Graphical solutions to one-phase free boundary problems

Max Engelstein, Xavier Fernández-Real, Hui Yu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)155-195
Number of pages41
JournalJournal fur die Reine und Angewandte Mathematik
Issue number804
StatePublished - Nov 1 2023

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© De Gruyter 2023.


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