Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners
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 $mathcal{P}imes mathbb{R}$ and $mathcal{N}imes mathbb{R}$ 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 $mathcal{P}imes mathbb{R}$ 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 ConcusFinn Conjecture for contact angles $0$ and $pi$, a capillary surface in positive gravity in $Omegaimesmathbb{R}$ 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 ConcusFinn Conjecture.

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