Graphical solutions to one-phase free boundary problems

2022/12/17 by Engelstein, Max, Fernández-Real, Xavier, Yu, Hui · 6 citations
#35R35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.08847

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.

Cited by

• On the asymptotic properties of solutions to one-phase free boundary problems
• Linearized equation and generic regularity in the Alt-Caffarelli problem
• New Homogeneous Solutions for the One-Phase Free Boundary Problem
• Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem
• C^∞ Regularity for the free boundary of one-phase Fractional Laplacian problem
• Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

Related

• Structure of one-phase free boundaries in the plane
• Existence and regularity in the fully nonlinear one-phase free boundary problem
• Existence and regularity of monotone solutions to a free boundary problem
• A gradient bound for free boundary graphs
• Free boundaries subject to topological constraints