The stability of the axially symmetric pendent drop

Wente, Henry C.

doi:10.2140/pjm.1980.88.421

Cited by 63 publications

(45 citation statements)

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“…If H is constant, then the contradiction comes from the touching principle resulting from the Hopf maximum principle (see Wente [10], §2).…”

Section: Iv) the Angle Between The Outer Normal To S And The Outer Nomentioning

confidence: 99%

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Uniqueness for certain surfaces of prescribed mean curvature

Vogel¹

1988

Pacific J. Math.

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The classical problem of liquid in a capillary tube concerns finding the minimum of a certain energy function, which leads to a surface of prescribed mean curvature. This paper weakens the hypotheses by considering local minima of the functional. It is shown that no new surfaces result.1. Introduction. Although the theorems in this paper have fairly general hypotheses, they are motivated by the theory of capillary surfaces. The classical capillary problem is to study the shape of the surface formed by capillary action when a tube of general cross-section Ω c R n is placed into an infinite reservoir. The assumption that has customarily been made is that the surface is energy minimizing over compact perturbations. It is natural to ask whether a surface can exist which is a local minimum for the functional but not a global minimum. In other words, is it possible, in the classical problem, for an "exotic" capillary surface to exist which is stable with respect to small compact perturbations, but unstable with respect to a large compact perturbation? The paper proves that under fairly weak assumptions, the answer is "no".Certainly there are other problems in capillarity with non-trivial local minima. The energy functional for the pendent drop (e.g. Wente [10]) has no minimum if arbitrarily large perturbations are allowed, since by falling to -oo a drop looses an infinite amount of potential energy. Another such situation is that of a drop trapped between two planes (Vogel [8]). For some values of the separation, the (stable) connected drop bridging the planes has a larger energy than a spherical cap on one plane.The reason the assumption that the surface is energy minimizing over all perturbations has been made in the classical problem is that a result of Miranda [5] then implies that the surface is a graph. The idea of the proof is that replacing the set U occupied by the liquid by the set U s = {(x, t): t < / 0°° φu{x> τ) dτ) will reduce perimeter and hence the relevant energy functional. Here t = 0 is chosen to lie below the

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“…If H is constant, then the contradiction comes from the touching principle resulting from the Hopf maximum principle (see Wente [10], §2).…”

Section: Iv) the Angle Between The Outer Normal To S And The Outer Nomentioning

confidence: 99%

“…The energy functional for the pendent drop (e.g. Wente [10]) has no minimum if arbitrarily large perturbations are allowed, since by falling to -oo a drop looses an infinite amount of potential energy. Another such situation is that of a drop trapped between two planes (Vogel [8]).…”

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confidence: 99%

Uniqueness for certain surfaces of prescribed mean curvature

Vogel¹

1988

Pacific J. Math.

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“…The situation is very similar to that considered in [19], Section 4, although we can't simply quote results from that paper. However, the approach we will use is certainly inspired by the approach in [19].…”

Section: Eigenvalue Conditionsmentioning

confidence: 53%

“…For example, in considering a liquid bridge between two parallel planes (e.g., [1], [6], [14], [15], [20]), it's pointless to seek a strict local minimum for energy, since translations of bridges parallel to the planes will leave energy and volume unchanged. A drop hanging from a horizontal plane ( [19]) is similarly indifferent to translations (although [19] includes a gravity term which we will neglect). A comparable situation occurs for a drop inside a sphere ( [11]) or attached to the outside of a sphere.…”

Section: Introductionmentioning

confidence: 99%

Local energy minimality of capillary surfaces in the presence of symmetry

Vogel¹

2002

Pacific J. Math.

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Consider the stationary capillary problem of a drop of liquid attached to a fixed surface, so that the drop minimizes an energy functional subject to a volume constraint. There are many such capillary problems in which, due to the symmetry of the fixed surface, one cannot hope for a capillary surface which is a strict local minimum for energy. A weaker concept which is sensible to consider is that of local minimality modulo the isometries of space which map the fixed surface into itself. In other words, it is reasonable to attempt to show that, given a capillary surface, any nearby comparison surface will have energy greater than or equal to the given surface, and if the energy is equal, the comparison surface is simply a translation or rotation of the given surface. Eigenvalue conditions are derived which imply that a capillary surface is a strict local minimum modulo isometries, and are applied to the specific example of a liquid bridge between two parallel planes.

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“…Using general formulas for eigenvalues of Sturm-Liouville problems, it is known (see [3,15]) that λ 0m is the minimum of the Rayleigh quotient…”

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confidence: 99%