On the biharmonic energy with Monge-Ampère constraints

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1 On the biharmonic energy with Monge-Ampère constraints Marta Lewicka University of Pittsburgh Paris LJLL May

Morphogenesis in growing tissues Observation: residual stress at free equilibria A model by Rodriguez-Hoger-McCulloch studied by Ben Amar et al.

2 Morphogenesis in growing tissues Observation: residual stress at free equilibria A model by Rodriguez-Hoger-McCulloch studied by Ben Amar et al., Marder-Papanicolau, also in plasticity: U R 3 reference configuration; A : U! R 3 3 sym decomposition u = FA u : U! R 3 deformation growth tensor, deta > 0; F elastic tensor Energy E(u)= Z U W(F)= Z U W ( u)a 1 (x) dx 2

3 Energy Non-Euclidean elasticity E(u)= Z Z U W(F)= U W ( u)a 1 (x) dx Assumptions on energy density W : R 3 3! R + : W(RF)=W(F) 8R 2 SO(3) frame indifference W(Id 3 )=minw = 0 normalisation W(F) c dist 2 (F,SO(3)), c > 0 non-degeneracy W(F)= 8detF apple 0, and W(F)! as detf! 0+ non-interpenetration Example: W 1 (F)= (F T F) 1/2 Id logdetf 2 W 2 (F)= (F T F) 1/2 Id detf 1 2 8detF > 0 and W 1 (F)=W 2 (F)= 8detF apple 0. 3

4 Residual stress Z E(u)= U W ( u)a 1 (x) dx E(u)=0, ( u)a 1 2 SO(3), u(x) 2 SO(3)A(x) 8a.e. x, ( u) T u = A T A and det u > 0 smooth Riemannian metric G = A T A : U! R 3 3. Theorem 9u (must be smooth) with E(u)=0, R ijkl (G) 0. Lemma (L-Pakzad 09) inf u2w 1,2 E(u) > 0, R ijkl (G) 6 0. Thin non-euclidean plates U = W h = W ( h/2,h/2) W R 2 open, bounded, simply connected given: G(x 0,x 3 )=A(x 0,x 3 ) T A(x 0,x 3 ) Z E h (u h )= 1 h W hw(( uh )A 1 ). How does infe h scale as h! 0? dimension reduction for nonlinear elasticity (A = Id): seminal analysis done by Friesecke-James-Muller 06 S h 4

5 Plan of lecture We will show that for A Id 3 +O(h g ) there holds: infe h h g+2. G For 0 < g < 2 we have:! I, where I is the 2-d energy: 1 h g+2eh I(v)= R W Q 2( 2 v) for v 2W 2,2 (W), det 2 v = 1 h g curl T curl A 2 2 Matching property: a method to construct the recovery sequence. Any 2nd order C 2,a isometry can be matched to an exact isometry for a metric with Gauss curvature k c > 0. Density of C 2,a solutions v to: det 2 v = f, in the set of solutions v 2 W 2,2. We prove interior regularity and convexity of v 2 W 2,2. We deduce density for f = const > 0. Uniqueness/multiplicity of minimizers to I(v). Limit scaling case g = 0, where A = O(1). 5

6 Monge-Ampère constrained morphogenesis model S g,b g : W! R 2 2 sym S g,b g : W! R 3 3 sym A h (x 0,x 3 )=Id 3 + h g S g (x 0 ) + h g/2 x 3 B g (x 0 ) given such that (S g) 2 2 = S g, (B g) 2 2 = B g G h =(A h ) T A h = Id 3 + 2h g S g (x 0 ) + 2h g/2 x 3 B g (x 0 ) + h.o.t. Id 2 + 2h g S g : Ideal 1st fundamental form I of the mid-surface. Let u h : W! R 3 satisfy: ( u h ) T u h = I u h (x 0,x 3 )=u h (x 0 )+x 3 N h (x 0 ), where N h = unit normal to u h (W). Then: x 3 h i u h, j u h i = 2II(e i,e j ) h g/2 B g : Ideal 2nd fundamental form II of the mid-surface. Compatibility through Gauss equation: detii =(deti)k(i) 1 h g 6

7 Monge-Ampère constrained morphogenesis model G h (x 0,x 3 )=Id 3 + 2h g S g (x 0 ) + 2h g/2 x 3 B g (x 0 ) + h.o.t. Id 2 + 2h g S g : Ideal 1st fundamental form of the mid-surface, h g/2 B g : Ideal 2nd fundamental form of the mid-surface. If these forms are not compatible (through Gauss-Codazzi equations), then: N h = h g/2 P, (P 6= B g ), E h (u h ) 1 h R and: W h ( u h ) T u h G h 2 R W h 2h g/2 x 3 (P B g ) 2 h g+2. 1 h We now consider scaling: 0 < g < 2 Limit scaling cases: g = 0 (Bhattacharya-L-Schaffner 14) g = 2 (L-Mahadevan-Pakzad 10). 7

8 The lower bound Theorem (L-Ochoa-Pakzad 14) If E h (u h ) apple Ch g+2, then 9R h 2 SO(3), c h 2 R 3 such that the following holds for the sequence: y h (x 0,x 3 ) := R h u h (x 0,hx 3 ) c h 2 W 1,2 (W 1,R 3 ) (i) y h (x 0,x 3 )! x 0 strongly in W 1,2 (ii) 1 h g/2 Z 1/2 1/2 yh (x 0,x 3 ) x 0 dx 3! (0,0,v) T strongly in W 1,2 (iii) v 2 W 2,2 (W,R) and det 2 v = 1 (iv) lim inf h!0 h g+2eh (u h ) I (v)= 1 12 proven for g 2 (0,2) Z curl T curl S g W Q 2 2 v + B g Gauss curvature k(id 2 + 2h g S g )= h g curl T curl S g +O(h 2g ) k( (id + h g/2 ve 3 ) T (id + h g/2 ve 3 )) = h g det 2 v +O(h 2g ) higher regularity v 2 W 2,2 due to nonlinear rigidity estimate FJM. 8

9 The upper bound Theorem (L-O-P 14) Let v 2W 2,2 (W,R) with det 2 v = curl T curl S g. Then there exists a recovery sequence u h 2 W 1,2 (W h,r 3 ) such that: (i) y h (x 0,x 3 )! x 0 (ii) 1 h g/2 Z 1/2 1/2 yh (x 0,x 3 ) x 0 dx 3! (0,0,v) T (iii) lim h!0 1 h g+2eh (u h )=I (v)= 1 12 Z W Q 2 2 v + B g proven for g 2 (1,2) also for g 2 (0,1], when f = curl T curl S g > 0, and one has: C 2,a ( W) \A f is dense in A f := {v 2 W 2,2 (W); det 2 v = f } the above holds, when f const > 0 and W is star-shaped. 9

10 The Monge-Ampère constrained energy 1 G I(v) if v 2 A! f h g+2eh f = curl + if v 62 A T curl S g f Condition A f 6= /0 is equivalent to: infe h apple Ch g+2 Condition: [ curl B g 6 0 or curl T curl S g + det B g 6 0 ] is equivalent to: infe h ch g+2 (c > 0) Gauss-Codazzi eqns corresponding to the ideal 1st fundamental form I = Id 2 + 2h g S g, and the ideal 2nd fundamental form II = h g/2 B g : curl II = hii : G(I)i h g/2 h g h 3g/2 det(ii)=(deti)k(i) h g curl T curl S g. Linearized Gauss-Codazzi eqns: curl B g = 0, curl T curl S g + det B g = 0 10

11 Construction of the recovery sequence Let v 2 A f, i.e.: v 2 W 2,2 (W) and det 2 v = f = curl T curl S g If u h (x 0 )=x 0 + h g/2 v(x 0 )e 3 then: ( u h ) T u h = Id 2 + h g v v If u h (x 0 )=x 0 + h g/2 v(x 0 )e 3 + h g w(x 0 ) then: ( u h ) T u h = Id 2 + h g v v + 2h g sym w + h.o.t. (we want) = Id + 2h g S g Define w 2 W 1,2 (W,R 2 ) by: sym w = 1 2 v v + S g. Then: u h (x 0,x 3 ) := x 0 + h g/2 v(x 0 )e 3 + h g w(x 0 ) + x 3 apple h g/2 v(x 0 ) 1 + higher order corrections. works for 1 < g < 2 techniques: truncation, approximation (Friesecke-James-Muller 06) another approach: Matching property. 11

12 Matching of 2nd order isometries for elliptic metrics Theorem (L-Ochoa-Pakzad 14) Assume: curl T curl S g c > 0 in W. If v 2 C 2,a ( W,R) and det 2 v = curl T curl S g then 9 w e 2 C 2,a ( W,R 3 ) equibounded, such that: 8e id + e(ve 3 )+e 2 w e T id + e(ve3 )+e 2 w e = Id 2 + 2e 2 S g. Corollary If A curl T curl S g \C 2,a (W) is W 2,2 dense in A curl T curl S g then there exists a more efficient recovery sequence: u h (x 0,x 3 )=x 0 + h g/2 v(x 0 )e 3 + h g w g/2 (x 0 ) + x 3 ~N g + higher order correction terms works for the full range 0 < g < 2. 12

13 Proof of the matching property Find w e = w 0 e + w 3 ee 3 in the isometry equation: Id 2 + e 2 (2sym w 0 e + v v) + 2e 3 sym( v w 3 e) + e 4 ( w 0T e w 0 e + w 3 e w 3 e)=id 2 + 2e 2 S g Match terms at e 2, 0 = curl T curl( v v 2S g )= 2(det 2 v + curl T curl S g ). Use implicit function theorem! (id + e 2 w 0 e) T (id + e 2 w 0 e) = Id 2 + e 2 2S g ( v + e w 3 e) ( v + e w 3 e). 13

14 Proof of the matching property Find w e = w 0 e + w 3 ee 3 in the isometry equation: Id 2 + e 2 (2sym w 0 e + v v) + 2e 3 sym( v w 3 e) + e 4 ( w 0T e w 0 e + w 3 e w 3 e)=id 2 + 2e 2 S g Match terms at e 2, 0 = curl T curl( v v 2S g )= 2(det 2 v + curl T curl S g ). Use implicit function theorem! (id + e 2 w 0 e) T (id + e 2 w 0 e) = Id 2 + e 2 2S g ( v + z) ( v + z) := g e (z). 14

15 Proof of the matching property Find w e = w 0 e + w 3 ee 3 in the isometry equation: Id 2 + e 2 (2sym w 0 e + v v) + 2e 3 sym( v w 3 e) + e 4 ( w 0T e w 0 e + w 3 e w 3 e)=id 2 + 2e 2 S g Match terms at e 2, 0 = curl T curl( v v 2S g )= 2(det 2 v + curl T curl S g ). Use implicit function theorem! (id + e 2 w 0 e) T (id + e 2 w 0 e) = Id 2 + e 2 2S g ( v + z) ( v + z) := g e (z) Need to solve: Gauss curvature k g e (z e ) = 0 Actually: k g e (z e ) = e 2 F (e,z e ). 15

16 Proof of the matching property F (e,z)= det( 2 v + 2 z [G k ij (I) k(v + z)]) (1 e 2 I ij i (v + z) j (v + z)) 4 deti + k(i)/e 2 (1 e 2 I ij i (v + z) j (v + z)) 2 where I = Id 2 + 2e 2 S g is the target metric on W.. 16

17 Proof of the matching property F (e,z)= det( 2 v + 2 z [G k ij (I) k(v + z)]) (1 e 2 I ij i (v + z) j (v + z)) 4 deti curl T curl S g + o(1) (1 e 2 I ij i (v + z) j (v + z)) 2 where I = Id 2 + 2e 2 S g is the target metric on W. F (0,0)= det 2 v curl T curl S g = 0 F / z(0,0) : z 7! cof 2 v : 2 z invertible since v convex.. 17

18 Proof of the matching property F (e,z)= det( 2 v + 2 z [G k ij (I) k(v + z)]) (1 e 2 I ij i (v + z) j (v + z)) 4 deti curl T curl S g +O(e 2 ) (1 e 2 I ij i (v + z) j (v + z)) 2 where I = Id 2 + 2e 2 S g is the target metric on W. F (0,0)= det 2 v curl T curl S g = 0 F / z(0,0) : z 7! cof 2 v : 2 z invertible since v convex. Hence for f T e f e = g e (z e ) 2 C 1, 9!f e 2 C 2 by Mardare 03 Actually f e 2 C 2,a ( W,R 2 ), equibounded and: f e = id + e 2 w 0 e with w 0 e 2 C 2,a equibounded. [Use continuity equation: 2 ij f e = G k ij (g e) k f e.] 18

19 Density: interior regularity v 2 W 2,2 (W,R), det 2 v = f, f 2 C (W,R), f c > 0. Question: 9? v n 2 C 2,a (W,R), det 2 v n = f, v n! v in W 2,2 (W)? Theorem (L-Mahadevan-Pakzad 13) v (or v) is locally convex in W. Uses only f c > 0. Based on V. Sverak s unpublished preprint 91 for v 2 W 2,. Convexity ) v is an Alexandrov s solution ) v 2 C (W) ( Example: W = B(0,1) R 2 x, v(x,y)= 2 e y2 /2 if x 0 x 2 e y2 /2 if x < 0 v 2 C 1,1 (W) and det 2 v = 2x 3 e y2 (1 y 2 ) smooth and non-negative but v is neither convex nor concave, v /2 C 2. 19

20 Sverak s method: proof of interior regularity u = v is a mapping of integrable dilatation: 8a.e. x 2 W u 2 (x) apple K(x)det u(x), K = u 2 c By Sverak and Iwaniec theorem: u = f h 1 2 L 1 (W) h : W 0! W homeomorphism, f : W 0! R 2 C holomorphic. Consequently: u is continuous and locally 1 1 outside of S := h(( f) 1 (0)) S is discrete, hence W \ S is open, connected, and of full measure Proposition If 9x 0 2 W \ S a Lebesgue point of 2 v with Lebesgue value A > 0, then v is locally convex in W \ S. This concludes the proof since S is discrete. 20

21 Sverak s method: proof of Proposition Classical result by Vodopyanov-Goldstein (u 2 W 1,2, det u > 0): u is monotonous: osc Br u i = osc Br u i. Logarithmic modulus of continuity: 8 B d B R W: osc Bd u i apple p 2p ln R 1 2k u i k d L 2 (B R ) Let x 0 2 W be a Lebesgue point of 2 v, A = 2 v(x 0 ). By blow-up method: 8e > 0 9r 0 > 0 8r < r 0 8a 2 B(x 0,r) kv(x) v(a)+ v(a) (x a)+ 1 2 (x a) A(x a) k C 0 (B r ) apple er2 Let x 0 2 W \ S as above with A > 0 (if necessary, change sign). Then v has a locally supporting hyperplane at x 0 : 8x 2 B r (x 0 ) v(x) v(x 0 )+ v(x 0 ) (x x 0 ) Recall that v is locally 1 1 in W \ S. Apply Theorem by J. Ball and use connectedness of W \ S. 21

22 Density: a special case v 2 W 2,2 (W,R), det 2 v = f, f 2 C (W,R), f c > 0. Question: 9? v n 2C 2,a (W,R), det( 2 v n )= f, v n! v in W 2,2 (W)? Corollary (to the regularity Theorem) If f c > 0 and W is star-shaped w.r.t. a ball, then: 9v n 2 C 2,a (W,R), det( 2 v n )= f, v n! v in W 2,2 (W). Proof: v l (x)= 1 l2v(lx) for 0 < l < 1. v l is smooth in W and v l! v in W 2,2 as l! 1. Clearly: det 2 v l = f. 22

23 Uniqueness of minimizers Problem: Uniqueness/multiplicity of minimizers/critical points of: Z I (v)= W 2 v 2, subject to det 2 v = f := curl T curl S g. (up to affine modifications) Let W = B(0,1) R 2 and f 1. There exists a non-trivial one parameter family of minimizers: v(x,y,q)=(sinq) x2 y 2 Let W = B(0,1) R 2 and f (cos q)xy. Then there is one unique minimizer: v(x,y)= x2 + y 2. 2 Observe: Uniqueness ) radial symmetry of the minimizer when W = B(0,1) and f = f (r) is radially symmetric. 23

24 A partial result Lemma (L-Pakzad 13) Let f (r) c > 0 be an L 2 (B(0,1)) function satisfying f (r) apple inf [0,r] f for a.e. r 2 [0,1], i.e. f is non-increasing. Then: the radial solution v f 2 W 2,2 (B(0,1)) is the unique minimizer of I (v) constrained to det 2 v = f. Let f = f (r) c > 0 be smooth. Then v f is a critical point of I. 24

25 G(x 0,x 3 )=G(x 0 ), The limit scaling g = 0 E h (u h )= 1 h Theorem (Bhattacharya-L-Schaffner 14) Z W hw(( uh ) p G 1 ), If E h (u h ) apple Ch 2, then 9c h 2 R 3 such that the following holds for the sequence: y h (x 0,x 3 ) := u h (x 0,hx 3 ) c h 2 W 1,2 (W 1,R 3 ) (i) y h (x 0,x 3 )! y(x 0 ) strongly in W 1,2 (ii) y 2 W 2,2 (W,R) and ( y) T y = G (iii) lim inf h!0 h 2Eh (u h ) I 2 (y)= 1 Z 24 Q 2 x 0,sym(( y) T ~b) dx 0 W where ~b 2 W 1,2 \ L (W,R 3 ) is the Cosserat vector: 1 y 2 y ~b T 1 y 2 y ~b = G When G(x 0 )=G e 3 e 3 then ~b = ~N = 1y 2 y 1 y 2 y, and ( y) T ~b = P y(w). 25

26 The limit scaling g = 0: the upper bound Theorem (B-L-S 14) Let y 2 W 2,2 (W,R 3 ) with ( y) T y = G 2 2. Then there exists u h 2 W 1,2 (W h,r 3 ) such that: (i) y h (x 0,x 3 )! y(x 0 ) 1 (ii) lim h!0 h 2Eh (u h )= 1 24 Z W Q 2 x 0,sym(( y) T ~b) dx 0 Corollary We have: infe h apple Ch 2 if and only if: 9 y 2 W 2,2 (W,R 3 ) ( y) T y = G 2 2. Corollary We have: infe h ch 2 if and only if: 69y 2 W 2,2 (W,R 3 ) ( y) T y = G 2 2, sym(( y) T ~b)=0. 26

27 The optimal energy scaling Theorem (B-L-S 14) We have: infe h ch 2 if and only if one of the following equivalent conditions DOES NOT hold: (i) 9 smooth y : W! R 3 ( y) T y = G 2 2, sym(( y) T ~b)=0. (ii) 9 smooth y : W! R 3 isometric immersion of G 2 2, apple with 2nd fundamental form: P y(w) = p1 G 3 11 G 3 12 G 33 G 3 12 G3 22 (iii) R = R3 221 = R (Riemann curvatures of G) When G(x 0 )=G e 3 e 3 then ~b = ~N and: (i), k(g 2 2 )=0, R(G)=0. Hence, if: 1 h 2 infeh! 0, then mine h = 0. 27

28 The optimal energy scaling: examples Let G(x 0 )=l(x 0 )Id 3. Then R = R3 221 = 0 and: R 1212 = l 2 k(lid 2 )= 1 2 ld(logl). Hence: (i), 1 h 2 infeh! 0, D(logl)=0. But: R(G)=0, (logl)=0. If (logl) 6= 0 then scaling of recovery sequence: E h (u h ) apple Ch x x 1 x 2 b 1 + x 1 b 3 Let G(x 1,x 2 )= 4 x 1 x x2 2 7 b 2 + x 2 b 3 5 b 1 + x 1 b 3 b 2 + x 2 b 3 ~b 2 where ~b = x 3 13, x3 2 3, x2 1 x2 2 2 T. Then: k(g2 2 )= 1 (1+x1 2 6= 0. +x2 2 )2 We have: (i), 1 h 2 infeh! 0 holds (as Db 3 = 0). But scalar Ricci curvature S(G)= 12 2x x = 0, hence infeh 6= 0. 28

29 Liquid glass model: A = l n v v + l n w w + l~n ~n where l > 1, director vector ~n = 1, ~n? = span(v,w) G(x 0 2n )=r n+1 Id 3 +(r 2 1)~n ~n, r = l n+1. Summary We rigorously derived, starting from nonlinear non-euclidean elasticity model, a 2-dimensional energy of the form: I(v)= R W Q 2( 2 v) subject to the constraint: v 2 W 2,2 (W), det 2 v = f We explained the matching property; we proved matching of 2nd order to exact C 2,a isometries for metrics with: k > 0 Question: Can C 2,a be relaxed to W 2,2?

30 Summary - ctd We explained the need of density of smooth solutions v of det 2 v = f in the set of solutions v 2 W 2,2. We proved interior regularity and convexity. We deduced density for f = const > 0. Question: Density in a general case? We discussed uniqueness/multiplicity of minimizers to I(v). Question: Uniqueness for f > 0, multiplicity for f < 0? The same as above for fully nonlinear model (g = 0)? We derived the nonlinear model: I 2 (y)= R W Q 2 x 0,sym(( y) T ~b) subject to: ( y) T y = G 2 2, where ~b = Cosserat vector. This model is relevant when: R R R Question: hierarchy of models in vanishing of curvatures? 29

31 THANK YOU. 30

PRESTRAINED ELASTICITY: FROM SHAPE FORMATION TO MONGE-AMPE RE ANOMALIES

PRESTRAINED ELASTICITY: FROM SHAPE FORMATION TO MONGE-AMPE RE ANOMALIES MARTA LEWICKA AND MOHAMMAD REZA PAKZAD 1. Introduction Imagine an airplane wing manufactured in a hyperbolic universe and imported