Graphical solutions to one-phase free boundary problems

TL;DR

This paper investigates viscosity solutions to one-phase free boundary problems, proving that in low dimensions, solutions with continuous free boundaries are half-plane solutions, and classifies certain monotone solutions.

Contribution

It provides a classification of solutions in low dimensions and extends Bernstein-type results to free boundary problems, also analyzing monotone solutions of semilinear equations.

Findings

Solutions with continuous free boundaries are half-plane solutions in low dimensions

Classified monotone solutions of semilinear equations with bump-type nonlinearity

Extended Bernstein's problem to free boundary contexts

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.