher dimensions is constructed, with the free boundary being of catenoid type. This solution is the first nontrivial smooth solution in higher dimensions for the one phase-free boundary problem, and serves as a higher dimensional analogy of a previously known solution in R-2.