Vol. 288, No. 1, 2017

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ISSN: 0030-8730
Radial limits of capillary surfaces at corners

Mozhgan (Nora) Entekhabi and Kirk E. Lancaster

Vol. 288 (2017), No. 1, 55–67
Abstract

Consider a solution f C2(Ω) of a prescribed mean curvature equation

div f 1 + |f|2 = 2H(x,f)  in Ω 2,

where 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 supxΩ|f(x)| and supxΩ|H(x,f(x))| are both finite. If α > π 2 , then the (nontangential) radial limits of f at O, namely

Rf(θ) = limr0f(rcosθ,rsinθ),

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 α (π 4 , π 2 ], 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 ] and the displayed inequalities do not hold.

Keywords
prescribed mean curvature, radial limits
Mathematical Subject Classification 2010
Primary: 35B40, 53A10, 76D45, 35J93
Milestones
Received: 8 April 2016
Revised: 24 October 2016
Accepted: 30 October 2016
Published: 8 April 2017
Authors
Mozhgan (Nora) Entekhabi
Department of Mathematics, Statistics and Physics
Wichita State University
Wichita, KS 67260-0033
United States
Kirk E. Lancaster
Department of Mathematics, Statistics, and Physics
Wichita State University
Wichita, KS 67260-0033
United States