Solutions of the Bernoulli one-phase problem with a defect

Abstract

We study the far-field behavior of solutions of the one-phase Bernoulli free boundary problem in the exterior of a ball, and of entire solutions with a single compactly supported inhomogeneity of the free boundary condition, which we call a defect. For solutions which blow down to a half-plane solution (proper solutions) we establish an asymptotic expansion at infinity: in dimension $d\ge 3$ the free boundary height converges to a limit at rate $|x{|}^{2-d}$ with a capacity-type coefficient, while in dimension $d=2$ the expansion carries a logarithmic term. A significant novelty is that the expansions are quantitative and uniform over all the proper solutions.

arXiv commentThis article was a part of the arXiv post arXiv:2512.11152v1, we have split that article into two parts. The companion article has already been posted as a replacement at arXiv:2512.11152v2, this posting serves as a replacement of the other half of arXiv:2512.11152