The Geometry of the Triple Junction between Three Fluids in …blanki/WichitaGeometry.pdf · 2016....

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The Geometry of the Triple Junction between ThreeFluids in Equilibrium

Ivan Blank

Kansas State University

[email protected]

Wichita State UniversityThe Twenty First

Midwest Geometry Conference

Ivan Blank (KSU) The Triple Junction March 13, 2016 1 / 42

Main Reference

Most of this talk is based on joint work with Ray Treinen and Alan Elcratand can be found on arXiv right now.

Alan Elcrat passed away on December 20, 2013.

The vast majority of this work was completed while he was still alive ...We are sure that it would have been finished far sooner if he had notpassed away.


Main Reference





Main Reference



The vast majority of this work was completed while he was still alive ...

We are sure that it would have been finished far sooner if he had notpassed away.


Main Reference





Figure : Alan looking dapper.


What are we studying?

Figure : Permissible sets Ej.


What are we studying?

Figure : Permissible sets Ej.


A Short History - I

The study of the floating drop problem goes back at least to 1806 whenLaplace formulated the problem with the assumption of symmetry, and ofcourse, the regularity of the interfaces between the fluids and also theregularity of the triple junction curve.

It wasn’t until 2004 when Elcrat, Neel, and Siegel showed the existence(and, under some assumptions, uniqueness) of solutions for Laplace’sformulation, and they still assumed the same conditions of symmetry andregularity.


A Short History - I

The study of the floating drop problem goes back at least to 1806 whenLaplace formulated the problem with the assumption of symmetry, and ofcourse, the regularity of the interfaces between the fluids and also theregularity of the triple junction curve.

It wasn’t until 2004 when Elcrat, Neel, and Siegel showed the existence(and, under some assumptions, uniqueness) of solutions for Laplace’sformulation, and they still assumed the same conditions of symmetry andregularity.


A Short History - II

In the 1970’s Jean Taylor classified the structure of the singularities ofsoap film clusters, and among other results was able to show that at triplejunction points the surfaces meet at 120.

Frank Morgan and collaborators worked on various other aspects of soapbubble clusters, including showing that the standard double bubble is theunique energy minimizer in a collaboration with Hutchings, Ritore, andRoss.

If you set all surface tensions to unity in our problem and ignore gravity,then you can recover the mathematical model for the soap bubble clusters.












The Tastier Version ...

Figure : Triple Point in My Food


The Tastier Version ...

Figure : Triple Point in My Food


The Formulation: Physics!

We consider three immiscible fluids in a container which minimize the sumof:

...1 Surface Energy. Surface tension.

...2 Wetting Energy. Fluids like (or dislike) to touch the walls of thecontainer.

...3 Potential Energy. Gravity.




...1 Surface Energy.

Surface tension....2 Wetting Energy. Fluids like (or dislike) to touch the walls of thecontainer.












...2 Wetting Energy.

Fluids like (or dislike) to touch the walls of thecontainer.













...3 Potential Energy.

Gravity.








The Formulation: Mathematics! Part I

.Definition (Permissible configurations)..

......

The triple of open sets Ej is said to be a permissible configuration ormore simply “permissible” if

1. The Ej are sets of finite perimeter.

Their characteristic functionsbelong to the space of functions of bounded variation... More on thissoon.

2. The Ej are disjoint.

3. The union of their closures is Ω .

In a case where volumes are prescribed, in order for sets to beV-permissible we will add to this list a fourth item:

4. The volumes are prescribed: |Ej | = vj for j = 0, 1, 2.




......


1. The Ej are sets of finite perimeter. Their characteristic functionsbelong to the space of functions of bounded variation... More on thissoon.








......










......










......








Structural Assumptions

1. Surface tension at interface of Ei and Ej is σij .

2. Coefficient of wetting energy of Ei on the boundary is βi .

3. Density of the ith fluid is ρi .

4. Gravitational constant is g .

5. Ω is our container and B1 ⊂⊂ Ω ⊂ IRn.

α0 := 12(σ01 + σ02 − σ12)

α1 := 12(σ01 + σ12 − σ02)

α2 := 12(σ02 + σ12 − σ01) ,

(1)

and we will assumeαj > 0, for all j (2)

throughout our paper, and note that this condition is frequently called thestrict triangularity hypothesis. It ensures that the fluids don’t like to mix!


The Formulation: Mathematics! Part II

We want to minimize:

FSWP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ βj

∫∂Ω

χEjdHn−1 + ρjg

∫Ej

z dV

)

Prescribing either:

Volume, or

Boundary Dirichlet data, or

Both.




FSWP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ βj

∫∂Ω

χEjdHn−1 + ρjg

∫Ej

z dV

)

Prescribing either:

Volume, or


Both.




FSWP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ βj

∫∂Ω

χEjdHn−1 + ρjg

∫Ej

z dV

)

Prescribing either:

Volume, or


Both.




FSWP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ βj

∫∂Ω

χEjdHn−1 + ρjg

∫Ej

z dV

)

Prescribing either:

Volume, or


Both.




FSWP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ βj

∫∂Ω

χEjdHn−1 + ρjg

∫Ej

z dV

)

Prescribing either:

Volume, or


Both.


Quirkiness of the Dirichlet Problem...

Wasn’t sure where to put this slide, but ...

With a Dirichlet problem we would prescribe the fluid touching aboundary, but ... because BV functions can have jump discontinuitiesalong decent surfaces (and some rather indecent surfaces), the meaning ofprescribing Dirichlet data requires a comment:

You can swap fluids immediately on entry into your domain if you arewilling to pay for an interface.




With a Dirichlet problem we would prescribe the fluid touching aboundary, but ...

because BV functions can have jump discontinuitiesalong decent surfaces (and some rather indecent surfaces), the meaning ofprescribing Dirichlet data requires a comment:













The Formulation: Mathematics! Part III

In fact we are most interested in the geometry of a triple junction awayfrom the boundary of the container, so by scaling inward we can ignore thewetting energy.

Thus, we really want to minimize:

FSP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ ρjg

∫Ej

z dV

)


The Formulation: Mathematics! Part III

In fact we are most interested in the geometry of a triple junction awayfrom the boundary of the container, so by scaling inward we can ignore thewetting energy.

Thus, we really want to minimize:

FSP(Ej) :=2∑

j=0

(αj

∫Ω|Dχ

Ej|+ ρjg

∫Ej

z dV

)


BV Background - I

In choosing the surface energy term, the choices are:

Measure the interfaces with (n− 1)-dimensional Hausdorff measure, or

Use sets of finite perimeter.

When measuring boundaries of sets, perimeters work much more nicelythan (n − 1)-dimensional Hausdorff measure.

Why?


BV Background - I





Why?


BV Background - I





Why?


BV Background - I





Why?


BV Background - II

First we recall some facts about the space of functions of boundedvariation.

.Definition (BV)..

......

We define BV (Ω) to be the subset of L1(Ω) with bounded variation,measured by∫

Ω|Df | = sup

∫Ωf divϕ : ϕ ∈ C 1

c (Ω; IRn), |ϕ| ≤ 1

.

Equivalently, we say f is in BV (Ω) if it has a distributional derivativewhich is a vector-valued finite Radon measure.

Insofar as the weak derivative of a function in BV does not need to belongto L1 it is clear that BV (Ω) is larger than the Sobolev space W 1,1(Ω).


BV Background - II

First we recall some facts about the space of functions of boundedvariation..Definition (BV)..

......


Ω|Df | = sup


c (Ω; IRn), |ϕ| ≤ 1

.




BV Background - II


......


Ω|Df | = sup


c (Ω; IRn), |ϕ| ≤ 1

.




BV Background - II


......


Ω|Df | = sup


c (Ω; IRn), |ϕ| ≤ 1

.




BV Background - III

Just like Sobolev Spaces, the space of functions of bounded variation has...1 Compactness theorems,

...2 Density theorems, and

...3 Trace theorems.


BV Background - III

Just like Sobolev Spaces, the space of functions of bounded variation has...1 Compactness theorems,...2 Density theorems,

and...3 Trace theorems.


BV Background - III

Just like Sobolev Spaces, the space of functions of bounded variation has...1 Compactness theorems,...2 Density theorems, and...3 Trace theorems.


BV Background - IV

Obnoxiously clever/Cleverly obnoxious example:

...1 Let xj enumerate the points with all rational coordinates in IRn.

...2 Let rj denote radii ↓ 0 with the property that:

∞∑j=1

Surface area of ball with radius rj = C (n)∞∑j=1

rn−1j < ∞.

...3 LetDk := ∪k

j=1Brj (xj) .

For all finite k we have

Perimeter(Dk) = Hn−1(∂Dk) .


BV Background - IV

Obnoxiously clever/Cleverly obnoxious example:...1 Let xj enumerate the points with all rational coordinates in IRn.

...2 Let rj denote radii ↓ 0 with the property that:

∞∑j=1


rn−1j < ∞.

...3 LetDk := ∪k

j=1Brj (xj) .




BV Background - IV

Obnoxiously clever/Cleverly obnoxious example:...1 Let xj enumerate the points with all rational coordinates in IRn....2 Let rj denote radii ↓ 0 with the property that:

∞∑j=1


rn−1j < ∞.

...3 LetDk := ∪k

j=1Brj (xj) .




BV Background - IV


∞∑j=1


rn−1j < ∞.

...3 LetDk := ∪k

j=1Brj (xj) .




BV Background - IV


∞∑j=1


rn−1j < ∞.

...3 LetDk := ∪k

j=1Brj (xj) .




BV Background - V

However ...

Notice that lower semicontinuity implies:

Perimeter(D∞) ≤ C (n)∞∑j=1

rn−1j < ∞

which certainly seems reasonable. On the other hand, because D∞contains a dense subset of IRn, it is clear that not only is

Hn−1(∂D∞) = ∞,

but in factLn(∂D∞) = ∞ .

So basically, perimeters behave much more nicely than Hausdorff measureof boundaries, and indeed:


BV Background - V

However ... Notice that lower semicontinuity implies:


rn−1j < ∞

which certainly seems reasonable.

On the other hand, because D∞contains a dense subset of IRn, it is clear that not only is

Hn−1(∂D∞) = ∞,




BV Background - V



rn−1j < ∞


Hn−1(∂D∞) = ∞,




BV Background - V



rn−1j < ∞


Hn−1(∂D∞) = ∞,




BV Background - V



rn−1j < ∞


Hn−1(∂D∞) = ∞,




General Background I

Working with sets of finite perimeter, in 1984 Massari proved:

.Theorem (Existence)..

......

Under the structural conditions that we give above, there exists aminimizer of FSWP .

Structural conditions? The most obvious thing is that the consants mustsatisfy conditions so that the fluids do not “want to mix.”

Also important for us was the fact that the Massari’s theorem allows thewetting term to be zero.

The fact that in this setting any interface between any two of the threefluids is smooth is by now classical and is certainly very well described byGiusti in his book Minimal Surfaces and Functions of Bounded Variation



Working with sets of finite perimeter, in 1984 Massari proved:.Theorem (Existence)..

......








......


Structural conditions?

The most obvious thing is that the consants mustsatisfy conditions so that the fluids do not “want to mix.”






......








......








......






General Background II

Almgren’s Volume Adjustment Lemma!

.Lemma (Almgren’s Volume Adjustment Lemma)..

......

Given any permissible triple Ej, there exists a C > 0, such that verysmall volume adjustments can be made at a cost to the energy which isnot more than C times the volume adjustment. Stated quantitatively:

∆FS ≤ C2∑

j=0

|∆Vj | (3)

where ∆Vj is the volume change of Ej .


General Background II

Almgren’s Volume Adjustment Lemma!.Lemma (Almgren’s Volume Adjustment Lemma)..

......

Given any permissible triple Ej, there exists a C > 0, such that verysmall volume adjustments can be made at a cost to the energy which isnot more than C times the volume adjustment. Stated quantitatively:

∆FS ≤ C2∑

j=0

|∆Vj | (3)

where ∆Vj is the volume change of Ej .


General Background III

Leonardi’s Elimination Theorem!

.Theorem (Leonardi’s Elimination Theorem)..

......

Under the assumptions above, if Ej is a V-minimizer, then Ej has theelimination property. Namely, there exists a constant η > 0, and a radiusr0 such that if 0 < ρ < r0, Br0 ⊂ Ω, and

|Ei ∩ Bρ(x)| ≤ ηρn , (4)

then|Ei ∩ Bρ/2(x)| = 0 . (5)


General Background III

Leonardi’s Elimination Theorem!.Theorem (Leonardi’s Elimination Theorem)..

......

Under the assumptions above, if Ej is a V-minimizer, then Ej has theelimination property. Namely, there exists a constant η > 0, and a radiusr0 such that if 0 < ρ < r0, Br0 ⊂ Ω, and

|Ei ∩ Bρ(x)| ≤ ηρn , (4)

then|Ei ∩ Bρ/2(x)| = 0 . (5)


What Is Open?

Obvious important open problems:

1. The regularity of the triple junction: We expect it to be a smoothcurve.

2. The interfaces should have “tangent half-planes” at the triplejunction, and the angles between these planes should satisfy the forcebalance relation given by Elcrat-Neel-Siegel.

3. To what extent can we ignore the gravitational term? You have allseen the “Jesus bugs” I’m sure.

All of these open problems, but especially the last one suggest doing“blow-up” arguments.


What Is Open?







What Is Open?







What Is Open?







What Is Open?




3. To what extent can we ignore the gravitational term?

You have allseen the “Jesus bugs” I’m sure.



What Is Open?







What Is Open?







Blow-up/Regularity Hierarchy

In order of increasing regularity...

0. There is no blow-up limit.

1. There exists a blow-up limit. (e.g. nice fractals and spirals)

2. Along any subsequence of rescalings, there exists a furthersubsequence which converges. (e.g. nice fractals and spirals)

3. All blow-up limits are cones (or more generally have some sort ofhomogeneity). (e.g. Spirals that are Reifenberg Vanishing sets.)

4. All blow-up limits are the same cones up to a rotation. (e.g. Spiralsthat are Reifenberg Vanishing sets.)

5. There exists a unique blow-up limit at each point and it is a specificcone.
























































Does 4 imply 5?

No!

Reifenberg Vanishing Sets giveexcellent examples.


Does 4 imply 5?

No!



Does 4 imply 5?

No!



Reifenberg vanishing sets

Take a sequence of rotations withangles ϵ/2, ϵ/3, ϵ/4, ... and usethe fact that the harmonic seriesdiverges.


Creating A Reifenberg Vanishing Set














Moral...

When thinking dyadically, there is aninfinite amount of space near zero, andwhen dealing with blowup limits you haveto think dyadically.


Moral...

When thinking dyadically, there is aninfinite amount of space near zero, andwhen dealing with blowup limits you haveto think dyadically.


Some Dyadic Rings...



Obstacle Problem Situations with Lower Regularity

People who know my work, know that the obstacle problem is near anddear to my heart.

In my dissertation [B 2001] I give an example of an obstacle problem(which is a minimization problem) where there are multiple blow up limitsat a point, but where the blow-ups are unique up to rotation. 4 but not 5...

More recently with my PhD student Zheng Hao [B-Hao 2015] we give anexample of an obstacle problem (still a minimization problem) where thereare different blow-up limits at the same point even after modding out byrotations. 3 but not 4...









In my dissertation [B 2001] I give an example of an obstacle problem(which is a minimization problem) where there are multiple blow up limitsat a point, but where the blow-ups are unique up to rotation.

4 but not 5...











More recently with my PhD student Zheng Hao [B-Hao 2015] we give anexample of an obstacle problem (still a minimization problem) where thereare different blow-up limits at the same point even after modding out byrotations.

3 but not 4...







New Results for the Three Fluid Problem - I

In this theorem we useχ

Ej,λi

to denote the characteristic function of Ej after it has been dilated by λ−1i .

.Theorem (Existence of blowup limits - BET)..

......

Assume that Ej is a D-minimizer or a V-minimizer of FSP in Ω. In eithercase, there exists a configuration (which we will denote by Ej ,0) and asequence of λi ↓ 0 such that for each j :

||χEj,λi

− χEj,0

||L1(B1) → 0 and DχEj,λi

∗ Dχ

Ej,0. (6)

Furthermore, the triple Ej ,0 is a D-minimizer of FS for whateverDirichlet data it has in the first case or a V-minimizer of FS for whatevervolume constraints it satisfies in the second case.

This theorem settles 1 and 2 from my list...




Ej,λi



......


||χEj,λi

− χEj,0


∗ Dχ

Ej,0. (6)


This theorem settles 1 and 2 from my list...




Ej,λi



......


||χEj,λi

− χEj,0


∗ Dχ

Ej,0. (6)


This theorem settles 1 and 2 from my list...Ivan Blank (KSU) The Triple Junction March 13, 2016 37 / 42

New Results for the Three Fluid Problem - II

.Theorem (Blowups are cones - BET)..

......

The blowup limits from the last theorem are cones, and each cone haspositive density at the origin.

A large part of the proof involves creating a scaled version of energy whichis a monotone function of how far you scale.

This theorem settles 3 from my list...

Figure : Limiting configuration as cones.




......




Figure : Limiting configuration as cones.




......




Figure : Limiting configuration as cones.Ivan Blank (KSU) The Triple Junction March 13, 2016 38 / 42

New Results for the Three Fluid Problem - III

Looking at these cones in the sphere centered at the origin, you can doanother blow up limit.

Figure : Cones in the tangent plane to the blow up sphere.


New Results for the Three Fluid Problem - III

Looking at these cones in the sphere centered at the origin, you can doanother blow up limit.

Figure : Cones in the tangent plane to the blow up sphere.


New Results for the Three Fluid Problem - IV

.Theorem (Classification of blowup limits - BET)..

......

The angles for the sectors in the tangent plane to the blowup spheresatisfy the angle conditions given by Elcrat, Neel, and Siegel. Namely:

sin γ01σ01

=sin γ02σ02

=sin γ12σ12

, (7)



New Results for the Three Fluid Problem - IV

.Theorem (Classification of blowup limits - BET)..

......

The angles for the sectors in the tangent plane to the blowup spheresatisfy the angle conditions given by Elcrat, Neel, and Siegel. Namely:

sin γ01σ01

=sin γ02σ02

=sin γ12σ12

, (7)



The End

Figure : Alan wins

Thank you for listening!


The End

Figure : Alan wins

Thank you for listening!


Open?

Note that 5 is still absolutely open!


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