Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

Abstract

The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions $f$ of Dirichlet problems for prescribed mean curvature equations in a domain $\Omega \subset \mathbb{R}^{2}$ 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 $\mathcal{O}$.