Scherk-Type Capillary Graphs

Abstract

Abstract. This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411-441) we clarify two outstanding questions.Solutions are constructed in the case α = π/2 for contact angle data (γ 1 , γ 2 ) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C 2 up to the corner. This shows that the best known regularity (C 1,ǫ ) is the best possible in some cases. Specific dependence of the Hölder exponent on the contact angle for our examples is given.Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469-482). It is shown that our examples are C 0,β up to and including the corner for any β < 1.Solutions with |γ −π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152-155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22-27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C 2/3 .

CMC capillary surfaces at reentrant corners

“…We will construct an appropriate "capillary graph" (for example, [Huff and McCuan 2006]) using a Weierstrass representation of minimal surfaces. We assume the domain is symmetric about the coordinate axes and starshaped as indicated in Figure 1.…”

“…The study of capillary surfaces in wedge domains can be tricky, as illustrated by [Keller et al 1991] and [Vreeburg 1990]. Concus and Finn [1991] mentioned that [Vreeburg 1990] was incorrect; we note that its argument is invalid since it does not distinguish between mean curvature and Gauss curvature and since it ignores [Tam 1986].…”

“…Vreeburg fails to consider the possibility of a jump discontinuity in the trace of the capillary surface on the boundary cylinder; see Section 2 and [Lancaster and Siegel 1996b]. Concus and Finn [1996b] noted that [Keller et al 1991] was incorrect. Keller, King and Merchant [1991] argued that the only ruled minimal surfaces in ‫ޒ‬ 3 were planes, ignoring helicoids, and seemed to be confused about the notion of principal curvature.…”

“…Concus and Finn [1996b] noted that [Keller et al 1991] was incorrect. Keller, King and Merchant [1991] argued that the only ruled minimal surfaces in ‫ޒ‬ 3 were planes, ignoring helicoids, and seemed to be confused about the notion of principal curvature. As with Vreeburg, they failed to consider the possibility of bounded, discontinuous (at the corner) capillary surfaces when γ 1 = γ 2 .…”

Capillary surfaces in wedge domains

“…Lieberman [1988] and Miersemann [1989] showed that such solutions are Hölder differentiable at O. Huff and McCuan [2006] gave examples to show that solutions need not be twice differentiable at O.…”