Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 32, pp. 1–11.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

DISCUSSION OF A UNIQUENESS RESULT IN “EQUILIBRIUM

CONFIGURATIONS FOR A FLOATING DROP”

RAYMOND TREINEN

Abstract. We analyze a uniqueness result presented by Elcrat, Neel, and

Siegel [1] for unbounded liquid bridges, and show that the proof they presented
is incorrect. We add a hypothesis to their stated theorem and prove that their

result holds under this condition. Then we use Chebyshev spectral methods to

approximate solutions to certain boundary value problems used to check this
hypothesis holds at least on a range of cases.

1. Introduction

In 2004 Elcrat, Neel, and Siegel published a collection of results on the floating
drop problem and the related floating bubble problem [1]. Physically, one can
visualize a drop of oil resting on a reservoir of water. The resulting free boundary
problem will not be described in detail here. This work has been held in high
esteem in the field of capillarity, which is evident in the review Robert Finn wrote
in Math Reviews for the paper [2]. We offer a select quote from that review here:

The problem of characterizing the configuration of a drop of liq-
uid floating in equilibrium on the surface of an infinite bath of
another liquid appeared initially in the second supplément to the
tenth book of Laplace’s Mécanique Céleste, in 1806, without de-
tailed treatment. It was later studied by Poisson in his “Nouvelle
Théorie. . . ”(1831), and discussed by Bowditch (1839) in his English
translation of Laplace’s treatise. These works were remarkable for
their time but far from complete, and there appears to be no fur-
ther mathematical discussion in the literature, prior to the present
study. That the problem was ignored so long despite its evident
theoretical and also practical interest is perhaps indicative of the
technical obstacles that have impeded a full formal description.

Also the present authors have not solved the problem completely,
but what they offer is impressive, with a direct hands-on approach.
The authors make clever use of new results that appeared in other
contexts during the past quarter century.

Given this perspective, it is unfortunate that the subject of this current paper
is a flaw in one of the proofs presented in that work. The theorem stated as [1,

2020 Mathematics Subject Classification. 35Q35, 76A02.

Key words and phrases. Capillarity; unbounded liquid bridges; uniqueness.
©2023. This work is licensed under a CC BY 4.0 license.
Submitted July 22, 2022. Published April 3, 2023.

1



2 R. TREINEN EJDE-2023/32

Theorem 3.2] treats the existence and uniqueness of a boundary value problem for
unbounded liquid bridges, and the proof of uniqueness is the topic of discussion
here. We state the theorem here as Conjecture 2.2. This flaw is seemingly in the
proof alone, and does not seem to be in the stated result. In what follows we will
analyze the presented proof and show how the published approach is seemingly not
able to be repaired. Then we offer an alternative approach and we give numerical
evidence that this alternate approach yields the (still unproven) result found in that
paper.

Before we proceed to the details, a short comment on the impact of this flaw
is appropriate. We note that the strongest results found in [1] are not effected by
this flaw, but those stronger results do not hold for the general cases of all physical
configurations of fluids. Specifically, they found that under some restrictions on the
associated surfaces tensions the floating drop problem is solved for any given drop
volume. This assumption implies all of the component surfaces for a floating drop
can be shown to be a graph over a base domain, and under this restriction all of
the results in [1] still hold. This assumption is discussed there, and with the goal of
avoiding the quite technical description of the floating drop problem, we refer the
reader to that work for the detailed criterion. It is when the problem was generalized
to admit all possible physical configurations that the result described below was
used. The results just described in that work are Theorem 4.1, Theorem 5.3,
Corollary 5.1 which have gaps, and Theorem 4.2 having no gap.

Finally, the layout of this paper is as follows. In Section 2 we present the un-
bounded liquid bridge and the theorem from [1]. In Section 3 we collect preliminary
results found in a paper by Vogel [7]. In Section 4 we analyze the proof given in
[1], and in Section 5 we prove a theorem on uniqueness with the addition of a hy-
pothesis. Then in Section 6 we describe numerical results that verify that this new
hypothesis holds over a subset of the total values needed.

2. Uniqueness of solutions for a boundary value problem involving
unbounded liquid bridges

We consider here the family of radially symmetric unbounded liquid bridges.
These bridges are modeled by solutions of the ordinary differential equations

dr

dφ
=
−r cosφ

ru+ sinφ
, (2.1)

du

dφ
=
−r sinφ

ru+ sinφ
(2.2)

over the range φ ∈ [0, π), that are subject to various initial values

r(φ0) = r0,

u(φ0) = u0

with 0 ≤ φ0 < π and r0, u0 > 0. Here the radial value is r = r(φ) and the height
of the interface above a fixed reference level is u = u(φ). A first observation is that
not all such initial values lead to an unbounded liquid bridge. Indeed, a special
curve of initial values

S = {(r0, u0) = (σ, T (σ)) : σ > 0}



EJDE-2023/32 FLOATING DROP 3

has been identified and studied in the papers of Siegel [3], Turkington [6], Vogel [7],
and Elcrat, Neel, and Siegel [1]. These initial values are precisely the initial values
for which the following hold for φ0 = π/2:

(1) r = r(φ) is (strictly) increasing for π/2 ≤ φ < π with

lim
φ↗π

r(φ) =∞,

and
(2) u = u(φ) is (strictly) decreasing for π/2 ≤ φ < π with

lim
φ↗π

u(φ) = 0.

An example of one such unbounded liquid bridge is shown in Figure 1.

0 0.5 1 1.5 2 2.5 3 3.5 4

r

-1

-0.5

0

0.5

1

1.5

2

2.5

u

Figure 1. An example unbounded liquid bridge with vertical
point at σ = 1, where the generating curve for the radially symmet-
ric surface is graphed. The reference height is indicated at u = 0
and the vertical line corresponds to the radius where the boundary
condition φ0 = 0 holds.

The function T = T (σ) is differentiable and increasing for 0 < σ <∞ with

lim
σ↘0

T (σ) = 0, lim
σ↘0

T ′(σ) =∞, lim
σ→∞

T (σ) =
√

2,

as illustrated in Figure 2.
Each solution of the system (2.1)-(2.2) subject to the initial conditions

r(π/2) = σ,

u(π/2) = T (σ)

is defined for φ ∈ [0, π), and each solution is called an unbounded liquid bridge
solution. Considering a fixed σ > 0 and for each φ0 ∈ [0, π), one can use these initial
conditions to find the unique point (r0, u0) = (r(φ0), u(φ0)) along the solution curve.
This then leads to the initial value problem of (2.1)-(2.2) with

r(φ0) = r0,



4 R. TREINEN EJDE-2023/32

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

r

0

0.2

0.4

0.6

0.8

1

1.2

1.4

u

T( )

Figure 2. The function T (σ) computed by the algorithm de-
scribed in Section 6 for σ ≥ 0.1 and smoothly interpolated from
the asymptotic behavior for smaller σ. This asymptotic expansion
was derived in [6] and can be found more simply stated in [7], and
plays no explicit role in the current work.

u(φ0) = u0

for φ ∈ [0, π), and this initial value problem is then uniquely solved by the same
unbounded liquid bridge we used to find those initial conditions.

One may consider multiple unbounded liquid bridges, leading to the possibilities
of other uniqueness results. Vogel [7] showed following result.

Theorem 2.1. Consider the solutions of (2.1)-(2.2) subject to

r(φ0) = r1,

u(φ0) = u0,

and also the solutions of (2.1)-(2.2) subject to

r(φ0) = r2,

u(φ0) = u0.

If both solutions are unbounded liquid bridges, then r1 = r2.

The following was claimed in ENS.

Conjecture 2.2 ([1, Theorem 3.2]). If the solutions of (2.1)-(2.2) subject to

r(φ0) = r0,

u(φ0) = u1,

and the solutions of (2.1)-(2.2) subject to

r(φ0) = r0,

u(φ0) = u2

are both unbounded liquid bridges, then u1 = u2.



EJDE-2023/32 FLOATING DROP 5

We follow the proof presented in [1], noting a specific inequality that was not
adequately justified. We will then follow with an observation that the reverse
inequality seems to have been verified in [1].

3. Preliminaries

The term profile curve will now be used to refer to the parametric image

{(r(φ), u(φ)) : 0 ≤ φ < π}

associated with an unbounded liquid bridge solution. The following results are from
Vogel [7].

Lemma 3.1. Let Γ = (r(φ), u(φ)) be a particular profile curve. Pick φ0 ∈ [0, π),
and let r0 = r(φ0). Let A be the set obtained by rotating the region bounded by
r = r0 and Γ, and let B be the set obtained by rotating the unbounded region
between Γ and the r-axis from r = r0 to r =∞ around the u-axis. Let |A| and |B|
denote the volume of each set. Then

|B| − |A| = 2πr0 sinφ0. (3.1)

This lemma is illustrated in Figure 3.

0 0.5 1 1.5 2 2.5 3 3.5 4

r

-1

-0.5

0

0.5

1

1.5

2

2.5

u

A

B

Figure 3. Generating curves describing the regions A and B from Lemma 3.1.

Lemma 3.2. No two distinct profile curves can cross twice.

The following result is contained in a remark in Vogel’s paper, and also is ex-
tended by his Theorem 2.1.

Lemma 3.3. Given two profile curves with vertical points at radii σ1 and σ2, if
σ2 > σ1, then u(φ0;σ2) > u(φ0;σ1). Conversely, if u(φ0;σ2) > u(φ0;σ1), then
σ2 > σ1.



6 R. TREINEN EJDE-2023/32

Next we will consider how the system (2.1)-(2.2) behaves as the location of the
vertical point moves by differentiating with respect to that parameter σ, the results
of which we denote by ṙ and u̇:

dṙ

dφ
= − cosφ

u̇r2 − ṙ sinφ

(ru+ sinφ)2
, (3.2)

du̇

dφ
= − sinφ

u̇r2 − ṙ sinφ

(ru+ sinφ)2
. (3.3)

The conditions that r(π/2;σ) = σ and u(π/2, σ) = T (σ) imply

ṙ(π/2, σ) = 1 and (3.4)

u̇(π/2, σ) = T ′(σ). (3.5)

Vogel shows solutions of this initial value problem satisfy:

Lemma 3.4. If u̇(0;σ) = 0, then ṙ(0;σ) < 0.

Lemma 3.5. If φ ∈ (0, π), then u̇(φ;σ) > 0.

4. The proof presented

The proof is by contradiction. Let Γ1 and Γ2 be two profile curves as above such
that r1(φ0) = r2(φ0) for some φ0 ∈ [0, π/2). We will assume that these curves exist
and are distinct. Upon possibly relabeling, we have u1(φ0) > u2(φ0).

Suppose φ0 is the largest such value of φ such that r1(φ0) = r2(φ0) and denote
that radius by ρ0. These are the leftmost such points, and ρ0 is the smallest such
radius. Let A1, B1, A2, and B2 be the corresponding sets from Lemma 3.1 for these
profile curves.

Consider the intersection points of these curves with r = ρ0. Define αU = u1(φ0)
and βU = u2(φ0) to be the upper intersection points, and define αL and βL to be
the lower intersection points. Define φ−1 ∈ (π/2, π) to be the inclination angle of
Γ1 at the crossing of r = ρ0 where u(φ−1 ) = αL, and similarly define φ−2 ∈ (π/2, π)
to be the angle where Γ2 crosses r = ρ0 and u2(φ−2 ) = βL. See Figure 4. We will
have some cases to consider, depending on these values.

Step 1 (The case that αL ≤ βL) This means αU > βU > βL ≥ αL. Since Γ1 and
Γ2 cannot cross more than once, in this case they cannot cross at all. Thus Γ1 lies
entirely outside the region bounded by Γ2 and the line r = ρ0. Thus σ1 < σ2 and
it follows that T (σ1) < T (σ2). This then implies that there is some φ1 ∈ (φ0, π/2)
such that u1(φ1) = u2(φ1). This contradicts Theorem 2.1 and eliminates the case
that αL ≤ βL.

Step 2 (The case that αL > βL) As above, if T (σ1) ≤ T (σ2), we contradict
Theorem 2.1. Thus T (σ1) > T (σ2) and σ1 > σ2. The goal here is to contradict
Lemma 3.1. To this end, we consider two more steps, the first of which is to show
that |B1| > |B2| and the second is to show that |A1| < |A2|.

Step 3 If αL ≤ βU , then Γ1 and Γ2 must cross somewhere above αL. Then,
as they cannot cross twice, they cannot cross below αL. Then, as αL > βL, this
implies the lower portion of Γ1 lies completely above the lower portion of Γ2 when
r > ρ0. Thus αL ≤ βU implies |B1| > |B2|.

If αL ≥ βU , then the part of Γ1 with r < ρ0 lies completely above the corre-
sponding part of Γ2. Thus, at u = βU , Γ1 has an inclination angle greater than
π/2, while Γ2 has an inclination angle less than π/2. Also, at r = ρ0, the lower



EJDE-2023/32 FLOATING DROP 7

0 0.5 1 1.5 2 2.5 3 3.5 4

r

-1

-0.5

0

0.5

1

1.5

2

2.5

u
2

(
0
,

U
)

(
0
,

U
)

(
0
,

L
)

(
0
,

L
)

1

Figure 4. The two (potentially) distinct curves Γ1 and Γ2.

part of Γ1 is above the lower part of Γ2. the goal is to show that Γ1 remains above
Γ2 for all r > ρ0. Assume the opposite: the two curves cross somewhere below
u = βU . At this crossing point the inclination angle of Γ2 would be less than that
of Γ2. At u = βU the inequality was reversed. This implies that there is a value
of u between βU and the value of u at the crossing point where both curves have
the same inclination angle. This leads to a contradiction Γ1 ≡ Γ2 using Siegel’s
uniqueness theorem [3] or alternatively Vogel’s Theorem 2.1. Thus the curves can-
not cross, and the lower part of Γ1 lies above the lower part of Γ2 for all r > ρ0.
Thus αL ≥ βU implies |B1| > |B2|, and we have established that |B1| > |B2| holds
for all possibilities in the case that αL > βL.

Step 4 The argument of [1] rests at this point on showing the volumes |A1| and
|A2| satisfy the inequality

|A1| ≤ |A2|.
It has been pointed out by one of the referees, however, that it is shown in [1] itself
that the angles φ−1 and φ−2 satisfy π/2 < φ−1 < φ−2 < π with

|A1| = 2πρ0(sinφ−1 − sinφ0) and |A2| = 2πρ0(sinφ−2 − sinφ0). (4.1)

Consequently, direct comparison of the quantities in (4.1) gives that the reverse
inequality

|A1| > |A2|
actually holds. In light of this, it was suggested by the other referee that we
remove the remaining detailed analysis of the proof presented in [1] where we showed
precisely where their argument breaks down.

5. A different approach

We first claim that given any φ0 ∈ [0, π/2] and any ρ0 > 0, there exists at least
one infinite bridge solution with r(φ0) = ρ0. We know from Vogel [7] that any such
solution must correspond to a unique σ > 0 so the initial conditions r(π/2) = σ



8 R. TREINEN EJDE-2023/32

and u(π/2) = T (σ) give the same solution. Moreover the entire family of infinite
liquid bridge solutions depends smoothly on σ for σ > 0. Vogel shows

lim
σ→0

u(0;σ)

r(0;σ)
= 0.

This along with the estimate

r(0;σ) = O
( 1√

log(1/σ)

)
implies u(0;σ)→ 0 as σ → 0. Vogel also shows√

σ

T (σ)
+ σ2 ≤ r(0;σ) ≤

√
2σ

T (σ)
+ σ2,

which implies r(0;σ) <∞ for σ <∞. We also have an upper bound on T by
√

2.
Starting with σ1 << 1 so that r(0;σ1) < ρ0 and smoothly increasing σ until

σ = ρ0 and thus r(φ; ρ0) > ρ0 for all φ 6= π/2, there is at least one σ ∈ (σ1, ρ0) for
which the desired condition r(φ0;σ) = ρ0 holds.

If there exist two distinct values σ2 and σ3 for which r(φ0;σ2) = r(φ0;σ3), then
it follows that there exists some σ ∈ (0,∞) for which ṙ(φ;σ) = 0. We conclude the
following result.

Theorem 5.1. If every solution of (2.1)-(2.2) with

r(π/2) = σ, (5.1)

u(π/2) = T (σ), (5.2)

for σ > 0 satisfies

ṙ(φ0;σ) > 0 for 0 ≤ φ ≤ π/2, (5.3)

then the uniqueness assertion of [1] holds. That is, Conjecture 2.2 holds with this
additional hypothesis that ṙ(φ;σ) > 0.

This author has not found a way to rigorously establish this new condition. In
the course of the rest of this paper we will give some numerical evidence that this
criteria holds.

6. Numerical study of ṙ

Given a radius σ > 0, we will first find the height T (σ) of the vertical point
on the unbounded liquid bridge there. In order to find this height, we will need
to solve (2.1)-(2.2) for φ ∈ [π/2, π) so that the solution has the required height
decay at infinity. We will adapt a recently developed Chebyshev spectral method
to achieve this. See [5] for an exposition of the methods used.

The equations (2.1)-(2.2) can be written as an equivalent system of three non-
linear ordinary differential equations, parametrized by the arclength s:

dr

ds
= cosψ,

du

ds
= sinψ,

dψ

ds
= u− sinψ

r
,



EJDE-2023/32 FLOATING DROP 9

where we still have r as the radius and u as the height of the interface, and we
introduce the inclination angle ψ, which merely satisfies ψ = φ−π. We will specify
boundary conditions at s = 0 and at some arclength s = ` > 0. The radius r(`)
meets a prescribed value b > 0, and the inclination angle ψ(`) = 0 is required there.
However, this value of the arclength ` is unknown, and the methods in [5] find this
arclength as part of solving the boundary value problem. The boundary conditions
are

r(0) = σ, (6.1)

r(`) = b, (6.2)

ψ(0) = −π/2, (6.3)

ψ(`) = 0. (6.4)

To summarize, s = 0 is the arclength where we prescribe the vertical point corre-
sponding to r = σ, and we prescribe a horizontal slope at the right endpoint r = b
where s = `.

The application of the Chebyshev spectral methods in [5] numerically solve
this nonlinear boundary value problem. We are primarily interested in the height
Tb(σ) := u(0; b) at the left endpoint.

We incrementally increase the prescribed value of b until the height Tb has incre-
mental changes within a small tolerance. We require 13 digits of relative error for
the boundary value problem, and the underlying Newton’s method requires 14 dig-
its of relative error. Then we use this converged value of Tb(σ) as an approximation
of T (σ). This left endpoint is at the radius r = σ, so we can sweep over a range of
those values to form the graph (σ, T (σ)). Our algorithm works well for values of σ
down to σ = 0.1, but smaller than that the problem becomes multi-scale in nature
and more specialized methods are required.

In light of the initial condition (3.5), we need a careful approach in our approx-
imation of T ′(σ). The interpolation of data points by Chebyshev polynomials is
known to be near optimal among polynomial interpolants. The basis of this inter-
polation is to pick the data points at so-called Chebyshev points. Roughly, if the
domain is [−1, 1], then those points are xj = cos(θj) where the angles θj are evenly
spaced in [0, π]. For domains such as the ones we are considering, as over a bounded
range of positive σ values, one translates and scales the problem. Then one can
compute the derivative of the interpolating polynomial with spectral accuracy by
using a so-called Chebyshev differentiation matrix D. If the grid values for T (σ)
are found at these Chebyshev points, and stored in a column vector T, then the
grid values of the derivative T ′(σ) are found as T′ = DT. See Trefethen [4] for
background.

Now we have the components necessary to construct numerical approximations
of the solutions of (3.2)-(3.3) with the boundary conditions (3.4)-(3.5) where we
need to append that system with the original (2.1)-(2.2) with the conditions that
r(π/2, σ) = σ and u(π/2, σ) = T (σ). Then we use Matlab’s ode45 to solve this
system from φ = π/2 to φ = 0 to obtain the “top” portion of the solution. Here
we ask for 11 digits of accuracy in both the absolute and relative error. We use
all values of σ considered and sweep out a region of the φṙ−plane in Figure 5.
This numerical experiment shows that the entirety of the foliation lies completely
in the upper half-plane, and thus ṙ is never negative for those σ values. We also
visualize this phenomenon by graphing the endpoint ṙ(0, σ) as a function of σ.



10 R. TREINEN EJDE-2023/32

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

rd
o
t

Figure 5. The solutions ṙ(φ) foliate a region of the φṙ−plane and
do not enter the lower half-plane.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

rd
o

t

Figure 6. The endpoint ṙ(0, σ) is graphed as a function of σ.
This is the lowest part of the curve ṙ(φ, σ) in many cases, but for
smaller values of σ the lowest part of the curve occurs at φ = π/2,
and this is graphed with a dotted line when relevant.

For many cases, this endpoint is the lowest part of the curve ṙ(φ, σ), however for
smaller values of σ the lowest part of the curve occurs at φ = π/2, and this is also
included in Figure 6. This figure gathers the most interesting parts of the graph of
minφ ṙ(φ, σ), though we have computed this function out to values of σ at 200, and
the function continues to be increasing and seemingly asymptotic to a height of 1.



EJDE-2023/32 FLOATING DROP 11

We have released the matlab code for these computations under an open source
license and hosted it on a software repository found at
https://github.com/raytreinen/Unbounded-Liquid-Bridges.git

Finally, the referees offered keen insight into the problem at hand, and the author
is grateful for their comments.

References

[1] David Siegel, Alan Elcrat, Robert Neel; Equilibrium configurations for a floating drop, J.

Math. Fluid Mech. 6 (2004), no. 4, 405–429.
[2] Robert Finn; Review of the article “equilibrium configurations for a floating drop. ” by alan

elcrat, robert neel, and david siegel, Mathematical Reviews MR2101889 (2005j:76017)

(2005).
[3] David Siegel; Height estimates for capillary surfaces, Pacific J. Math. , 88 (1980), no. 2,

471–515.

[4] Lloyd N. Trefethen; Spectral methods in matlab, SIAM (2000) 10, xviii+165.
[5] Ray Treinen; Spectral methods for capillary surfaces described by bounded generating curves,

https://arxiv.org/abs/2205.02931.

[6] Bruce Turkington; Height estimates for exterior problems of capillarity type, Pacific J. Math.
88 (1980), no. 2, 517–540.

[7] Thomas I. Vogel; Symmetric unbounded liquid bridges, Pacific J. Math. 103 (1982), no. 1,

205–241.

Raymond Treinen

Department of Mathematics, Texas State University, 601 University Dr., San Marcos,

TX 78666, USA
Email address: rt30@txstate.edu


	1. Introduction
	2. Uniqueness of solutions for a boundary value problem involving unbounded liquid bridges
	3. Preliminaries
	4. The proof presented
	5. A different approach
	6. Numerical study of 
	References