Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 160, pp. 1-25.
Title: Vertical blow ups of capillary surfaces in $R^3$,
Part 2: Nonconvex corners
Authors: Thalia Jeffres (Wichita State Univ., Kansas, USA)
Kirk Lancaster (Wichita State Univ., Kansas, USA)
Abstract:
The goal of this note is to continue the investigation started in
Part One of the structure of "blown up" sets of the form
$\mathcal{P}\times \mathbb{R}$ and
$\mathcal{N}\times \mathbb{R}$ when
$\mathcal{P}, \mathcal{N} \subset \mathbb{R}^{2}$ and
$\mathcal{P}$ (or $\mathcal{N}$) minimizes
an appropriate functional and the domain has a nonconvex corner.
Sets like $\mathcal{P}\times \mathbb{R}$ can be the limits of
the blow ups of subgraphs of solutions of capillary surface or
other prescribed mean curvature problems, for example. Danzhu Shi
recently proved that in a wedge domain $\Omega$ whose boundary
has a nonconvex corner at a point $O$ and assuming the correctness
of the Concus-Finn Conjecture for contact angles $0$ and $\pi$, a
capillary surface in positive gravity in $\Omega\times\mathbb{R}$
must be discontinuous under certain conditions. As an application,
we extend the conclusion of Shi's Theorem to the case where the
prescribed mean curvature is zero without any assumption about the
Concus-Finn Conjecture.
Submitted August 3, 2007. Published December 09, 2008.
Math Subject Classifications: 49Q20, 53A10, 76B45.
Key Words: Blow-up sets; capillary surface; Concus-Finn conjecture.