Graphical solutions to one-phase free boundary problems
Analysis of PDEs
2023-11-13 v2
Abstract
We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein's problem for minimal surfaces. As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity.
Cite
@article{arxiv.2212.08847,
title = {Graphical solutions to one-phase free boundary problems},
author = {Max Engelstein and Xavier Fernández-Real and Hui Yu},
journal= {arXiv preprint arXiv:2212.08847},
year = {2023}
}