Graphical solutions to one-phase free boundary problems

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.

Automated summary

This study focuses on viscosity solutions to classical one-phase problem and its thin counterpart. In low dimensions, when the free boundary is the graph of a continuous function, the solution is proven to be the half-plane solution. This result answers a one-phase free boundary analog of Bernstein's problem for minimal surfaces in significant dimensions, and also classifies monotone solutions of semilinear equations with a bump-type nonlinearity.