Radial limits of capillary surfaces at corners

Abstract

Consider a solution f ∈ C 2 (Ω) of a prescribed mean curvature equationwhere Ω is a domain whose boundary has a corner at O = (0, 0) ∈ ∂Ω and the angular measure of this corner is 2α, for some α ∈ (0, π). Suppose sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite. If α > π 2, then the (nontangential) radial limits of f at O,were recently proven by the authors to exist, independent of the boundary behavior of f on ∂Ω, and to have a specific type of behavior.Suppose, the contact angle γ(·) that the graph of f makes with one side of ∂Ω has a limit (denoted γ2) at O and π − 2α < γ2 < 2α.We prove that the (nontangential) radial limits of f at O exist and the radial limits have a specific type of behavior, independent of the boundary behavior of f on the other side of ∂Ω. We also discuss the case α ∈ 0, π 2 .

A Generalization of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner"

“…In §2.1 of [4], a specific torus is constructed which depends solely on M 2 and which is used as a comparison surface; one should compare this with, for example, [12] where several types of comparison surfaces are used or [3] where an unduloid is used as a comparison surface. We shall use this torus as our comparison surface here.…”

“…Then q is also the modulus of continuity of functions (i.e. h + , h − β , h + β ) whose graphs are obtained by rotations and translations in the horizontal plane of T (see [4], page 59).…”

“…Let S 0 = gra(f ) = {(x, y, f (x, y)) : (x, y) ∈ Ω * } and allow S to be the closure of S 0 in R 3 . As in §2.2 of [4], there exists an isothermal parametrization [4], there exists a connected arc…”

On cusp solutions to a prescribed mean curvature equation

“…The hypotheses of [3] include the assumption that H satisfies one of the conditions which guarantees that cusp solutions do not exist; the following Corollary is a consequence of Theorem 1 and [3]. (A second corollary, similar to Corollary 1, follows by applying Theorem 1 to Theorems 1 & 2 of [4]. )…”

“…For solutions of boundary value problems which satisfy appropriate conditions, Rf (θ) can be proven to exist for θ ∈ [−α, α] \ J, where J is a countable subset of (−α, α) (e.g. [3,4,8,10,11,12,13]). We know of no examples in which J = ∅ and we ask if J = ∅ always holds; this is related to the existence of cusp solutions.…”