Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 152, pp. 1-24.
Title: Vertical blow ups of capillary surfaces in $\mathbb{R}^3$,
Part 1: convex corners
Authors: Thalia Jeffres (Wichita State Univ., Kansas, USA)
Kirk Lancaster (Wichita State Univ., Kansas, USA)
Abstract:
One technique which is useful in the calculus of
variations is that of "blowing up". This technique can
contribute to the understanding of the boundary behavior of
solutions of boundary value problems, especially when they involve
mean curvature and a contact angle boundary condition. Our goal in
this note is to investigate 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; sets
like $\mathcal{P}\times \mathbb{R}$ can be the limits of the blow
ups of subgraphs of solutions of mean curvature problems, for example.
In Part One, we investigate "blown up" sets when the domain has
a convex corner. As an application, we illustrate the second author's
proof of the Concus-Finn Conjecture by providing a simplified proof
when the mean curvature is zero.
Submitted January 9, 2007. Published November 13, 2007.
Math Subject Classifications: 49Q20, 53A10, 76B45.
Key Words: Blow-up sets; capillary surface; Concus-Finn conjecture.