Radial limits of bounded nonparametric prescribed mean curvature surfaces

Abstract

Consider a solution f ∈ C 2 (Ω) of a prescribed mean curvature equationwhere Ω ⊂ IR 2 is a domain whose boundary has a corner at O = (0, 0) ∈ ∂Ω. If sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite and Ω has a reentrant corner at O, then the radial limits of f at O,f (r cos(θ), r sin(θ)), are shown to exist and to have a specific type of behavior, independent of the boundary behavior of f on ∂Ω \ {O}. If sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite and the trace of f on one side has a limit at O, then the radial limits of f at O exist and have a specific type of behavior.

Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

“…As noted previously (e.g., [9]), the "gliding hump" construction (which depends on the existence of classical solutions of (1.1)-(1.2)) cannot be successfully used when β − α > π. When β − α < π and (2.2) holds, local barriers for (1.1)-(1.2) do not exist on ∂ ± Ω and the "gliding hump" construction in [20] and [21,Theorem 3] cannot be directly used in Ω.…”

“…Since We may, if we wish, extend C by adding to C a curve from x 0 to a point on ∂Ω \ E ρ(p −1 (τ ),o) (o). Now we modify the argument in the proof of [9,Theorem 2] to show that Rf (θ) = z 2 for all θ ∈ (α, β); that is, we shall show that the nontangential limit of f at O exists and equals z 2 . Let α , β ∈ (α, β) with α < β .…”

“…Here and throughout this note, we adopt the sign convention that the curvature of ∂Ω is nonnegative when Ω is convex and we denote by Λ(x) the curvature of ∂Ω at points x ∈ ∂Ω at which ∂Ω is smooth. Our primary interest is in the existence and behavior of the radial limits at O, (1.1); the existence of radial limits when H ≡ 0 was established in [18] (see also [6,8,19]) and this was extended to general H in [7] (see also [9,10,21]). When Γ is a C 2,λ open subset of ∂Ω for some λ ∈ (0, 1), O ∈ Γ, H ≡ 0 and f is a variational solution of (1.1)-(1.2), the following is known: [11,Theorem 1.1]);…”

“…(iii) If Rf (θ) exists for some θ 0 ∈ [α, β], then Rf (θ) exists for every θ ∈ (α, β) (i.e., [9,Theorem 2]);…”

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].…”