Boundary Behavior of Rotationally Symmetric Prescribed Mean Curvature Hypersurfaces in $\pmb {\varvec{{\mathbb {R}}}}^{4}$

Abstract

In a domain $\Omega \subset {\mathbb {R}}^{n}$ whose boundary is smooth except on a set ${\mathcal {C}}$ of codimension (in $\partial \Omega$) k,  the behavior of a nonparametric prescribed mean curvature hypersurface $z=f\left( \mathbf{x}\right)$ in the vertical cylinder $\Omega \times {\mathbb {R}}$ at a point ${\mathcal {O}}\in {\mathcal {C}}$ can be largely unknown when $n\ge 3,$ depending on the type of boundary condition f satisfies. In a previous note, the authors considered an example in which $n=3$ and $k=2;$ that is, when ${\mathcal {C}}$ is a (nonconvex) conical point on $\partial \Omega .$ Here, we consider prescribed mean curvature boundary value problems which are rotationally symmetric and investigate the behavior of variational solutions near a point $P\in {\mathcal {C}}$ when $n=3$ and $k=1.$