Vertical blow UPS of capillary surfaces in ℝ3, Part 2: Nonconvex corners

Electronic Journal of Differential Equations 12/2008; 2008.
Source: DOAJ


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 Concus-Finn 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 Concus-Finn Conjecture.

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    ABSTRACT: Consider a nonparametric capillary or prescribed mean curvature surface z=f(x,y) defined in a cylinder Ω×ℝ over a two-dimensional region Ω that has a boundary corner point at O with an opening angle of 2α. Suppose 2α≤π and the contact angle approaches limiting values γ 1 and γ 2 in (0,π) as O is approached along each side of the opening angle. Our results yield a proof of the Concus-Finn conjecture, which provides the last piece of the puzzle of determining the qualitative behavior of a capillary surface at a convex corner. We find that – if (γ 1 ,γ 2 ) satisfies 2α+|γ 1 -γ 2 |>π, then f is bounded but discontinuous at O and has radial limits at O from all directions in Ω and, these radial limits behave in a prescribed way; – if (γ 1 ,γ 2 ) satisfies |γ 1 +γ 2 -π|>2α, then f is unbounded in every neighborhood of O; and – otherwise f is continuous at O.
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