A Ginzburg-Landau Type Problem for Nematics with Highly Anisotropic Elastic Term

Transcription

1 A Ginzburg-Landau Type Problem for Nematics with Highly Anisotropic Elastic Term Peter Sternberg In collaboration with Dmitry Golovaty (Akron) and Raghav Venkatraman (Indiana) Department of Mathematics Indiana University, Bloomington IMA Jan 2018

2 Director-Based Theory Thin Film Regime Suppose that a nematic occupies a thin domain with cross-section Ω R 2 and n : Ω S 1. The director field n(x) represents local orientation of nematic molecules near x Ω. To formulate a continuum variational theory, need an energy functional that takes into account Elastic distortions of the director field n in Ω Interactions of the nematic with the walls of the container, i.e. the boundary or anchoring conditions satisfied by the director field n on Ω.

3 Director-Based Theory Thin Film Regime Suppose that a nematic occupies a thin domain with cross-section Ω R 2 and n : Ω S 1. The director field n(x) represents local orientation of nematic molecules near x Ω. To formulate a continuum variational theory, need an energy functional that takes into account Elastic distortions of the director field n in Ω Interactions of the nematic with the walls of the container, i.e. the boundary or anchoring conditions satisfied by the director field n on Ω.

4 Director-Based Theory Thin Film Regime Suppose that a nematic occupies a thin domain with cross-section Ω R 2 and n : Ω S 1. The director field n(x) represents local orientation of nematic molecules near x Ω. To formulate a continuum variational theory, need an energy functional that takes into account Elastic distortions of the director field n in Ω Interactions of the nematic with the walls of the container, i.e. the boundary or anchoring conditions satisfied by the director field n on Ω.

5 Director-Based Theory Thin Film Regime Suppose that a nematic occupies a thin domain with cross-section Ω R 2 and n : Ω S 1. The director field n(x) represents local orientation of nematic molecules near x Ω. To formulate a continuum variational theory, need an energy functional that takes into account Elastic distortions of the director field n in Ω Interactions of the nematic with the walls of the container, i.e. the boundary or anchoring conditions satisfied by the director field n on Ω.

6 Oseen-Frank Model Oseen-Frank elastic energy density (Frank, 1958): f OF (n, n) := K 1 2 ( div n)2 + K 2 2 ( curl n n)2 + K 3 curl n n K 2 + K 4 (tr ( n) 2 ( div n) 2) 2 Splay Twist Bend Saddle Splay

7 Example: Oseen-Frank with strong anchoring: Minimize F OF [n] := Ω { K1 2 ( div n)2 + K 2 2 ( curl n n)2 + K } 3 curl n n 2 2 in H 1 ( Ω, S 1) subject to the appropriate Dirichlet boundary data g, i.e., n Ω = g such as n Ω = outer normal for the homeotropic anchoring. Invoking the identity: (div n) 2 + (curl n) 2 = n 2 + null Lagrangian and assuming that K 2 = K 3 we need only retain two of the elastic terms, say n 2 and (div n) 2.

8 Example: Oseen-Frank with strong anchoring: Minimize F OF [n] := Ω { K1 2 ( div n)2 + K 2 2 ( curl n n)2 + K } 3 curl n n 2 2 in H 1 ( Ω, S 1) subject to the appropriate Dirichlet boundary data g, i.e., n Ω = g such as n Ω = outer normal for the homeotropic anchoring. Invoking the identity: (div n) 2 + (curl n) 2 = n 2 + null Lagrangian and assuming that K 2 = K 3 we need only retain two of the elastic terms, say n 2 and (div n) 2.

9 Relaxed model Relax the constraint n S 1 by replacing n by u R 2 with a penalty for u deviating from 1. The resulting functional resembles Ericksen model for a director u/ u and variable degree of orientation (scalar) u. Upon rescaling, we arrive at a functional that will be the focus of this talk: E ε (u) = 1 2 Ω 1 ε ( u 2 1) 2 + ε u 2 + L(div u) 2 dx. Here L > 0 is independent of ε > 0, whereas ε 1, so splay is penalized much more heavily than bending. Admissible competitors u must lie in H 1 (Ω; R 2 ) and satisfy an S 1 -valued Dirichlet condition u = g on Ω for some g H 1/2 ( Ω; S 1 ). Notation: We ll write u H 1 g (Ω; R 2 ) for such competitors.

10 Relaxed model Relax the constraint n S 1 by replacing n by u R 2 with a penalty for u deviating from 1. The resulting functional resembles Ericksen model for a director u/ u and variable degree of orientation (scalar) u. Upon rescaling, we arrive at a functional that will be the focus of this talk: E ε (u) = 1 2 Ω 1 ε ( u 2 1) 2 + ε u 2 + L(div u) 2 dx. Here L > 0 is independent of ε > 0, whereas ε 1, so splay is penalized much more heavily than bending. Admissible competitors u must lie in H 1 (Ω; R 2 ) and satisfy an S 1 -valued Dirichlet condition u = g on Ω for some g H 1/2 ( Ω; S 1 ). Notation: We ll write u H 1 g (Ω; R 2 ) for such competitors.

11 Motivation for the model We are interested in capturing singular structures such as vortices and domain walls (both smooth and non-smooth) arising in nematic liquid crystal models that one might associate with a large disparity in the value of the elastic constants. The model we look at here is a kind of toy problem meant to isolate certain key features while for the time being ignoring other complicating factors associated with, say, the full Landau-deGennes Q-tensor theory or with double well potentials.

12 Motivation for the model We are interested in capturing singular structures such as vortices and domain walls (both smooth and non-smooth) arising in nematic liquid crystal models that one might associate with a large disparity in the value of the elastic constants. The model we look at here is a kind of toy problem meant to isolate certain key features while for the time being ignoring other complicating factors associated with, say, the full Landau-deGennes Q-tensor theory or with double well potentials.

13 Two asymptotic limits for the term L (div u) 2 dx Note that if one takes L = 0 in E ε, then this is precisely the famous Brezis-Bethuel-Helein [BBH] problem, multiplied by ε: 1 1 inf u Hg 1 (Ω;R 2 ) 2 Ω ε ( u 2 1) 2 + ε u 2 dx, whose minimizers are characterized by Ginzburg-Landau vortices. On the other hand, if we formally consider the limit L so that competitors u are required to be divergence-free, then writing u = ( v) for some scalar v : Ω R we find that E ε takes the form Eε AG (v) := Ω ε ( v 2 1) 2 + ε D 2 v 2 dx, which is the well-known Aviles-Giga energy, whose minimizers in the limit ε 0 are characterized by wall-type singularities.

14 Two asymptotic limits for the term L (div u) 2 dx Note that if one takes L = 0 in E ε, then this is precisely the famous Brezis-Bethuel-Helein [BBH] problem, multiplied by ε: 1 1 inf u Hg 1 (Ω;R 2 ) 2 Ω ε ( u 2 1) 2 + ε u 2 dx, whose minimizers are characterized by Ginzburg-Landau vortices. On the other hand, if we formally consider the limit L so that competitors u are required to be divergence-free, then writing u = ( v) for some scalar v : Ω R we find that E ε takes the form Eε AG (v) := Ω ε ( v 2 1) 2 + ε D 2 v 2 dx, which is the well-known Aviles-Giga energy, whose minimizers in the limit ε 0 are characterized by wall-type singularities.

15 Two asymptotic limits for the term L (div u) 2 dx Note that if one takes L = 0 in E ε, then this is precisely the famous Brezis-Bethuel-Helein [BBH] problem, multiplied by ε: 1 1 inf u Hg 1 (Ω;R 2 ) 2 Ω ε ( u 2 1) 2 + ε u 2 dx, whose minimizers are characterized by Ginzburg-Landau vortices. On the other hand, if we formally consider the limit L so that competitors u are required to be divergence-free, then writing u = ( v) for some scalar v : Ω R we find that E ε takes the form Eε AG (v) := Ω ε ( v 2 1) 2 + ε D 2 v 2 dx, which is the well-known Aviles-Giga energy, whose minimizers in the limit ε 0 are characterized by wall-type singularities.

16 Singular structures in this model: a GL vortex Ginzburg-Landau type vortex u ε = ρ ε (r)(cos θ, sin θ) expensive! L (div u) 2 dx L ln ε so E ε (u ε ). Ω

17 Singular structures in this model: a zero divergence vortex A divergence-free vortex u ε = ρ ε (r)( sin θ, cos θ) div u ε 0 so E ε (u ε ) ε ln ε 0

18 Singular structures: a domain wall Figure: u and u. Note: Continuity of normal component across the (vertical) jump.

19 The right space of competitors for a limiting problem Given that energy-bounded sequences E ε (w ε ) < C satisfy the bounds div w ε L2 (Ω) < C and it makes sense to seek a limiting problem defined for Ω ( w ε 2 1) 2 dx < C ε 2, u H div (Ω; S 1 ) := {u L 2 (Ω; S 1 ) : div u L 2 (Ω)}. Key point: Functions u H div (Ω; S 1 ) are allowed to have jump discontinuities across a curve provided u n is continuous. (In particular, the normal trace is well-defined.) Since u = 1 on either side of the jump, this means across the jump set the tangential component simply switches signs: u + τ = u τ, where u ± denote the traces on either side of the jump set.

20 The right space of competitors for a limiting problem Given that energy-bounded sequences E ε (w ε ) < C satisfy the bounds div w ε L2 (Ω) < C and it makes sense to seek a limiting problem defined for Ω ( w ε 2 1) 2 dx < C ε 2, u H div (Ω; S 1 ) := {u L 2 (Ω; S 1 ) : div u L 2 (Ω)}. Key point: Functions u H div (Ω; S 1 ) are allowed to have jump discontinuities across a curve provided u n is continuous. (In particular, the normal trace is well-defined.) Since u = 1 on either side of the jump, this means across the jump set the tangential component simply switches signs: u + τ = u τ, where u ± denote the traces on either side of the jump set.

21 Towards a Γ-Convergence result: Compactness With only a minor modification of the compactness proof of DeSimone, Kohn, Müller, Otto (2001) for the Aviles-Giga functional, one has: Theorem Assume {v ε } H 1 (Ω) satisfies the uniform energy bound 1 1 sup E (v ε ) = sup ε>0 ε>0 2 Ω ε ( v 2 1) 2 + ε v 2 + L(div v) 2 dx <. Then there exists a subsequence (still denoted by v ε ) and a function v H div (Ω; S 1 ) such that v ε v defined as div v ε div v weakly in L 2 v ε v in L p (Ω; R 2 ) for all p < [DKMO]. Note: Under this convergence, if v ε = g on Ω then v n = g n.

22 Asymptotic cost of a horizontal domain wall along y = 0 To smoothly approximate, say, a horizontal wall across which u jumps from ( 1 a(x) 2, a(x) ) to ( 1 a(x) 2, a(x) ) in an energetically efficient way, a natural ansatz is: u ε (x, y) = (ζ(x, y ε ), a) with ζ(x, ± ) = ± 1 a 2 where a = a(x) = normal (here 2nd) component of u(x, 0). The optimal such profile ζ(x, y) is given by the heteroclinic connection (hyperbolic tangent profile) minimizing F (ζ) := A direct calculation yields E ε (u ε ) 1 6 (ζ y ) 2 + (1 a 2 ζ 2 ) 2 dy, ζ(x, ± ) ± 1 a 2. J u u + u 3 dh 1 + L 2 Ω (div u) 2 dx dy where J u denotes the jump set of u; in this example J u = (0, 1) {0}.

23 Asymptotic cost of a horizontal domain wall along y = 0 To smoothly approximate, say, a horizontal wall across which u jumps from ( 1 a(x) 2, a(x) ) to ( 1 a(x) 2, a(x) ) in an energetically efficient way, a natural ansatz is: u ε (x, y) = (ζ(x, y ε ), a) with ζ(x, ± ) = ± 1 a 2 where a = a(x) = normal (here 2nd) component of u(x, 0). The optimal such profile ζ(x, y) is given by the heteroclinic connection (hyperbolic tangent profile) minimizing F (ζ) := A direct calculation yields E ε (u ε ) 1 6 (ζ y ) 2 + (1 a 2 ζ 2 ) 2 dy, ζ(x, ± ) ± 1 a 2. J u u + u 3 dh 1 + L 2 Ω (div u) 2 dx dy where J u denotes the jump set of u; in this example J u = (0, 1) {0}.

24 Asymptotic cost of a horizontal domain wall along y = 0 To smoothly approximate, say, a horizontal wall across which u jumps from ( 1 a(x) 2, a(x) ) to ( 1 a(x) 2, a(x) ) in an energetically efficient way, a natural ansatz is: u ε (x, y) = (ζ(x, y ε ), a) with ζ(x, ± ) = ± 1 a 2 where a = a(x) = normal (here 2nd) component of u(x, 0). The optimal such profile ζ(x, y) is given by the heteroclinic connection (hyperbolic tangent profile) minimizing F (ζ) := A direct calculation yields E ε (u ε ) 1 6 (ζ y ) 2 + (1 a 2 ζ 2 ) 2 dy, ζ(x, ± ) ± 1 a 2. J u u + u 3 dh 1 + L 2 Ω (div u) 2 dx dy where J u denotes the jump set of u; in this example J u = (0, 1) {0}.

25 These types of wall constructions are well-known from earlier studies in many different contexts: smectic-a liquid crystals, thin film blistering, micromagnetics,... Within the math community, there are many contributors including: Aviles/Giga, Jin/Kohn, Conti/DeLellis, Ignat, James, Poliakovsky, Alouges/Riviere/Serfaty, and many others...

26 The Γ-limit: What a uniform energy bound does not yield is that the limit lies in BV (cf. example by Ambrosio/De Lellis/Montegazza) However, we make this assumption and propose a candidate for the Γ-limit: For u H div (Ω; S 1 ) BV (Ω; S 1 ) with u n = g n on Ω, let E 0 (u) be given by E 0 (u) := 1 6 J u Ω u + u 3 dh L 2 J u Ω Ω u Ω g 3 dh 1 ( div u ) 2 dx, where u + and u denote the traces of u on J u Ω, and u Ω trace of u along Ω. denotes the

27 Γ-convergence Theorem Let u H div (Ω; S 1 ) BV (Ω; S 1 ) with u Ω n = g on Ω (i) If u ε H 1 g (Ω, R 2 ) is a sequence of functions such that u ε u, then lim inf ε 0 E ε(u ε ) E 0 (u). (ii) There exists w ε H 1 g (Ω; R 2 ) with w ε u satisfying lim sup E ε (w ε ) = E 0 (u). ε 0 The proof uses the ideas from Jin/Kohn and Alouges/Riviere/Serfaty (lower semicontinuity) and Conti/De Lellis (recovery sequence).

28 Criticality Conditions for E 0 Theorem Suppose that u BV (Ω, S 1 ) H div (Ω, S 1 ) such that u Ω n = g n on Ω is a critical point of E 0. Denote by J u its jump set. Then u div u = 0 holds weakly on Ω\J u, where u = ( u 2, u 1 ). Furthermore, if the traces div u + and div u on J u are sufficiently smooth, then L [div u] + 4 ( 1 (u ν u ) 2) 1/2 (u νu ) = 0 on J u Ω, where [a] = a + a represents the jump of a across J u and ν u is the unit normal to J u pointing from the + side of J u to the side. One can also derive criticality conditions associated with variations of the jump set itself that involve curvature of J u.

29 Criticality Conditions for E 0 Theorem Suppose that u BV (Ω, S 1 ) H div (Ω, S 1 ) such that u Ω n = g n on Ω is a critical point of E 0. Denote by J u its jump set. Then u div u = 0 holds weakly on Ω\J u, where u = ( u 2, u 1 ). Furthermore, if the traces div u + and div u on J u are sufficiently smooth, then L [div u] + 4 ( 1 (u ν u ) 2) 1/2 (u νu ) = 0 on J u Ω, where [a] = a + a represents the jump of a across J u and ν u is the unit normal to J u pointing from the + side of J u to the side. One can also derive criticality conditions associated with variations of the jump set itself that involve curvature of J u.

30 A method of characteristics approach in the bulk Corollary Suppose u is smooth and critical for E 0. Then writing u locally in terms of a lifting as u(x, y) = ( cos θ(x, y), sin θ(x, y) ) and defining the scalar v := div u one has that the criticality condition u div u = 0 on Ω\J u is equivalent to the following system for the two scalars θ and v: { sin θ vx + cos θ v y = 0, sin θ θ x + cos θ θ y = v.

31 Integrating the characteristic system one finds: x t = sin θ, y t = cos θ, θ t = v v t = 0 Characteristics are circular arcs that carry constant values of divergence and curvature of each such circular arc is given by that constant divergence. In case the divergence is zero, the corresponding characteristic is a straight line.

32 First example: a periodic strip To understand how bulk divergence versus walls contribute to the total energy E 0, we first consider a basic example of a rectangle with periodic boundary conditions on the left and right sides: Let Ω = [ T, T ] [ H, H] and set { g( T, y) = g(t, y), y [ H, H], g(x, ±H) = (±1, 0), x [ T, T ].

33 1D versions of E ε and E 0 Let A 0 := {u = u(y) H 1 (( H, H); R 2 ), u(±h) = (±1, 0)}. and consider the variational problem inf u A 0 Eε 1D (u), where Eε 1D (u) := 1 H ε u H ε ( u 2 1) 2 + L(u 2) 2 dy. and the Γ-limit restricted to 1D competitors: E0 1D (u) =: L H (u 2 2) 2 dy + 1 H 6 [u 1 ](y j ) 3. y j J u1 In 1D the jump set only involves jumps in u 1 since u 2 H 1 and J u1 consists of a set of points {y j }.

34 Improved Compactness in 1D Theorem Let u ε = ( u (1) ε (y), u (2) ε (y) ) be an energy-bounded sequence, i.e. E 1D ε (u ε ) C. Then, up to extraction of subsequences, one has u (1) ε u 1 in L 3 ( H, H) for some function u 1 such that u1 3 BV ( H, H) and one has u(2) ε u 2 in C 0,γ for all γ < 1/2. Furthermore, ( u 1 (y), u 2 (y) ) = 1 a.e.

35 Minimizers of the 1D Γ-limit Theorem (i) If L/H < 2, the problem inf E 1D A 0 0 (u) has a unique solution u = (u1, u 2 ) where u 1 has exactly one jump located at y = 0 and u2 is continuous on [ H, H] and linear on the subintervals [ H, 0] and [0, H]. The infimum of the energy is E0 1D (u ) = L H 1 L 3 (ii) If L/H > 2 then the minimizer has the form { u ( 1, 0) for y ( H, y (y) = ], (1, 0) for y (y, H), where y [ H, H] is arbitrary and the infimum of the energy is E 1D 0 (u ) = 4/ H 3

36 L = 0.3, H = 0.5, T = 0.5, ε = Figure: u and u.

37 L = 0.5, H = 0.5, T = 0.3, ε = Figure: u and u.

38 L = 0.5, H = 0.5, T = 0.3, ε = Figure: Level curves for the divergence of u.

39 Theorem Consider the minimization problem for E 0 in the rectangle Ω = ( T, T ) ( H, H), subject to the boundary conditions u(x, ±H) = (±1, 0). There exist constants L and L such that whenever L/H (L 0, L 1 ) and T = H T (L/H) where T (L/H) solves a certain algebraic equation, we have inf E 0 (u) 2T < inf A 0 E 1D 0 (u). Here the infimum on the left is taken over all u H div (Ω; S 1 ) BV (Ω; S 1 ) such that u n = 0 on the top and bottom sides of the the rectangle y = ±H and u is 2T -periodic in x.

40 Characteristics solution construction H y II T I III T x Figure: Regions corresponding to different characteristics families. Typical characteristics for each region are indicated by dashed lines.

41 L = 0.5, H = 0.5, T = 0.3, ε = y x Figure: Level curves for the divergence of u.

42 E/T D analytical, cross-tie 1D analytical 1D numerical 2D numerical, cross-tie 2D numerical, cross-tie II L/H Figure: Energy per unit length. For L/H between about 1.27 and 2.14 the minimizer is not 1D.

43 Examples for Ω = D (disk) under various boundary conditions I. Examples where minimizers have no walls: Theorem For boundary conditions u n = g n on D with (i) g = ê θ (tangential b.c.) or (ii) g = (x, y) (hedgehog b.c.) the global minimizer of E 0 in H div (D; S 1 ) BV (D; S 1 ) is (i) u ê θ and (ii) u ± (r, θ) = rê r ± 1 r 2 ê θ, respectively. In both examples, there is a divergence-free vortex at the origin and no jump set. In (i) E 0 (ê θ ) = 0. In (ii) all of the minimizing energy comes from the divergence as the minimizer unwinds from ê θ to ê r.

44 Examples for Ω = D (disk) under various boundary conditions II. An example with nontrivial wall structure: Take as Dirichlet condition: These degree -1 boundary conditions induce walls. g(x, y) = (x, y). At least for some parameter regimes of L, we can capture these analytically again using the conservation law approach. We conclude with some numerical experiments with this boundary condition, letting L increase.

45 Examples for Ω = D (disk) under various boundary conditions II. An example with nontrivial wall structure: Take as Dirichlet condition: These degree -1 boundary conditions induce walls. g(x, y) = (x, y). At least for some parameter regimes of L, we can capture these analytically again using the conservation law approach. We conclude with some numerical experiments with this boundary condition, letting L increase.

46 Computed solution for g(x, y) = (x, y) with L = 0.2 Figure: L = 0.2. The field u and level curves of u.

47 Computed solution for g(x, y) = (x, y) with L = 1.0 Figure: L = 1.0. The field u and level curves of u.

48 Computed solution for g(x, y) = (x, y) with L = 1.3 Figure: L = 1.3. The field u and level curves of u.

49 Computed solution for g(x, y) = (x, y) with L = 2.0 Figure: L = 2.0. The field u and level curves of u.

50 Computed solution for g(x, y) = (x, y) with L = 10.0 Figure: L = The field u and level curves of u.