Geometric evolution equations with triple junctions. junctions in higher dimensions

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Geometric evolution equations with triple junctions in higher dimensions

Harald Garcke

University of Regensburg

joint work with Daniel Depner (University of Regensburg) and Yoshihito Kohsaka (Muroran IT)

February 2014

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1 Surface cluster

2 First variation

3 Mean curvature flow with junctions

4 Known results (curves and graphs)

5 An existence result

6 Numerical approximation of flows with triple junctions by a method introduced by Barrett, G., N¨urnberg

Harald Garcke Geometric evolution equations with triple junctions

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Cluster of hypersurfaces

Given smooth hypersurfaces

Γ₁, . . . , Γ_N in R^{d +1} with (piecewise) smooth boundaries

Boundaries ∂Γ_i at piecewise smooth junctions

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Energy for such structures Γ = (Γ₁, . . . , Γ_N)

F (Γ) =

N

X

i =1

Z

Γ_i

γ_id H^d =

N

X

i =1

γ_iH^d(Γ_i) Consider gradient flow of F

First variation of F

Given vector field ζ : R^{d +1} → R^{d +1} we set

Γ_i(ε) := {x + εζ(x ) | x ∈ Γ_i}

Remark: Triple junctions stay triple junctions under the variation

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Transport theorem gives d

d ε Z

Γ_i(ε)

1d H^d = − Z

Γ_i(ε)

V_iH_id H^d + Z

∂Γ_i(ε)

v_∂Γ,id H^{d −1}

Quantities:

V_i : normal velocity of Γ_i(ε) H_i : mean curvature

v_∂Γ,i : outer conormal velocity of boundary

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d

d εF (Γ(ε)) =

N

X

i =1

d d ε

Z

Γ_i(ε)

γ_id H^d = X

i

Z

Γ_i(ε)

−γ_iV_iH_id H^d +X

i

Z

∂Γ_i(ε)

γ_iv_∂Γ,id H^{d −1}

Steepest descent w.r.t. L²-metric

V_i = γ_iH_i on Γ_i mean curvature flow in bulk.

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Boundary conditions on triple junctions Boundary term on junction

v_∂Γ_i = ζ(x ) · τ_i τ_i outer unit conormal on ∂Γ_i On triple junction with i = 1, 2, 3

3

X

i =1

γ_iτ_i

!

· ζ(x) = 0

ζ arbitrary ⇒

3

P

i =1

γ_iτ_i = 0

symmetric case (γ₁ = γ₂ = γ₃) ⇒ 120^◦ angle condition

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Gradient flow (symmetric case)

V_i = H_i on Γ_i 120^◦ angle condition on triple junction

Typical solution (Numerics by Barrett, G., N¨urnberg)

equal area non-equal area

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Lower dimensional singularities

E.g. Four triple junctions meet at quadruple point Equal surface energies:

cos θ = −¹₃ (tetrahedron angles) θ : angle between junctions

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Double bubble conjecture

Hutchings, Morgan, Ritor´e, Ros (2000) proved:

The standard double bubble provides the least-perimeter way to enclose two prescribed volumes in R³

standard double bubble use an instability argument involving

the second variation of area to rule this out

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Allen-Cahn systems

Mean curvature flow with triple junctions arises as singular limit of Allen-Cahn systems:

Search u : Ω → R^N such that

ε∂_tu = ε∆u − ¹_εDψ(u) , ε > 0

ψ : R^N → R potential with at least three global minima (multi-well potential)

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Well-posedness

Existence of varifold solutions : Brakke, 1978

In this setting no uniqueness, no satisfactory regularity results

Question:

Do classical solutions exist ? What is the best formulation ?

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Local well-posedness for curves

Bronsard Reitich, 1993 Idea:

Parametrize curves over fixed intervals

Xⁱ : [0, 1] × [0, T ] → R² , i = 1, 2, 3 (p, t) 7→ Xⁱ(p, t)

Solve

X_tⁱ = X_ppⁱ

|X_p|² de Turck like trick (instead of X_tⁱ = X_ssⁱ , X_s arc length derivative)

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How to handle triple junctions Require

I

X¹(p, t) = X²(p, t) = X³(p, t) at boundary points p = 0, 1 II

P3 i =1

X_pⁱ

|X_pⁱ| = 0 (⇔ angle condition)

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How to prove local existence Linearize

Check Lopatinski-Shapiro conditions Use fixed point argument

Fourth order curve flow with junctions V = −∆_sH in the plane + junctions

Local existence: Garcke and Novick-Cohen

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Long time behaviour

Mantegazzu, Novaga, Tortorelli

Situation with Dirichlet boundary conditions

Solution exists as long as lengths Lⁱ, i = 1, . . . , N are bounded from below.

Schn¨urer etal / Bellettini Novaga

Asymptotic behaviour of lens-shape geometries

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Higher dimensional case Additional difficulty:

junctions have dimension d − 1 > 0 tangential degree of freedom

No general existence result for smooth solutions known before Special case of graphs: local existence of smooth solutions (Freire, 2010)

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Graph situation

MCF with triple junction is a free boundary problem

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Another graph situation

The projection of γ(t) on the plane is the free boundary

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Another graph situation

The resulting PDEs and the boundary conditions are highly nonlinear

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Triple junction flow of graphs

Complete set of equations

∂_tgⁱ

p1 + |∇gⁱ|² = div ∇gⁱ

p1 + |∇gⁱ|²

!

in D_i(t)

2

X

i =1

1

p|∇gⁱ|² + 1(−∇g_{i ,}1) − 1

p|∇g³|² + 1(−∇g_3,1) = 0 on γ(t) g₁ = g₂ = g₃ on γ(t)

Three second order PDEs Four boundary conditions

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Local existence for graphs Theorem (Freire)

Given C^3+α initial surface with triple junctions which initially fulfill - angle condition

- H¹ + H² + H³ = 0

Then there exists a unique local solution in C^2+α,1+^α²

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Generic initial data are not graphs

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MCF with boundary conditions Related simpler problem:

Consider mean curvature flow with 90^◦ boundary condition

V = H on Γ

^(Γ, ∂Ω) = 90^◦ at outer boundary ∂Ω

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MCF with boundary conditions

Use curvilinear coordinate system Search for w (σ, t).

w solves second order parabolic PDE + Neumann type boundary condition, see

Stahl 1996 Depner 2010

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Evolving triple junctions are more difficult to describe in higher dimensions

E.g. tangential motion in general necessary

normal velocitites V₁, V₂ given

V₃ fixed through V₁ + V₂ + V₃ = 0 triple junction has to move tangentially

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Parametrizing evolving suface clusters

Γ¹_∗, Γ²_∗, Γ³_∗ initial hyper-surfaces such that γ := ∂Γ¹_∗ = ∂Γ²_∗ = ∂Γ³_∗

ν_∗¹, ν_∗², ν_∗³ triple junction outer conormals on γ N_∗¹, N_∗², N_∗³ normals to Γ¹_∗, Γ²_∗, Γ³_∗

Parametrize new surface over Γ_∗ = (Γ¹_∗, Γ²_∗, Γ³_∗):

Ψⁱ(σ, w , r ) := σ + wN_∗ⁱ (σ) + r τ_∗ⁱ(σ)

(σ, t) 7→ Ψⁱ(σ, ρⁱ(σ, t), µⁱ(prⁱ(σ), t)) =: Φⁱ(σ, t) τ_∗ⁱ : extension of conormal with support local to γ

prⁱ : Γⁱ_∗ → ∂Γⁱ_∗ pr_|∂Γⁱ _i

∗ = id

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PDE formulation Degrees of freedom

(ρ¹, ρ², ρ³) change in normal direction on (Γ¹_∗, Γ²_∗, Γ³_∗)

(µ¹, µ², µ³) change in tangential direction on (∂Γ¹_∗, ∂Γ²_∗, ∂Γ³_∗) Equations:

MCF: V ⁱ = Hⁱ on Γⁱ_∗

Persistence of triple junction:

σ + ρ¹N_∗¹ + µ¹ν_∗¹ = σ + ρ²N_∗² + µ²ν_∗² = σ + ρ³N_∗³ + µ³ν_∗³ on γ 4 conditions angle conditions on γ 2 conditions

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Persistence of triple junction gives:

ρ¹ + ρ² + ρ³ = 0 on γ µⁱ = ^√¹

3(ρ^k − ρ^j) on γ This implies

(µ¹, µ², µ³) on γ are given by (ρ¹, ρ², ρ³) on γ

Parametrization σ + ρⁱ(σ, t)N_∗ⁱ + µⁱ(prⁱ(σ), t)τ_∗ⁱ implies non local dependence on boundary values

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PDE formulation (NPDE)

ρⁱ_t = Fⁱ(ρⁱ, ρ ◦ pr) + bⁱ(ρⁱ, ρ ◦ pr)n ³ P

j=1

P^ij(ρ, ρ ◦ pr)F^j(ρ^j, ρ ◦ pr)o Fⁱ second order non-local differential operator

(P^ij)_ij matrix leading to a coupling of the three equations via non-local boundary terms

(BC₁) ρ¹ + ρ² + ρ³ = 0 on γ (BC₂) angle conditions on γ

nonlinear conditions involving first derivatives of ρ and µ

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Strategy for local existence result 1. Linearize

2. Show existence/uniqueness of linear equation in energy spaces 3. Show C^2+α,1+^α² regularity of solutions (Solonnikov theory) 4. Use a contraction argument (Baconneau, Lunardi)

Special feature: Non-local term includes derivatives of highest order

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1. Linear problem

The linear problem (in angle condition I only show principal part)

∂_tuⁱ = ∆_Γi

∗uⁱ + |Πⁱ_∗|²uⁱ on Γⁱ_∗

∂_ν¹

∗u¹ = ∂_ν²

∗u² = ∂_ν³

∗u³ on γ

u¹ + u² + u³ = 0 on γ

Equations are formulated on each surface coupled through boundary conditions on triple junction

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Existence of weak solution (linearized problem) Testing procedure gives

d dt

P

i

R

Γⁱ_∗ 1

2(uⁱ)² + P

i

R

Γⁱ_∗

|∇_Γ^∗

i uⁱ|² = R

Γ^∗_i

P

i

(∂_νⁱ

∗uⁱ)uⁱ + l.o.t.

= R

Γ^∗_i

∂_ν¹

∗u¹(u¹ + u² + u³) + l.o.t.

= 0 + l.o.t.

Solution in energy space exists

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Regularity of solutions

Use results by Solonnikov (locally)

Situation in a neighborhood U of p ∈ γ

Write equation locally with the same parameter domain D = B₁(0) ∩ R^d+

system on same domain of definition

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Are the complementarity (Lopatinski-Shapiro) conditions fulfilled ? Use suitable coordinates !

In a fixed point we get:

∂_tu_i − ∆u_i = 0 + b.c. u₁ + u₂ + u₃ = 0, ∂_du₁ = ∂_du₂ = ∂_du₃ Lopatinski-Shapiro condition:

(ϕ₁, ϕ₂, ϕ₃) = 0 is the only bounded solution of the ODE system λϕ_j − ϕ⁰⁰_j + |ξ|²ϕ_j = 0 for z ∈ R⁺ ,

ϕ₁ + ϕ₂ + ϕ₃ = 0 for z = 0 , ϕ⁰₁ = ϕ⁰₂ = ϕ⁰₃ for z = 0

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Lopatinski-Shapiro condition

λϕ_j − ϕ⁰⁰_j + |ξ|²ϕ_j = 0 for z ∈ R⁺ , ϕ₁ + ϕ₂ + ϕ₃ = 0 for z = 0 ,

ϕ⁰₁ = ϕ⁰₂ = ϕ⁰₃ for z = 0 . Multiply by ϕ_j and integrate

λ

∞

R

0

|ϕ_j|² + P

j

(−

∞

R

0

ϕ⁰⁰_j ϕ_j + |ξ|²|ϕ_j|²) = 0 We compute

−

∞

R

0

ϕ⁰⁰_j ϕ_j =

∞

R

0

|ϕ⁰_j|² − [ϕ⁰_jϕ_j]^∞₀ P

j

(ϕ⁰_jϕ_j)(0) = ϕ⁰₁(0)P

j

ϕ_j(0) = 0 Since Re λ > 0 we obtain (ϕ₁, ϕ₂, ϕ₃) ≡ 0

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Solonnikov theory gives C^2+α,1+^α² -solvability of linear problem Use contraction argument for the fully nonlinear, non-local

PDE on the surface cluster (similar as Baconneau, Lunardi) Remark: Here it is important that the non-local term vanishes in the linearization

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Theorem (Depner, G., Kohsaka, 2014, ARMA)

Let (Γ¹₀, Γ²₀, Γ₀³) be a C^2+α surface cluster with a C^2+α triple junction curve γ

We assume the compatibility conditions - (Γ¹₀, Γ²₀, Γ³₀) fufill the angle conditions - H₀¹ + H₀² + H₀³ = 0

Then there exists a local C^2+α,1+^α² solution of Vⁱ = Hⁱ

+angle conditions with initial data (Γ¹₀, Γ²₀, Γ³₀)

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Remarks

1. Baconneau, Lunardi and Freire have a loss of regularity.

How to avoid this ?

Parametrize over a smooth cluster close by.

2. Generalizations to more than three surfaces are no problem

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Remarks

3. Completely missing is (Huisken for MCF)

continuation criterium

(maybe possible in terms of curvature and surface area of cluster patches)

blow-up analysis at first singular time

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Quadruple junctions are more difficult

A triple bubble with a quadruple point

∂Γ is only piecewise smooth !

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Higher order flows (surface diffusion) G.+Novick-Cohen

V_i = −∆_Γ_iH_i on Γ_i angle condition

H₁ + H₂ + H₃ = 0 continuity of chem.pot.

∇_Γ₁H₁ · ν_∂Γ₁ = ∇_Γ₂H₂ · ν_∂Γ₂ = ∇_Γ₃H₃ · ν_∂Γ₃ = 0 flux balance Properties:

d dt

X

i

Z

Γ_i

1d Hⁿ ≤ 0 surface area decreasing volume preserving

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Stationary solutions

0 = d dt

X

i

Z

Γ_i

1d Hⁿ = −X

i

Z

Γ_i

|∇H_i|²

(Γ₁, Γ2, Γ₃) stationary ⇒ H_i constant

b.c. ⇒ H₁ + H₂ + H₃ = 0 Stationary solutions: Double bubbles, triple bubles,...

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Surface diffusion: no junctions

Escher Simonett showed for V = −∆_sH

Spheres are (up to rescaling and translation) asymptotically stable under surface diffusion

Method: Center manifold theory

Needed since equilibria are not discrete

(indeed they build a (d + 2)-dimensional set translation + rescaling possible)

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Stability for surface diffusion with junctions

Problem: Fully nonlinear boundary conditions lead to nonlinear phase manifold

Usual stability theory in Banach-spaces not applicable

Try to use generalized principle of linearized stability for PDEs with nonlinear BC’s (Pr¨uss, Simonett, Zacher)

Some ingredients: ∂_tu + A(u(t))u(t) = F (u(t)) Equilibria form a C¹-manifold E

tangent space for E is given by N(A₀) (A₀ linearization) σ(A₀) \ {0} ⊂ C⁺

0 is semi-simple eigenvalue

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Work in progress (Arab, Abels, G.)

Mercedes star is stable under surface diffusion.

Double bubble is stable under surface diffusion

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Conclusions

Local well-posedness for mean curvature flow with triple junctions can be shown in higher dimensions also in the parametric case

Results for longer times are missing in higher dimensions (Continuation results, classification of singularities)

The situation with codimension 2 singularities is more difficult Semi-group theory for parabolic PDEs with fully nonlinear

BC’s can help to show stability results

The approach of Barrett, G., N¨urnberg can easily deal with junctions