Existence and behavior of the radial limits of a bounded capillary surface at a corner

A proof of the Concus–Finn conjecture

“…At a nonconvex corner O (that is, α > π/2), [Shi and Finn 2004] shows that information about ∂ ∩ B (O) and γ in B (O) for some > 0 need not be sufficient to determine the continuity at O of a solution of (3) and (2). Lancaster and Siegel [1996a] investigated the behavior of bounded solutions of (3) and (2) at corners, both convex and nonconvex corners, and they noted in [1996a; 1996b] that the conclusions in [1996a] carry over to solutions of (1) and (2) when H satisfies some minor restrictions (that is, H (x, z) is either real-analytic or strictly increasing in z); in this case, a bounded solution…”

“…Thus, under mild restrictions on H , Figure 2 continues to illustrate the behavior at O of solutions of (1) and (2). (See Remark 3.1 for a comment about [Lancaster and Siegel 1996a]. )…”

“…Consider β ∈ (−α, α) fixed and set u(x) = f (x) − R f (β), as in [Lancaster and Siegel 1996a]. Set δ 0 = 2δ * .…”

“…(To see this, we may represent the minimal surface ᏹ * in isothermal coordinates as the (downward oriented) parametric surface X : B(O, 1) → ‫ޒ‬ 3 (for example, [Courant 1977;Lancaster 1985;Elcrat and Lancaster 1986;Lancaster and Siegel 1996a]) and obtain the Weierstrass ( f, g)-representation of ᏹ * , where…”

“…The proof of this theorem uses the conformal (or isothermal) representation of a prescribed mean curvature surface discussed in [Lancaster and Siegel 1996a] and properties of two-dimensional quasiconformal maps to obtain a contradiction to the assumption that the solution f is continuous at the origin. This proof uses Kenmotsu's theorem [1979], Theorem 2.1, Gehring's lemma [1973] and properties of solutions of Riemann-Hilbert problems to obtain a contradiction, illustrated in Figure 3, of the Phragmén-Lindelof theorem.…”