A gradient bound for free boundary graphs
Daniela De Silva, David Jerison
TL;DR
This paper establishes a gradient bound for free boundary graphs in a one-phase free boundary problem, showing such free boundaries are smooth if they minimize energy, and offers a new proof for the classical minimal surface gradient bound.
Contribution
It introduces a novel gradient bound for free boundary graphs and demonstrates their smoothness when energy-minimizing, extending classical minimal surface results.
Findings
Energy-minimizing free boundary graphs are smooth.
A new proof of the classical minimal surface gradient bound.
Extension of gradient bounds to free boundary problems.
Abstract
We prove an analogue for a one-phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy-minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Advanced Mathematical Modeling in Engineering