On cusp solutions to a prescribed mean curvature equation

Abstract

The nonexistence of "cusp solutions" of prescribed mean curvature boundary value problems in Ω × IR when Ω is a domain in IR 2 is proven in certain cases and an application to radial limits at a corner is mentioned.

Positive solutions of the prescribed mean curvature equation with exponential critical growth

“…More precisely, the authors investigate the boundary behavior of variational solutions of problem (P) at smooth boundary points where certain boundary curvature conditions are satisfied. In [11] the authors show a nonexistence result. To the best of our knowledge, the main result in this paper is the first work on the problem of medium curvature in dimension two and non-linearity with critical exponential growth.…”

Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

“…Proof of Theorem 2.5. All of the claims in the theorem except those in the last sentence follow from [9, Theorem 1] and [5] (when β − α > π) and [9, Theorem 2] and [5] (when β − α < π). (When β − α = π, all of the claims follow from Theorem 2.3 and [11].…”

“…(r cos θ, r sin θ), and set Rf (α) = lim ∂ − Ω x→O f * (x) and Rf (β) = lim ∂ + Ω x→O f * (x) when these limits exist, where f * denotes the trace of f on ∂Ω. In[9] (together with[5]), the following two results were proven.Proposition 2.1. (see [9, Theorem 1] and [5]) Let f ∈ C 2 (Ω) ∩ L ∞ (Ω) satisfy (1.1) and suppose β − α > π.…”

“…From the Courant-Lebesgue Lemma and the general comparison principle (see[12, Theorem 5.1]), we see that Y is uniformly continuous on E ρ(δ 1 ,w 0 ) (w 0 ) and so extends to a continuous function on the closure of E ρ(δ 1 ,w 0 ) (w 0 ). From Steps 2, 4 and 5 of[21] and with[5] replacing Step 3 of[21], we see that there exist…”

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

“…[8]) is used to prove that radial limits exist for almost every direction, (iv) cusp solutions are excluded (e.g. [1]) and (v) the behavior of the radial limit function is determined. The only step which does not follow from previous work is (ii) and so the proof of Theorem 1 comes down to establishing (ii).…”