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Geometric variational problems involving competition between line and surface energy

Eliot Fried

Mathematical Soft Matter Unit Okinawa 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 (M´

• em. 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 of various lengths are dipped into and extracted from soapy water.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various lengths are dipped into and extracted from soapy water.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various lengths are dipped into and extracted from soapy water.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various lengths are dipped into and extracted from soapy water.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various lengths are dipped into and extracted from soapy water.

Experiments and photos by Aisa Biria

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various 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.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various 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.

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

Experimental motivation

In an inventive generalization of experiments conducted by Plateau (M´

• em. 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 of various 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 energetically favorable to reduce the area of the film in favor of bending the loop away from circular.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 3 / 24

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

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.

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

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.

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

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 system comprised by the loop and the film is given by E :=

• C

1 2aκ2 +

• S

σ, where κ denotes the curvature of C.

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

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 system comprised by the loop and the film is given by E :=

• C

1 2aκ2 +

• S

σ, 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 minimizers

• f the functional
• C

1 2a|κ − κ0|2 +

• S

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

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 4 / 24

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

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 system prefers a flat configuration with circular boundary if ν := R3σ a < 3.

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

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 system prefers a flat configuration with circular boundary if ν := R3σ a < 3. On this basis, they reasoned that a bifurcation to a flat, oval configuration should occur at ν = R3σ/a = 3.

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

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 system prefers a flat configuration with circular boundary if ν := R3σ a < 3. On this basis, they reasoned that a bifurcation to a flat, oval configuration should occur at ν = R3σ/a = 3. They also performed numerical experiments that seem to support this assertion.

Giomi & Mahadevan (Proc. R. Soc. A 468 (2012), 1851–1864) Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 5 / 24

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

Equilibrium conditions of Giomi & Mahadevan

The first variation condition δF = 0 requires that:

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

Equilibrium conditions of Giomi & Mahadevan

The first variation condition δF = 0 requires that:

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

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0.

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0.

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

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0.

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

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• 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,

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• 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,

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• 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, and cos ϑ = p · n|C, where n is a unit normal to S.

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• 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, and cos ϑ = 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).

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

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, β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• 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, and cos ϑ = 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 to determine a closed space curve: Efimov (Usp. Mat. Nauk 2 (1947), 193– 194), Fenchel (Bull. Amer. Math. Soc. 57 (1951), 44–54).

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 6 / 24

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

Planar specialization: Pressurized cylindrical tube

Considered by: L´ evy (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) Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 7 / 24

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}.

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.

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. . .

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 2R3 dθ + 2π R σ|xr(r, θ) × xθ(r, θ)| dr dθ.

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 2R3 dθ + 2π R σ|xr(r, θ) × xθ(r, θ)| dr dθ. Notice that the highest derivatives of x appear in the boundary term. . .

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 8 / 24

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

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 of E is, at best, determined up to such a transformation.

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

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 of E is, at best, determined up to such a transformation. For the simple choice xg(r, θ) = rˆ r(θ), which represents a circular disc of radius R and can thus be thought of as a base state, E specializes to yield a reference value of the energy E[xg] = π(1 + ν)a R , ν = R3σ a = πR2σ πa/R > 0.

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

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 of E is, at best, determined up to such a transformation. For the simple choice xg(r, θ) = rˆ r(θ), which represents a circular disc of radius R and can thus be thought of as a base state, E specializes to yield a reference value of the energy E[xg] = π(1 + ν)a R , ν = R3σ a = πR2σ πa/R > 0. For a circular disc of radius R, ν represents the ratio of the surface energy πR2σ of the film to the bending energy πa/R of the loop and

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

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 of E is, at best, determined up to such a transformation. For the simple choice xg(r, θ) = rˆ r(θ), which represents a circular disc of radius R and can thus be thought of as a base state, E specializes to yield a reference value of the energy E[xg] = π(1 + ν)a R , ν = R3σ a = πR2σ πa/R > 0. For a circular disc of radius R, ν represents the ratio of the surface energy πR2σ of the film to the bending energy πa/R of the loop and is the only dimensionless parameter entering the problem.

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

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 of E is, at best, determined up to such a transformation. For the simple choice xg(r, θ) = rˆ r(θ), which represents a circular disc of radius R and can thus be thought of as a base state, E specializes to yield a reference value of the energy E[xg] = π(1 + ν)a R , ν = R3σ a = πR2σ πa/R > 0. For a circular disc of radius R, ν represents the ratio of the surface energy πR2σ of the film to the bending energy πa/R of the loop and is the only dimensionless parameter entering the problem. There are three conceivable ways to adjust ν, the simplest of which is to alter R while holding σ and a fixed.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 9 / 24

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

Equilibrium conditions for the recast problem

The first variation condition δE = 0 yields two equilibrium conditions, a scalar 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,

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

Equilibrium conditions for the recast problem

The first variation condition δE = 0 yields two equilibrium conditions, a scalar 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-order

• rdinary-differential equation

[νxθ ×n + (xθθθ − λxθ)θ]r =R = 0, to be satisfied on the boundary of the disc of radius R.

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

Equilibrium conditions for the recast problem

The first variation condition δE = 0 yields two equilibrium conditions, a scalar 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-order

• rdinary-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.

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

Equilibrium conditions for the recast problem

The first variation condition δE = 0 yields two equilibrium conditions, a scalar 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-order

• rdinary-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 satisfaction

• f the constraint

|xθ|r =R = R, which must be imposed along with the equilibrium conditions.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 10 / 24

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

The problem for x and λ involves a second-order partial differential equation subject to a fourth-order boundary condition. Notwithstanding the important contributions of Agmon, Douglis & Nirenberg (Comm. Pure Appl. Math. 12 (1959), 623–727; 18 (1964), 35–92), this problem provides involves many new mathematical challenges, as do related problems that we will mention. The partial-differential equation is equivalent to H = 0.

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

The problem for x and λ involves a second-order partial differential equation subject to a fourth-order boundary condition. Notwithstanding the important contributions of Agmon, Douglis & Nirenberg (Comm. Pure Appl. Math. 12 (1959), 623–727; 18 (1964), 35–92), this problem provides involves many new 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 β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0,

where β is a constant force, per unit length, related to λ and κ via β = a R2

• λ + 3R2κ2

2

• .

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

The problem for x and λ involves a second-order partial differential equation subject to a fourth-order boundary condition. Notwithstanding the important contributions of Agmon, Douglis & Nirenberg (Comm. Pure Appl. Math. 12 (1959), 623–727; 18 (1964), 35–92), this problem provides involves many new 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 β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0,

where β is a constant force, per unit length, related to λ and κ via β = a R2

• λ + 3R2κ2

2

• .

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

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

The problem for x and λ involves a second-order partial differential equation subject to a fourth-order boundary condition. Notwithstanding the important contributions of Agmon, Douglis & Nirenberg (Comm. Pure Appl. Math. 12 (1959), 623–727; 18 (1964), 35–92), this problem provides involves many new 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 β′ a t −

• κ′′ + 1

2κ3 −

• τ 2 + β

a

• κ − σ sin ϑ

a

• p

−

• 2κ′τ + κτ ′ + σ cos ϑ

a

• b = 0,

where β is a constant force, per unit length, related to λ and κ via β = a R2

• λ + 3R2κ2

2

• .

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

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 11 / 24

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

Second variation condition for the recast problem

A parametrization x satisfying the equilibrium conditions is stable if the second-variation condition δ2F = 2π (|uθθ|2 + λ|uθ|2)|r =R dθ + 2π R 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.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 12 / 24

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.

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 solution ξ(ρ, θ) = ρˆ r(θ) is a disk of (dimensionless) radius unity and the correspond- ing value of the Lagrange multiplier λ is λ = −(1 + ν).

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 solution ξ(ρ, θ) = ρˆ 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 and w of the flat, circular solution: u = vˆ r + wˆ r × ˆ θ.

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 solution ξ(ρ, θ) = ρˆ 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 and w of the flat, circular solution: u = vˆ r + wˆ r × ˆ 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 2

θ − v 2)]ρ=1 dθ ≥ 0,

2π [w 2

θθ − (1 + ν)w 2 θ]ρ=1 dθ +

2π R ν

• ρw 2

ρ + 1

ρw 2

θ

• dρdθ ≥ 0.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 13 / 24

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

The left-hand side of the first inequality admits a minimum if and only if ι ≥ 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 choice v(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 if γ ≥ 0 for every solution of ν

• wρρ + 1

ρwρ + 1 ρ2 wθθ

• + γw = 0,

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0, which is the case if ν ≤ 6. Evaluating the inequality for the particular choice w(ρ, θ) = ρ2 sin 2θ shows that the foregoing condition is also necessary.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 14 / 24

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

The left-hand side of the first inequality admits a minimum if and only if ι ≥ 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 choice v(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 if γ ≥ 0 for every solution of ν

• wρρ + 1

ρwρ + 1 ρ2 wθθ

• + γw = 0,

[wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0, which is the case if ν ≤ 6. Evaluating the inequality for the particular choice w(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.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 14 / 24

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 λ possesses a nontrivial solution branch that bifurcates from the flat, circular solution branch only if the linearized equations have a nontrivial solution. To linearize about the flat, circular solution, consider ξ = ρˆ r + η + wˆ r × ˆ θ, λ = −(1 + ν) + ǫ, where η obeys η · (ˆ r × ˆ θ) = 0 and is, thus, planar.

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 λ possesses a nontrivial solution branch that bifurcates from the flat, circular solution branch only if the linearized equations have a nontrivial solution. To linearize about the flat, circular solution, consider ξ = ρˆ r + η + wˆ r × ˆ θ, λ = −(1 + ν) + ǫ, where η obeys η · (ˆ r × ˆ θ) = 0 and is, thus, planar. The linearized problem for η and ǫ is [ηθθθθ + (1 + ν)ηθθ − ν(ˆ r · ηθ)ˆ θ]ρ=1 + ǫˆ r − ǫθˆ θ = 0, ˆ θ · ηθ|ρ=1 = 0,

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 λ possesses a nontrivial solution branch that bifurcates from the flat, circular solution branch only if the linearized equations have a nontrivial solution. To linearize about the flat, circular solution, consider ξ = ρˆ r + η + wˆ r × ˆ θ, λ = −(1 + ν) + ǫ, where η obeys η · (ˆ r × ˆ θ) = 0 and is, thus, planar. The linearized problem for η and ǫ is [ηθθθθ + (1 + ν)ηθθ − ν(ˆ r · ηθ)ˆ θ]ρ=1 + ǫˆ r − ǫθˆ θ = 0, ˆ θ · ηθ|ρ=1 = 0, while that for w is ρ(ρwρ)ρ + wθθ = 0, [wθθθθ + (1 + ν)wθθ + νwρ]ρ=1 = 0.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 15 / 24

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

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

The in-plane problem has nontrivial solutions η(1, θ) = m(D1 sin mθ + D2 cos mθ)ˆ r + m(D1 cos mθ − D2 sin mθ)ˆ θ, ǫ(θ) = −3νm(D1 sin mθ + D2 cos mθ),

• 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 solution

• branch. All remaining modes m ≥ 3 describe unstable bifurcations.

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

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 16 / 24

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

Obervations

As ν increases monotonically from some value ν0 < 3, a stable bifurcation to a flat, noncircular shape occurs at ν = 3. Any other bifurcation solution branch that emanates from the flat, circular solution branch is unstable. Any stable nonplanar solution branch must emanate from the stable branch

• f flat but noncircular solutions.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 17 / 24

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

Obervations

As ν increases monotonically from some value ν0 < 3, a stable bifurcation to a flat, noncircular shape occurs at ν = 3. Any other bifurcation solution branch that emanates from the flat, circular solution branch is unstable. Any stable nonplanar solution branch must emanate from the stable branch

• f 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

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 17 / 24

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

• ut-of-plane bifurcation suppressed

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 18 / 24

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

• ut-of-plane bifurcation suppressed

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

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 18 / 24

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

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.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 19 / 24

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

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 filament with an elliptical cross-section having minor axis in the plane of the loop.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 19 / 24

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• a1(κ2 − ¯

κ2)2 + a2(κ1 − ¯ κ1)2 + b(ω − ¯ ω)2 . Here:

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the previously encountered vector curvature of C.

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the previously encountered vector curvature of C. ω is a measure of twist density given by ω = t · (d × d′).

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the 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.

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the 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.

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the 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 an inextensible, unshearable rod.

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

To gain control over bifurcation pathways, it suffices to replace

• C

1 2aκ2 by

• C

1 2

• 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, where t is the unit tangent of C, d is a unit vector field orthogonal to t and

• riented along the minor axis of the cross-section of the filament, and κ is

the 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 an inextensible, unshearable rod. Taking a1 = a2 = a > 0, b = 0, and ¯ κ1 = ¯ κ2 = ¯ ω = 0 reduces it to

• C

1 2aκ2.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 20 / 24

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

Sample results: Moderate curvature mismatch regime

Set ¯ κ1 = ¯ ω = 0 and introduce the curvature mismatch ζ = 1 − R¯ κ2.

ζ = 1 ζ = 1 2 ζ = 0

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

Sample results: Moderate curvature mismatch regime

Set ¯ κ1 = ¯ ω = 0 and introduce the curvature mismatch ζ = 1 − R¯ κ2.

ζ = 1 ζ = 1 2 ζ = 0

For ζ < 1

2, the flat, circular solution branch becomes unstable at

ν = R3σ a1 = min

• 3, 6ζ(α2 + α3 − ζ) + 18α2α3

α2 + 4α3 − ζ

• ,

α2 = a2 a1 , α3 = b a1 .

relative importance of transverse modes relative importance of twisting modes Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 21 / 24

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

Dissipative dynamics

Instead of gradient flow versions of the equilibrium conditions, consider xθ × nr + nθ × xr = η0(n · ∆˙ x)n

• n [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 = (η1 ˙ κ1xθ × d − η2 ˙ κ2d)θ and 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 = (η3 ˙ ωxθ)θ

• n [0, 2π], where ηi ≥ 0, i = 0, 1, 2, 3 are viscosities and λ1, λ2, and λ3 are

multipliers associated with the constraints on x and d.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 22 / 24

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.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 23 / 24

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 if and only if ν ≤ 3.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 23 / 24

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 if and only if ν ≤ 3. A bifurcation analysis reveals that: A stable bifurcation from the trivial solution branch to the flat noncircular solution branch occurs at ν = 3. Any other bifurcation solution branch that emanates from the trivial solution branch — in particular any nonplanar solution branches — are unstable. Any stable nonplanar solution branch must emanate from the flat, noncircular solution branch.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 23 / 24

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 if and only if ν ≤ 3. A bifurcation analysis reveals that: A stable bifurcation from the trivial solution branch to the flat noncircular solution branch occurs at ν = 3. Any other bifurcation solution branch that emanates from the trivial solution branch — in particular any nonplanar solution branches — are unstable. Any stable nonplanar solution branch must emanate from the flat, noncircular solution branch. Numerical studies indicate that: A stable bifurcation from the flat, noncircular solution branch to a nonplanar solution 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.

Eliot Fried | RIKEN–Oaska–OIST Joint Workshop 2016 23 / 24

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

Replacing the simple bending energy

• C

1 2aκ2 by the energy for an inextensible,

unshearable Kirchhoff rod allows for different bifurcation pathways.

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

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

Replacing the simple bending energy

• C

1 2aκ2 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 sound alternative to more conventional gradient flow approaches.

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

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

Replacing the simple bending energy

• C

1 2aκ2 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 sound alternative 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.

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

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

Replacing the simple bending energy

• C

1 2aκ2 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 sound alternative 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 problem and its various generalizations provides a new set of challenges that might lead to further progress.

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