AMS 11 12 16

Structures and universality:

Rough solutions in geometric analysis and

the calculus of variations.

AMS Sectional meeting, Raleigh November 12, 2016

Shankar Venkataramani

Alan Newell

Toby Shearman

Dislocation

Energy driven pattern formation

Heated from below

Cooled from above

E

=

Z

W

(

Du

) +

✏

2

Z

F

(

D

2

u

)

Part I

convection rolls and pattern

formation

Pr = 1.4

courtesy of Eberhard Bodenschatz

From W. Meevasana and G. Ahlers,

θ

Line vs. point defects

Concentration of Energy

Convex and Concave Disclinations

From J. Liu and G. Ahlers,

Phys. Rev. E 55, 6950 (1997).

Twist

A Dislocation

Image Courtesy J. Lega

Square: Smooth function

θ

(

x, y

)

Circle: Smooth function

θ

(x, y)

Overlap:

∃

ϵ

∈

{

1

,

−

1

}

k

∈

Z

Identify

The “phase” space

“Tangent” bundle of the phase space

Symmetry allows more ”freedom” for directors

Vectors vs. directors

Defect solutions and effective energies

Basic field is

k

= “

r

✓

”.

E

↵

ective energy:

Analogy with Mumford-Shah functional. Sum of bulk part, line energy and

point defect energy.

Quantization:

r ⇥

k

=

X

i

⇡

d

i z

Part II

N

(

p

)

·

d

r

(

p

) = 0

I

≡

ds

2

=

d

r

(

p

)

·

d

r

(

p

)

Geometry: The Gauss Normal map

p

x

y

u

v

w

Φ

Elastic energy of a thin sheet

E

=

!

∥

γ

∥

2

+

ϵ

2

∥

κ

∥

2

κ

= ˆ

n

·

D

2

Φ

γ

= (

D

Φ

)

T

·

D

Φ

−

g

Stretching energy

Bending energy

Elastic energy

E

t

[ ] =

S

[ ] +

t

2

B

[

H, K

]

=

Z

⌦

Q

( )

dxdy

+

t

2

Z

⌦

(4

H

2

2

K

)

dxdy,

Lewicka and Pakzad (2011).

–limit:

lim

t

!

0

t

2

E

t

=

(R

⌦

(4

H

2

2

K

)

dxdy

2

W

2

,

2

iso

+

1

otherwise

x

y

u

v

w

Immersion

:

⌦

!

R

3

of the center surface.

Reference Riemannian metric

g

.

Hilbert’s Theorem

I

Hilbert (1901): In

E

there is no complete, analytic surface

of constant negative curvature.

I

Efimov (1968): In

E

there is no complete,

C

surface with

curvature

K



c <

0

.

I

Nash (1954): In

E

there exist complete,

C

surfaces with

K



c <

0

“Theorem” (A Quantitative Hilbert’s Theorem (TS, SV))

For the elastic energy given previously,

i)

inf

C1,1

E

.

exp

R

1/2

ii)

inf

What is a C^{1,1} solution?

Solutions:

Small slopes approximation:

det(

⇥⇥

w

) = 1

w

=

1

2

⇣

ax

2

y

a

⌘

.

4

see Fig. 4(e-f). Note that, if a hyperbolic surface is C2, every point is locally a (regular) saddle (as in Fig. 4(a)) and there-fore cannot contain branch points. Non-C2 immersions are therefore qualitatively di↵erent from C2 immersions in that

they admit 3-saddles (“monkey saddles”) and higher order saddles, which can mediate a local refinement of the buckling wavelength (See Fig. 5).

FIG. 4. (a-b) Small slope isometric immersions w0

4(x1, x2) and

w0₄(x₁, x₂) for constant Gaussian curvature K = 1. w0₄(x₁, x₂) is

con-structed by taking odd periodic reflections of the piece of w0

4(x1, x2)

bounded between the green lines. The mesh on both of these sur-faces correspond to their asymptotic lines. (c-d) Projection of the asymptotic lines of w0

4(x1, x2) and w04(x1, x2) onto the x1, x2 plane.

(e-f) Direction of the gradient rw along circles centered at the ori-gin. The regular saddle in (a) corresponds to a gradient field with winding number -1, so the gradient map is 1 to 1. The 4-saddle in (b) has winding number -3, so the gradient map is a 3 sheeted covering near the origin.

Multiple branch points can be introduced on the surface by replicating the above process at any point, not just the origin. For example, consider the surface w0₂(x₁, x₂) = x₁x₂ which is

ruled by the asymptotic lines x₁, x₂ = const. A branch point

can be added at (x₁, x₂) = (1/ p2, 1/ p2) by removing the

sec-tor x₁, x₂ 1/ p2 and in this region fitting three rotated and

translated copies of w0₆(x₁, x₂) = x₂(x₁ p3x₂) so that the

resulting surface has continuous partial derivatives across the cut; see Fig 5(a). Three more branch points b₂,1, b2,2, b2,3 at a radial distance of 1/4 from b_1,1 can be added along rays

emanating from b₁,1 that bisect the lines of inflection; see Fig

5(b). This construction can be continued so that at the n-th it-eration 3n new branch points are added at a radial distance of (1/2)n from the previous branch points. The surface w(x₁, x₂)

formed in the limit n _{! 1} is a fractal with an infinite number of subwrinkles in the region x₁ 0, x₂ 0, x2₁ + x2₂ 1, and

it satisfies [w, w] = 1. The solution can be extended by odd

periodic reflections to give a small-slopes isometric immer-sion of the unit disk with K = 1. To illustrate the wrinkling

behavior near the edge we map w to a strip geometry through a conformal map h[x + iy] = w[ex+iy]; see Figs. 5(c-d).

FIG. 5. Finite bending energy solutions to the Monge-Ampere equa-tion [w0_,w0_{] = 1. (a) Three subwrinkle solution created by}

insert-ing three rotated and translated copies of the solution w0

6(x1, x2) =

x₂(x₁ p3x₂) onto the solution w0

2(x1, x2) = x1x2 at a branch point.

(b) Nine subwrinkle solution created by inserting nine copies of

w0

12(x1, x2) = x2(x1 (2 +

p

3)x₂) at three branch points added onto the three subwrinkle solution. (c) Extension of the nine subwrinkle solution to the full circular domain. (d) The nine subwrinkle solution mapped to the strip geometry by a conformal map.

The existence of self-similar isometric immersions has im-plications to the modeling of non-Euclidean elastic sheets. As for the strip with = 1, the solution w0₂(x₁, x₂) is

har-monic yet the extension of w0₂(x₁, x₂) to an exact

isomet-ric immersion has divergent bending energy for R ' 1.25

with the bending content concentrated near the singular point

x₁ = x₂ ⇡ 1.25/ p2 [22]. We can isometrically immerse disks

with larger R by a global refinement of the wavelength i.e taking n > 2. These solutions increase the bending energy

globally. An energetically favorable alternative might be to introduce a branch point in the n = 2 solution near the

singu-lar point, and locally refining the wavelength instead. Indeed, numerics for = 1/3 in the strip geometry indicate that, even

within the small slopes approximation, localized self similar wrinkling profiles may be energetically preferred over global refinement of the wavelength [2, 23].

Crumpled sheets have an energy scale t5/3 _{which is} inter-mediate between the stretching and bending energies [32, 33]. In contrast, the existence of W2,2 _{isometric immersions for}

Index of a branch point

C

2

isometries are not

dense in

W

2,2

isometries!

Geodesics and Asymptotic lines: C

2

surfaces

Piecewise quadratic surfaces

w

(x, y

) =

(

xy

y

2

cot(✓

+

))

0



✓



✓

+

xy

+

y

2

cot(✓

))

✓



✓



0

Let us consider solutions of det(D

2

w

) =

1

w

is

C

1

,

1

.

All the straight lines

through any point

lie in a common plane.

Gemmer, J., Sharon, E., Shearman, T. & Venkataramani, S. C., Isometric immersions, energy

Construction of

C

1

,

1

Isometries

Lelieuvre Formulae:

I

Using these connections between

r

and

N

, restricting

ourselves to

K

=

1

surfaces:

N

⇥

N

= 0

I

Equivalently

(

N

⇥

N

)

+ (

N

⇥

N

)

= 0

and

r

=

N

⇥

N , r

=

N

⇥

N

Distributional Sine-Gordon equation

!

=

v

dv

u

du

d!

=

uv

du

^

dv

Intrinsic form of Sine-Gordon:

d

!

= sin( )

du

^

dv

On an appropriate covering space:

uv

“ = ” sin( )

2⇡

X

i

Optimal control for Coupled Pendulums

Minimize

k

k1

on the unit square [0

,

1]

over solutions of

=

sin .

For a simple pendulum,

the minimum scales

⇡