Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

TL;DR

This paper studies the boundary behavior of solutions to a prescribed mean curvature equation in planar domains, showing that radial limits exist near boundary points under various curvature and boundary data conditions.

Contribution

It provides new results on the existence of radial limits for nonparametric PMC surfaces at boundary points with intermediate curvature conditions.

Findings

Radial limits of solutions exist near boundary points under certain curvature assumptions.

Boundary curvature influences the boundary behavior of solutions.

Results depend on the prescribed boundary data and local boundary curvature properties.

Abstract

We investigate the boundary behavior of the variational solution $f$ of a Dirichlet problem for a prescribed mean curvature equation in a domain $\Omega\subset{\bf R}^{2}$ near a point $\mathcal{O}\in\partial\Omega$ under different assumptions about the curvature of $\partial\Omega$ on each side of $\mathcal{O}.$ We prove that the radial limits at $\mathcal{O}$ of $f$ exist under different assumptions about the Dirichlet boundary data $\phi,$ depending on the curvature properties of $\partial\Omega$ near $\mathcal{O}.$