A one-sided two phase Bernoulli free boundary problem
Analysis of PDEs
2023-09-06 v1
Abstract
We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a two-phase condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is regular surface. Moreover, we show that the remaining singular set has Hausdorff dimension at most as in the case of the classical one-phase problem, being the dimension of the space.
Cite
@article{arxiv.2309.01749,
title = {A one-sided two phase Bernoulli free boundary problem},
author = {Lorenzo Ferreri and Bozhidar Velichkov},
journal= {arXiv preprint arXiv:2309.01749},
year = {2023}
}