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

Lancaster, Kirk E.; Siegel, David

doi:10.2140/pjm.1996.176.165

Cited by 36 publications

(135 citation statements)

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“…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…”

Section: If (1) Specifically Is (3)mentioning

confidence: 99%

“…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]. )…”

Section: If (1) Specifically Is (3)mentioning

confidence: 99%

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

Section: Image Of the Gauss Mapmentioning

confidence: 99%

“…(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…”

Section: Image Of the Gauss Mapmentioning

confidence: 99%

“…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.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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A proof of the Concus–Finn conjecture

Lancaster¹

2010

Pacific J. Math.

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Consider a nonparametric capillary or prescribed mean curvature surface z = f (x, y) defined in a cylinder × ‫ޒ‬ over a two-dimensional region that has a boundary corner point at O with an opening angle of 2α. Suppose 2α ≤ π and the contact angle approaches limiting values γ 1 and γ 2 in (0, π ) as O is approached along each side of the opening angle. Our results yield a proof of the Concus-Finn conjecture, which provides the last piece of the puzzle of determining the qualitative behavior of a capillary surface at a convex corner. We find that• if (γ 1 , γ 2 ) satisfies 2α + |γ 1 − γ 2 | > π , then f is bounded but discontinuous at O and has radial limits at O from all directions in and, these radial limits behave in a prescribed way;• if (γ 1 , γ 2 ) satisfies |γ 1 + γ 2 − π | > 2α, then f is unbounded in every neighborhood of O; and• otherwise f is continuous at O.

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Section: If (1) Specifically Is (3)mentioning

confidence: 99%

“…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]. )…”

Section: If (1) Specifically Is (3)mentioning

confidence: 99%

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

Section: Image Of the Gauss Mapmentioning

confidence: 99%

Section: Image Of the Gauss Mapmentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

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A proof of the Concus–Finn conjecture

Lancaster¹

2010

Pacific J. Math.

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Canthotaxis/Wetting Barriers/Pinning Lines

Langbein

2002

Springer Tracts in Modern Physics

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Scherk-Type Capillary Graphs

Huff

McCuan

2005

J. math. fluid mech.

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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 .

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

Cited by 36 publications

References 21 publications

A proof of the Concus–Finn conjecture

A proof of the Concus–Finn conjecture

Canthotaxis/Wetting Barriers/Pinning Lines

Scherk-Type Capillary Graphs

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