Geometric evolution equations with triple junctions. junctions in higher dimensions

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1 Geometric evolution equations with triple junctions in higher dimensions University of Regensburg joint work with and Daniel Depner (University of Regensburg) Yoshihito Kohsaka (Muroran IT) February 2014

2 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., Nürnberg

3 Cluster of hypersurfaces Given smooth hypersurfaces Γ 1,..., Γ N in R d+1 with (piecewise) smooth boundaries Boundaries Γ i at piecewise smooth junctions

4 Energy for such structures Γ = (Γ 1,..., Γ N ) N F(Γ) = γ i dh d = Γ i i=1 Consider gradient flow of F N γ i H d (Γ i ) i=1 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

5 First variation of F Transport theorem gives d 1dH d = dε Γ i (ε) Γ i (ε) V i H i dh d + Γ i (ε) v Γ,i dh d 1 Quantities: V i : normal velocity of Γ i (ε) H i : mean curvature v Γ,i : outer conormal velocity of boundary

6 First variation of F d N dε F(Γ(ε)) = d γ i dh d = dε i=1 Γ i (ε) i + γ i v Γ,i dh d 1 i Γ i (ε) Γ i (ε) γ i V i H i dh d Steepest descent w.r.t. L 2 -metric mean curvature flow in bulk. V i = γ i H i on Γ i

7 Boundary conditions on triple junctions Boundary term on junction v Γi = ζ(x) τ i τ i outer unit conormal on Γ i On triple junction with i = 1, 2, 3 ( 3 ) γ i τ i i=1 ζ(x) = 0 ζ arbitrary 3 i=1 γ i τ i = 0 symmetric case (γ 1 = γ 2 = γ 3 ) 120 angle condition

8 Gradient flow (symmetric case) V i = H i on Γ i 120 angle condition on triple junction Typical solution (Numerics by Barrett, G., Nürnberg) equal area non-equal area

9 Lower dimensional singularities E.g. Four triple junctions meet at quadruple point Equal surface energies: cos θ = 1 3 (tetrahedron angles) θ : angle between junctions

10 Double bubble conjecture Hutchings, Morgan, Ritoré, Ros (2000) proved: The standard double bubble provides the least-perimeter way to enclose two prescribed volumes in R 3 standard double bubble use an instability argument involving the second variation of area to rule this out

11 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 t u = ε u 1 ε Dψ(u), ε > 0 ψ : R N R potential with at least three global minima (multi-well potential)

12 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?

13 Local well-posedness for curves Bronsard Reitich, 1993 Idea: Parametrize curves over fixed intervals X i : [0, 1] [0, T ] R 2, i = 1, 2, 3 (p, t) X i (p, t) Solve X i t = X i pp X p 2 de Turck like trick (instead of X i t = X i ss, X s arc length derivative)

14 How to handle triple junctions Require I II 3 i=1 X 1 (p, t) = X 2 (p, t) = X 3 (p, t) at boundary points p = 0, 1 X i p X i p = 0 ( angle condition)

15 How to prove local existence Linearize Check Lopatinski-Shapiro conditions Use fixed point argument Fourth order curve flow with junctions V = s H in the plane + junctions Local existence: Garcke and Novick-Cohen

16 Long time behaviour Mantegazzu, Novaga, Tortorelli Situation with Dirichlet boundary conditions Solution exists as long as lengths L i, i = 1,..., N are bounded from below. Schnürer etal / Bellettini Novaga Asymptotic behaviour of lens-shape geometries

17 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)

18 Graph situation MCF with triple junction is a free boundary problem

19 Another graph situation The projection of γ(t) on the plane is the free boundary

20 Another graph situation The resulting PDEs and the boundary conditions are highly nonlinear

21 Triple junction flow of graphs Complete set of equations ( ) t g i 1 + g i = div g i g i 2 in D i (t) 2 i=1 1 g i ( g i,1) 1 g ( g 3,1) = 0 on γ(t) g 1 = g 2 = g 3 on γ(t) Three second order PDEs Four boundary conditions This fixes degree of freedom for triple junction

22 Local existence for graphs Theorem (Freire) Given C 3+α initial surface with triple junctions which initially fulfill - angle condition - H 1 + H 2 + H 3 = 0 Then there exists a unique local solution in C 2+α,1+ α 2

23 Generic initial data are not graphs Goal: Show existence result in these cases

24 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

25 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

26 Evolving triple junctions are more difficult to describe in higher dimensions E.g. tangential motion in general necessary normal velocitites V 1, V 2 given V 3 fixed through V 1 + V 2 + V 3 = 0 triple junction has to move tangentially

27 Parametrizing evolving suface clusters Γ 1, Γ 2, Γ 3 initial hyper-surfaces such that γ := Γ 1 = Γ 2 = Γ 3 ν, 1 ν, 2 ν 3 triple junction outer conormals on γ N 1, N 2, N 3 normals to Γ 1, Γ 2, Γ 3 Parametrize new surface over Γ = (Γ 1, Γ 2, Γ 3 ): Ψ i (σ, w, r) := σ + wn i (σ) + rτ i (σ) (σ, t) Ψ i (σ, ρ i (σ, t), µ i (pr i (σ), t)) =: Φ i (σ, t) τ i : extension of conormal with support local to γ pr i : Γ i Γ i pr i = id Γ i We do not allow for motion tangential to triple junction

28 PDE formulation Degrees of freedom (ρ 1, ρ 2, ρ 3 ) change in normal direction on (Γ 1, Γ 2, Γ 3 ) (µ 1, µ 2, µ 3 ) change in tangential direction on ( Γ 1, Γ 2, Γ 3 ) Equations: MCF: Persistence of triple junction: V i = H i on Γ i σ + ρ 1 N 1 + µ 1 ν 1 = σ + ρ 2 N 2 + µ 2 ν 2 = σ + ρ 3 N 3 + µ 3 ν 3 on γ 4 conditions angle conditions on γ 2 conditions

29 Persistence of triple junction gives: This implies ρ 1 + ρ 2 + ρ 3 = 0 on γ µ i = 1 3 (ρ k ρ j ) on γ (µ 1, µ 2, µ 3 ) on γ are given by (ρ 1, ρ 2, ρ 3 ) on γ Parametrization σ + ρ i (σ, t)n i + µ i (pr i (σ), t)τ i implies non local dependence on boundary values

30 PDE formulation (NPDE) { ρ i t = F i (ρ i, ρ pr) + b i (ρ i 3 }, ρ pr) P ij (ρ, ρ pr)f j (ρ j, ρ pr) F i second order non-local differential operator (P ij ) ij matrix leading to a coupling of the three equations via non-local boundary terms (BC 1 ) ρ 1 + ρ 2 + ρ 3 = 0 on γ (BC 2 ) angle conditions on γ nonlinear conditions involving first derivatives of ρ and µ j=1

31 Strategy for local existence result 1. Linearize 2. Show existence/uniqueness of linear equation in energy spaces 3. Show C 2+α,1+ α 2 regularity of solutions (Solonnikov theory) 4. Use a contraction argument (Baconneau, Lunardi) Special feature: Non-local term includes derivatives of highest order

32 1. Linear problem The linear problem (in angle condition I only show principal part) t u i = Γ i u i + Π i 2 u i on Γ i ν 1 u 1 = ν 2 u 2 = ν 3 u 3 on γ u 1 + u 2 + u 3 = 0 on γ Equations are formulated on each surface coupled through boundary conditions on triple junction

33 Existence of weak solution (linearized problem) Testing procedure gives d 1 dt 2 (ui ) 2 + i i Γ i Γ i Γ i u i 2 = Γ i Solution in energy space exists ( ν i u i )u i + l.o.t. i = ν 1 u 1 (u 1 + u 2 + u 3 ) + l.o.t. Γ i = 0 + l.o.t.

34 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 = B 1 (0) R d + system on same domain of definition

35 Are the complementarity (Lopatinski-Shapiro) conditions fulfilled? Use suitable coordinates! In a fixed point we get: t u i u i = 0 + b.c. u 1 + u 2 + u 3 = 0, d u 1 = d u 2 = d u 3 Lopatinski-Shapiro condition: (ϕ 1, ϕ 2, ϕ 3 ) = 0 is the only bounded solution of the ODE system λϕ j ϕ j + ξ 2 ϕ j = 0 for z R +, ϕ 1 + ϕ 2 + ϕ 3 = 0 for z = 0, ϕ 1 = ϕ 2 = ϕ 3 for z = 0 for all ξ R d 1 and λ C with Re λ > 0

36 Lopatinski-Shapiro condition λϕ j ϕ j + ξ 2 ϕ j = 0 for z R +, ϕ 1 + ϕ 2 + ϕ 3 = 0 for z = 0, ϕ 1 = ϕ 2 = ϕ 3 for z = 0. Multiply by ϕ j and integrate λ ϕ j 2 + ( j We compute 0 0 ϕ j ϕ j = 0 0 ϕ j ϕ j + ξ 2 ϕ j 2 ) = 0 ϕ j 2 [ϕ j ϕ j] 0 (ϕ j ϕ j)(0) = ϕ 1 (0) ϕ j (0) = 0 j j Since Re λ > 0 we obtain (ϕ 1, ϕ 2, ϕ 3 ) 0

37 Solonnikov theory gives C 2+α,1+ α 2 -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

38 Theorem (Depner, G., Kohsaka, 2014, ARMA) Let (Γ 1 0, Γ2 0, Γ3 0 ) be a C 2+α surface cluster with a C 2+α triple junction curve γ We assume the compatibility conditions - (Γ 1 0, Γ2 0, Γ3 0 ) fufill the angle conditions - H0 1 + H2 0 + H3 0 = 0 Then there exists a local C 2+α,1+ α 2 with initial data (Γ 1 0, Γ2 0, Γ3 0 ) V i = H i solution of +angle conditions

39 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

40 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

41 Quadruple junctions are more difficult A triple bubble with a quadruple point Γ i is only piecewise smooth! Regularity theory more difficult

42 Higher order flows (surface diffusion) G.+Novick-Cohen V i = Γi H i on Γ i angle condition H 1 + H 2 + H 3 = 0 continuity of chem.pot. Γ1 H 1 ν Γ1 = Γ2 H 2 ν Γ2 = Γ3 H 3 ν Γ3 = 0 flux balance Properties: d dt volume preserving i Γ i 1dH n 0 surface area decreasing

43 Stationary solutions 0 = d dt i Γ i 1dH n = i Γ i H i 2 (Γ 1, Γ2, Γ 3 ) stationary H i constant b.c. H 1 + H 2 + H 3 = 0 Stationary solutions: Double bubbles, triple bubles,... Numerical simulation Barrett, G., Nürnberg

44 Surface diffusion: no junctions Escher Simonett showed for V = s H 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)

45 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 (Prüss, Simonett, Zacher) Some ingredients: t u + A(u(t))u(t) = F (u(t)) Equilibria form a C 1 -manifold E tangent space for E is given by N(A 0 ) (A 0 linearization) σ(a 0 ) \ {0} C + 0 is semi-simple eigenvalue

46 Work in progress (Arab, Abels, G.) Mercedes star is stable under surface diffusion. Double bubble is stable under surface diffusion

47 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., Nürnberg can easily deal with junctions