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.
Title
Graphical solutions to one-phase free boundary problems
Published in
Journal Fur Die Reine Und Angewandte Mathematik
Volume
2023
Issue
804
Pages
155-195
Date
2023-10-27
Publisher
Walter De Gruyter Gmbh, Berlin
ISSN
0075-4102
1435-5345
Grant
NSF: DMS 2000288
NSF CAREER: 2143719
Swiss National Science Foundation (SNF): 200021_182565
Swiss State Secretariat for Education, Research and lnnovation (SERI): MB22.00034
AEI project: PID2021-125021NAI00
Presidential Young Professor Fund (National University of Singapore)
Record creation date
2024-02-16