Publication

Graphical solutions to one-phase free boundary problems

Xavier Fernandez-Real Girona
2023
Journal Articles

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.

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Graphical solutions to one-phase free boundary problems

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