Bernstein Functions and Radial Limits of Prescribed Mean Curvature Surfaces

TL;DR

This paper investigates the boundary behavior of solutions to prescribed mean curvature equations, focusing on how boundary curvature affects radial limits at boundary points, including nonconvex corners, and extends classical results.

Contribution

It provides new sufficient conditions for the existence of tangential radial limits of solutions at boundary points, even with irregular boundary data and contact angles, complementing classical theorems.

Findings

Sufficient conditions for tangential radial limits at boundary points.

Existence of limits even with unbounded boundary data and contact angles.

Extension of Leon Simon's 1976 theorem on least area surfaces.

Abstract

The radial limits at a point ${\bf y}$ of the boundary of the domain $\Omega\subset {\bf R}^{2}$ of a bounded variational solution $f$ of Dirichlet or contact angle boundary value problems for a prescribed mean curvature equation are studied with an emphasis on the effects of assumptions about the curvatures of the boundary $\partial\Omega$ on each side of the point ${\bf y}.$ For example, at a nonconvex corner ${\bf y},$ we previously proved that all nontangential radial limits of $f$ at ${\bf y}$ exist, here we provide sufficient conditions for the tangential radial limits to exist, even when the Dirichlet data $\phi\in L^{\infty}(\partial\Omega)$ has no one-sided limits at ${\bf y}$ or the contact angle $\gamma\in L^{\infty}(\partial\Omega:[0,\pi])$ is not bounded away from $0$ or $\pi.$ We also provide a complement to a 1976 Theorem by Leon Simon on least area surfaces.