Geometric variational problems involving …Geometric variational problems involving competition...

Geometric variational problems involving competitionbetween line and surface energy

Eliot Fried

Mathematical Soft Matter UnitOkinawa Institute of Science and Technology Graduate University

Collaborators: Yi-chao Chen, Giulio Giusteri, Abdul Majid

Geometric variational problems involving competition between line and surface energy | Outline

1 Euler–Plateau problem

2 Planar specialization

3 Recasting of the Euler–Plateau problem in parametric form

4 Stability of flat, circular solutions

5 Bifurcation from flat, circular solutions

6 Extensions of the Euler–Plateau problem

7 Synopsis and discussion

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 2 / 24

Geometric variational problems involving competition between line and surface energy | Euler–Plateau problem

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (Mem. Acad.Sci. Belgique 23 (1849), 1–151), Giomi & Mahadevan (Proc. R. Soc. A 468(2012), 1851–1864) explored what happens when closed loops of fishing line ofvarious lengths are dipped into and extracted from soapy water.

Experiments and photos by Aisa Biria

For a loop of given length, a flat circular disk has maximal area and, thus,maximum surface energy.

A circular loop has minimum bending energy.

If the length of the loop exceeds a certain threshold, it becomes energeticallyfavorable to reduce the area of the film in favor of bending the loop awayfrom circular.

Formulation of Giomi & Mahadevan

Following the conventional approach to the Plateau problem, the soap film is mod-eled as a surface S with uniform tension σ > 0.

Motivated by Euler’s work on the elastica, the fishing line is modeled as an inex-tensible ring with uniform flexural rigidity a > 0 and rectilinear rest configuration,the midline of which coincides with the boundary C = ∂S of S.

If gravitational effects are negligible, then the net potential-energy E of the systemcomprised by the loop and the film is given by

12aκ2 +

∫Sσ,

where κ denotes the curvature of C.

Overlooked contribution: Granted that C is free of self-contact, Bernatzki & Ye(Ann. Glob. Anal. Geom. 19 (2001), 1–9) established the existence of minimizersof the functional ∫

12a|κ− κ0|2 +

∫Sσ,

where κ denotes the vector curvature of C and κ0 is an intrinsic vector curvature.

12aκ2 +

∫Sσ,

12a|κ− κ0|2 +

∫Sσ,

12aκ2 +

∫Sσ,

12a|κ− κ0|2 +

∫Sσ,

12aκ2 +

∫Sσ,

12a|κ− κ0|2 +

∫Sσ,

Analytical and numerical results of Giomi & Mahadevan

Using an energy comparison argument based on a particular trial solution,Giomi & Mahadevan find that for a ring of length L = 2πR the systemprefers a flat configuration with circular boundary if

ν :=R3σ

On this basis, they reasoned that a bifurcation to a flat, oval configurationshould occur at ν = R3σ/a = 3. They also performed numerical experimentsthat seem to support this assertion.

Giomi & Mahadevan (Proc. R. Soc. A 468 (2012), 1851–1864)

ν :=R3σ

On this basis, they reasoned that a bifurcation to a flat, oval configurationshould occur at ν = R3σ/a = 3.

They also performed numerical experimentsthat seem to support this assertion.

ν :=R3σ

On this basis, they reasoned that a bifurcation to a flat, oval configurationshould occur at ν = R3σ/a = 3. They also performed numerical experimentsthat seem to support this assertion.

Equilibrium conditions of Giomi & Mahadevan

The first variation condition δF = 0 requires that:

At all points on the surface S, H = 0.

At all points on the boundary C = ∂S,

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

2κ′τ + κτ ′ +σ cosϑ

)b = 0.

β is a Lagrange multiplier needed to ensure the inextensibility of C,a prime signifies differentiation with respect to arclength along C,{t, p, b} is the Frenet frame of C,κ and τ are the curvature and torsion of C, andcosϑ = p · n|C , where n is a unit normal to S.

For σ = 0, the condition on C is classical. See, for example, Langer & Singer(J. Diff. Geom. 20 (1984), 10–22).

Requiring κ and τ to be smooth and periodic does not generally suffice todetermine a closed space curve: Efimov (Usp. Mat. Nauk 2 (1947), 193–194), Fenchel (Bull. Amer. Math. Soc. 57 (1951), 44–54).

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

β is a Lagrange multiplier needed to ensure the inextensibility of C,

a prime signifies differentiation with respect to arclength along C,{t, p, b} is the Frenet frame of C,κ and τ are the curvature and torsion of C, andcosϑ = p · n|C , where n is a unit normal to S.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

β is a Lagrange multiplier needed to ensure the inextensibility of C,a prime signifies differentiation with respect to arclength along C,

{t, p, b} is the Frenet frame of C,κ and τ are the curvature and torsion of C, andcosϑ = p · n|C , where n is a unit normal to S.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

β is a Lagrange multiplier needed to ensure the inextensibility of C,a prime signifies differentiation with respect to arclength along C,{t, p, b} is the Frenet frame of C,

κ and τ are the curvature and torsion of C, andcosϑ = p · n|C , where n is a unit normal to S.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

β is a Lagrange multiplier needed to ensure the inextensibility of C,a prime signifies differentiation with respect to arclength along C,{t, p, b} is the Frenet frame of C,κ and τ are the curvature and torsion of C,

andcosϑ = p · n|C , where n is a unit normal to S.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0.

Geometric variational problems involving competition between line and surface energy | Planar specialization

Planar specialization: Pressurized cylindrical tube

Considered by:

Levy (J. Math. Pures Appl. 10 (1884), 5–42)

Halphen (Fonctions elliptiques, Gauthier-Villars et fils, Paris, 1888)

Greenhill (Math. Ann. 52 (1889), 465–500)

Carrier (J. Math. Phys. 26 (1947), 94–103)

Tadjbakhsh & Odeh (J. Math. Anal. Appl. 18 (1967), 59–74)

Flaherty, Keller & Rubinow (SIAM J. Appl. Math. 23 (1972), 446–455)

Watanabe & Takagi (Japan J. Indust. Appl. Math. 25 (2008), 331–372)

Giomi (Soft Matter 9 (2013), 8121–8139)ν = 3.250 ν = 4.750 ν = 5.247

Flaherty, Keller & Rubinow (SIAM J. Appl. Math. 23 (1972), 446–455)

Geometric variational problems involving competition between line and surface energy | Recasting of the Euler–Plateau problem in parametric form

Recast Euler–Plateau problem (with Yi-chao Chen)

Suppose that S admits a (sufficiently) smooth parametrization

S = {x ∈ R3 : x = x(r , θ), 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}.

Then C = ∂S is parametrized according to

C = {x ∈ R3 : x = x(R, θ), 0 ≤ θ ≤ 2π},

where, to ensure that the boundary is inextensibile, x must satisfy

|xθ(R, θ)| = R, 0 ≤ θ ≤ 2π.

Periodicity requires that x(r , 0) = x(r , 2π) for 0 < r ≤ R and so on. . .

E can then be represented as a functional of x:

E [x] =

∫ 2π

a|xθθ(R, θ)|2

2R3dθ +

∫ 2π

σ|xr (r , θ)× xθ(r , θ)| dr dθ.

Notice that the highest derivatives of x appear in the boundary term. . .

S = {x ∈ R3 : x = x(r , θ), 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}.

C = {x ∈ R3 : x = x(R, θ), 0 ≤ θ ≤ 2π},

|xθ(R, θ)| = R, 0 ≤ θ ≤ 2π.

E [x] =

∫ 2π

a|xθθ(R, θ)|2

2R3dθ +

∫ 2π

S = {x ∈ R3 : x = x(r , θ), 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}.

C = {x ∈ R3 : x = x(R, θ), 0 ≤ θ ≤ 2π},

|xθ(R, θ)| = R, 0 ≤ θ ≤ 2π.

E [x] =

∫ 2π

a|xθθ(R, θ)|2

2R3dθ +

∫ 2π

S = {x ∈ R3 : x = x(r , θ), 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}.

C = {x ∈ R3 : x = x(R, θ), 0 ≤ θ ≤ 2π},

|xθ(R, θ)| = R, 0 ≤ θ ≤ 2π.

E [x] =

∫ 2π

a|xθθ(R, θ)|2

2R3dθ +

∫ 2π

S = {x ∈ R3 : x = x(r , θ), 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}.

C = {x ∈ R3 : x = x(R, θ), 0 ≤ θ ≤ 2π},

|xθ(R, θ)| = R, 0 ≤ θ ≤ 2π.

E [x] =

∫ 2π

a|xθθ(R, θ)|2

2R3dθ +

∫ 2π

Invariance and scaling of the recast energy functional

Given an orthogonal linear transformation Q and a vector c, it follows that

E [Qx + c] = E [x]

and, thus, that E is invariant under rigid transformations. Any minimizer ofE is, at best, determined up to such a transformation.

For the simple choice xg (r , θ) = r r(θ), which represents a circular disc ofradius R and can thus be thought of as a base state, E specializes to yielda reference value of the energy

E [xg ] =π(1 + ν)a

R, ν =

a=πR2σ

πa/R> 0.

For a circular disc of radius R, ν represents the ratio of the surface energyπR2σ of the film to the bending energy πa/R of the loop and is the onlydimensionless parameter entering the problem.

There are three conceivable ways to adjust ν, the simplest of which is toalter R while holding σ and a fixed.

E [Qx + c] = E [x]

E [xg ] =π(1 + ν)a

R, ν =

a=πR2σ

πa/R> 0.

E [Qx + c] = E [x]

E [xg ] =π(1 + ν)a

R, ν =

a=πR2σ

πa/R> 0.

For a circular disc of radius R, ν represents the ratio of the surface energyπR2σ of the film to the bending energy πa/R of the loop and

is the onlydimensionless parameter entering the problem.

E [Qx + c] = E [x]

E [xg ] =π(1 + ν)a

R, ν =

a=πR2σ

πa/R> 0.

E [Qx + c] = E [x]

E [xg ] =π(1 + ν)a

R, ν =

a=πR2σ

πa/R> 0.

Equilibrium conditions for the recast problem

The first variation condition δE = 0 yields two equilibrium conditions, ascalar second-order partial-differential equation

n · (xθ×nr + nθ×xr ) = 0, n =xr ×xθ|xr ×xθ|

to be satisfied on the interior of the disc of radius R,

and a vector fourth-orderordinary-differential equation

[νxθ×n + (xθθθ − λxθ)θ]r =R = 0,

to be satisfied on the boundary of the disc of radius R.

The unknowns are the parametrization x and a Lagrange multiplier λ.

The ratio aλ/R2 is the reactive force density needed to ensure satisfactionof the constraint

|xθ|r =R = R,

which must be imposed along with the equilibrium conditions.

to be satisfied on the interior of the disc of radius R, and a vector fourth-orderordinary-differential equation

|xθ|r =R = R,

The problem for x and λ involves a second-order partial differential equationsubject to a fourth-order boundary condition. Notwithstanding the importantcontributions of Agmon, Douglis & Nirenberg (Comm. Pure Appl. Math. 12(1959), 623–727; 18 (1964), 35–92), this problem provides involves manynew mathematical challenges, as do related problems that we will mention.

The partial-differential equation is equivalent to H = 0.

The ordinary-differential equation is equivalent to

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0,

where β is a constant force, per unit length, related to λ and κ via

3R2κ2

A smoothly periodic x satisfying the equilibrium conditions determines κ,τ , and ϑ. Since the closed-curve problem remains unresolved, the converseassertion is not generally true.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0,

3R2κ2

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0,

3R2κ2

A smoothly periodic x satisfying the equilibrium conditions determines κ,τ , and ϑ.

Since the closed-curve problem remains unresolved, the converseassertion is not generally true.

(κ′′ +

2κ3 −

(τ 2 +

)κ− σ sinϑ

)b = 0,

3R2κ2

Second variation condition for the recast problem

A parametrization x satisfying the equilibrium conditions is stable if thesecond-variation condition

δ2F =

∫ 2π

(|uθθ|2 + λ|uθ|2)|r =R dθ

∫ 2π

ν( |P(ur ×xθ + xr ×uθ)|2

|xr ×xθ|+ 2m · (ur ×uθ)

)dr dθ ≥ 0

holds for all admissible variations u = δx, where

P = I−m⊗m

is the perpendicular projector onto the tangent space of S.

To be admissible, u must satisfy

xθ · uθ|r =R = 0.

Geometric variational problems involving competition between line and surface energy | Stability of flat, circular solutions

Stability of flat, circular solutions

Let ρ = r/R and ξ(ρ, θ) = x(r , θ)/R.

Then the flat, circular solutionξ(ρ, θ) = ρ r(θ) is a disk of (dimensionless) radius unity and the correspond-ing value of the Lagrange multiplier λ is

λ = −(1 + ν).

Express the variation u in terms of radial and transverse perturbations v andw of the flat, circular solution:

u = v r + w r× θ.

The second-variation condition then yields decoupled inequalities for the ra-dial and transverse perturbations v and w :∫ 2π

[(vθθ + v)2 − ν(v 2θ − v 2)]ρ=1 dθ ≥ 0,

∫ 2π

[w 2θθ − (1 + ν)w 2

θ ]ρ=1 dθ +

∫ 2π

(ρw 2

ρw 2θ

)dρdθ ≥ 0.

Let ρ = r/R and ξ(ρ, θ) = x(r , θ)/R. Then the flat, circular solutionξ(ρ, θ) = ρ r(θ) is a disk of (dimensionless) radius unity and the correspond-ing value of the Lagrange multiplier λ is

λ = −(1 + ν).

u = v r + w r× θ.

[(vθθ + v)2 − ν(v 2θ − v 2)]ρ=1 dθ ≥ 0,

∫ 2π

[w 2θθ − (1 + ν)w 2

θ ]ρ=1 dθ +

∫ 2π

(ρw 2

ρw 2θ

)dρdθ ≥ 0.

λ = −(1 + ν).

u = v r + w r× θ.

[(vθθ + v)2 − ν(v 2θ − v 2)]ρ=1 dθ ≥ 0,

∫ 2π

[w 2θθ − (1 + ν)w 2

θ ]ρ=1 dθ +

∫ 2π

(ρw 2

ρw 2θ

)dρdθ ≥ 0.

λ = −(1 + ν).

u = v r + w r× θ.

[(vθθ + v)2 − ν(v 2θ − v 2)]ρ=1 dθ ≥ 0,

∫ 2π

[w 2θθ − (1 + ν)w 2

θ ]ρ=1 dθ +

∫ 2π

(ρw 2

ρw 2θ

)dρdθ ≥ 0.

The left-hand side of the first inequality admits a minimum if and only ifι ≥ 0 for every solution of

[vθθθθ + (2 + ν)vθθ + (1 + ν − ι)v ]ρ=1 = 0,

which is the case if ν ≤ 3. Evaluating the inequality for the particular choicev(1, θ) = sin 2θ shows that the foregoing condition is also necessary.

The left-hand side of the second inequality admits a minimum if and only ifγ ≥ 0 for every solution of

(wρρ +

ρwρ +

ρ2wθθ

)+ γw = 0,

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0,

which is the case if ν ≤ 6. Evaluating the inequality for the particular choicew(ρ, θ) = ρ2 sin 2θ shows that the foregoing condition is also necessary.

Conclusion

The trivial solution is stable if and only if ν ≤ 3.

The left-hand side of the first inequality admits a minimum if and only ifι ≥ 0 for every solution of

[vθθθθ + (2 + ν)vθθ + (1 + ν − ι)v ]ρ=1 = 0,

which is the case if ν ≤ 3. Evaluating the inequality for the particular choicev(1, θ) = sin 2θ shows that the foregoing condition is also necessary.

The left-hand side of the second inequality admits a minimum if and only ifγ ≥ 0 for every solution of

(wρρ +

ρwρ +

ρ2wθθ

)+ γw = 0,

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0,

which is the case if ν ≤ 6. Evaluating the inequality for the particular choicew(ρ, θ) = ρ2 sin 2θ shows that the foregoing condition is also necessary.

Conclusion

The trivial solution is stable if and only if ν ≤ 3.

Geometric variational problems involving competition between line and surface energy | Bifurcation from flat, circular solutions

Bifurcation from flat, circular solutions

By the implicit function theorem, the boundary-value problem for ξ and λpossesses a nontrivial solution branch that bifurcates from the flat, circularsolution branch only if the linearized equations have a nontrivial solution.

To linearize about the flat, circular solution, consider

ξ = ρr + η + w r× θ, λ = −(1 + ν) + ε,

where η obeys η · (r× θ) = 0 and is, thus, planar.

The linearized problem for η and ε is

[ηθθθθ + (1 + ν)ηθθ − ν(r · ηθ)θ]ρ=1 + εr − εθθ = 0,

θ · ηθ|ρ=1 = 0,

while that for w is

ρ(ρwρ)ρ + wθθ = 0,

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0.

ξ = ρr + η + w r× θ, λ = −(1 + ν) + ε,

θ · ηθ|ρ=1 = 0,

while that for w is

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0.

ξ = ρr + η + w r× θ, λ = −(1 + ν) + ε,

θ · ηθ|ρ=1 = 0,

while that for w is

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0.

Without loss of generality, we neglect rigid transformations. Then:

The in-plane problem has nontrivial solutions

η(1, θ) = m(D1 sinmθ + D2 cosmθ)r + m(D1 cosmθ − D2 sinmθ)θ,

ε(θ) = −3νm(D1 sinmθ + D2 cosmθ),

}m ≥ 2,

where D1 and D2 are constants and ν must obey

ν = m2 − 1 ≥ 3.

The out-of-plane problem has nontrivial solutions

w(ρ, θ) = ρn(C1 cos nθ + C2 sin nθ), n ≥ 2,

where C1 and C2 are constants and ν must obey

ν = n(n + 1) ≥ 6.

Conclusion

The mode m = 2 describes a stable bifurcation to a flat, noncircular solutionbranch. All remaining modes m ≥ 3 describe unstable bifurcations.

All choices of the mode n ≥ 2 describe unstable out-of-plane bifurcations.

Obervations

As ν increases monotonically from some value ν0 < 3, a stable bifurcationto a flat, noncircular shape occurs at ν = 3.

Any other bifurcation solution branch that emanates from the flat, circularsolution branch is unstable.

Any stable nonplanar solution branch must emanate from the stable branchof flat but noncircular solutions.

Numerical results (with Abdul Majid)

ν = 4.30 ν = 4.42 ν = 4.50

ν = 4.65 ν = 4.77 ν = 4.92

Obervations

As ν increases monotonically from some value ν0 < 3, a stable bifurcationto a flat, noncircular shape occurs at ν = 3.

Any other bifurcation solution branch that emanates from the flat, circularsolution branch is unstable.

Any stable nonplanar solution branch must emanate from the stable branchof flat but noncircular solutions.

Numerical results (with Abdul Majid)

ν = 4.30 ν = 4.42 ν = 4.50

ν = 4.65 ν = 4.77 ν = 4.92

Geometric variational problems involving competition between line and surface energy | Extensions of the Euler–Plateau problem

Questions (with Giulio Giusteri)

Can the theory be modified to suppress in-plane or out-of-plane bifurcations?

ν = 3 ν ≈ 4.42

in-plane bifurcation suppressed

out-of-plane bifurcation suppressed

Is there a dissipative dynamical generalization of the theory that is ‘nice’ in thesense that it is both physically sound and mathematically useful?

Questions (with Giulio Giusteri)

Can the theory be modified to suppress in-plane or out-of-plane bifurcations?

ν = 3 ν ≈ 4.42

in-plane bifurcation suppressed

out-of-plane bifurcation suppressed

Is there a dissipative dynamical generalization of the theory that is ‘nice’ in thesense that it is both physically sound and mathematically useful?

Suppressing in-plane or out-of-plane bifurcations

In-plane bifurcations can be suppressed by making the loop from a filament with:

a circular cross-section having intrinsic curvature or twist density;

an elliptical cross-section having major axis in the plane of the loop.

Out-of-plane bifurcations can be suppressed by making the loop from a filamentwith an elliptical cross-section having minor axis in the plane of the loop.

Suppressing in-plane or out-of-plane bifurcations

In-plane bifurcations can be suppressed by making the loop from a filament with:

a circular cross-section having intrinsic curvature or twist density;

an elliptical cross-section having major axis in the plane of the loop.

Out-of-plane bifurcations can be suppressed by making the loop from a filamentwith an elliptical cross-section having minor axis in the plane of the loop.

To gain control over bifurcation pathways, it suffices to replace∫C

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 and κ2 are measures of curvature given by

κ1 = (t× d) · κ and κ2 = d · κ ,

where t is the unit tangent of C, d is a unit vector field orthogonal to t andoriented along the minor axis of the cross-section of the filament, and κ isthe previously encountered vector curvature of C.

ω is a measure of twist density given by

ω = t · (d× d′).

a1 > 0 and a2 ≥ 0 are flexural rigidities and b ≥ 0 is the twisting rigidity.

κ1 ≥ 0 and κ2 ≥ 0 are intrinsic curvatures and ω is the intrinsic twist density.

This is Kirchhoff’s (J. reine angew. Math. 56 (1859), 285–313) energy for aninextensible, unshearable rod. Taking a1 = a2 = a > 0, b = 0, and κ1 = κ2 =ω = 0 reduces it to

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

Here:κ1 and κ2 are measures of curvature given by

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

This is Kirchhoff’s (J. reine angew. Math. 56 (1859), 285–313) energy for aninextensible, unshearable rod.

Taking a1 = a2 = a > 0, b = 0, and κ1 = κ2 =ω = 0 reduces it to

12 aκ2.

12 aκ2 by∫

(a1(κ2 − κ2)2 + a2(κ1 − κ1)2 + b(ω − ω)2

κ1 = (t× d) · κ and κ2 = d · κ ,

ω = t · (d× d′).

12 aκ2.

Sample results: Moderate curvature mismatch regime

Set κ1 = ω = 0 and introduce the curvature mismatch ζ = 1− Rκ2.

ζ = 1ζ = 1

2ζ = 0

For ζ < 12 , the flat, circular solution branch becomes unstable at

ν =R3σ

a1= min

6ζ(α2 + α3 − ζ) + 18α2α3

α2 + 4α3 − ζ

}, α2 =

a1, α3 =

relative importance of transverse modes relative importance of twisting modes

Sample results: Moderate curvature mismatch regime

Set κ1 = ω = 0 and introduce the curvature mismatch ζ = 1− Rκ2.

ζ = 1ζ = 1

2ζ = 0

For ζ < 12 , the flat, circular solution branch becomes unstable at

ν =R3σ

a1= min

6ζ(α2 + α3 − ζ) + 18α2α3

α2 + 4α3 − ζ

}, α2 =

a1, α3 =

relative importance of transverse modes relative importance of twisting modes

Dissipative dynamics

Instead of gradient flow versions of the equilibrium conditions, consider

xθ × nr + nθ × xr = η0(n ·∆x)n

on [0,R)× [0, 2π] in conjunction with

(a1(κ2 − κ2)xθ × d + a2(κ1 − κ1)d)θθ

− (a1(κ2 − κ2)d× xθθ + b(ω − ω)d× dθ)θ

+ σxθ × n− (λ1xθ + λ3d)θ = (η1κ1xθ × d− η2κ2d)θ

a1(κ2 − κ2)xθθ × xθ + a2(κ1 − κ1)xθθ

− b((ω − ω)xθθ × d + (ω − ω)θxθ × d + 2(ω − ω)xθ × dθ)

+ λ2d + λ3xθ = (η3ωxθ)θ

on [0, 2π], where ηi ≥ 0, i = 0, 1, 2, 3 are viscosities and λ1, λ2, and λ3 aremultipliers associated with the constraints on x and d.

Geometric variational problems involving competition between line and surface energy | Synopsis and discussion

The Euler–Plateau problem has been recast to avoid the closed-curve problem.

A stability analysis reveals that a circular loop spanned by a flat film is stable ifand only if ν ≤ 3.

A bifurcation analysis reveals that:

A stable bifurcation from the trivial solution branch to the flat noncircularsolution branch occurs at ν = 3.

Any other bifurcation solution branch that emanates from the trivial solutionbranch — in particular any nonplanar solution branches — are unstable.

Any stable nonplanar solution branch must emanate from the flat, noncircularsolution branch.

Numerical studies indicate that:

A stable bifurcation from the flat, noncircular solution branch to a nonplanarsolution branch occurs at ν ≈ 4.42.

A stable bifurcation from the noncircular solution branch to a planar figure-eight like configuration occurs at ν ≈ 4.92.

Replacing the simple bending energy∫C

12 aκ2 by the energy for an inextensible,

unshearable Kirchhoff rod allows for different bifurcation pathways.

A framework for dissipative dynamics has been provided as a physically soundalternative to more conventional gradient flow approaches.

The Plateau problem inspired significant advances in differential geometry, varia-tional calculus, analysis, and various other areas of mathematics.

The class of geometrical variational problems including the Euler–Plateau problemand its various generalizations provides a new set of challenges that might leadto further progress.

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