On the Monge-Amp` ere equation via prestrained elasticity Marta - PowerPoint PPT Presentation

On the Monge-Amp` ere equation via prestrained elasticity Marta Lewicka University of Pittsburgh — 27 January 2017, ICERM, Providence — Workshop for Professor Susan Friedlander 1 / 23

An old story: isometric immersions (equidimensional) Assume that u : R n � Ω ! R n satisfies: ∇ u ( x ) T ∇ u ( x ) = Id n Equation of isometric immersion: h ∂ i u , ∂ j u i = δ ij = h e i , e j i (For u 2 C 1 , this is equivalent to u preserving length of curves) � R ; R T R = Id Equivalent to: ∇ u 2 O ( n ) = = SO ( n ) [ SO ( n ) J J = diag { � 1 , 1 ,..., 1 } Liouville (1850), Reshetnyak (1967): u 2 W 1 , ∞ and ∇ u 2 SO ( n ) a.e. in Ω ) ∇ u ⌘ const ) u ( x ) = Rx + b rigid motion 2 / 23

An old story: isometric immersions (equidimensional) Gromov (1973): Convex integration: 9 u 2 W 1 , ∞ such that ( ∇ u ) T ∇ u = Id a.e. in Ω , and ∇ u takes values in SO ( n ) and in SO ( n ) J , in every open U ⇢ Ω . Even more: 9 u arbitrarily close to any u 0 with 0 < ( ∇ u 0 ) T ∇ u 0 < Id Example: 0 ) 2 < 1 Given u 0 : ( 0 , 1 ) ! R with ( u 0 k ) 2 = 1 uniformly ! u 0 with ( u 0 � want: u k more oscillations as k ! ∞ Hevea project: Inst. Camille Jordan, Lab J. Kuntzmann, Gipsa-Lab (France) 3 / 23

Isometric immersions of Riemann manifold ( Ω , G ) sym , + ) . Look for u : Ω ! R n so that ( ∇ u ) T ∇ u = G in Ω Let G 2 C ∞ ( Ω , R n ⇥ n Theorem (Gromov 1986) Let u 0 : Ω ! R n be smooth short immersion, i.e.: 0 < ( ∇ u 0 ) T ∇ u 0 < G in Ω . Then: 8 ε > 0 9 u 2 W 1 , ∞ k u � u 0 k C 0 < ε and ( ∇ u ) T ∇ u = G. Theorem (Myers-Steenrod 1939, Calabi-Hartman 1970) Let u 2 W 1 , ∞ satisfy ( ∇ u ) T ∇ u = G and det ∇ u > 0 a.e. in Ω . (For example, u 2 C 1 enough). Then ∆ G u = 0 and so u is smooth. In fact, u is unique up to rigid motions, and: 9 u , Riem ( G ) ⌘ 0 in Ω . W ( F ) ⇠ dist 2 ( F , SO ( 3 )) Z � � ( ∇ u ) G � 1 / 2 ( x ) E ( u ) = W d x G 2 C ∞ ( Ω , R 3 ⇥ 3 sym , + ) incompatibility Ω metric tensor E ( u ) = 0 , ∇ u ( x ) 2 SO ( 3 ) G 1 / 2 ( x ) 8 a . e . x , ( ∇ u ) T ∇ u = G and det ∇ u > 0 4 / 23

Non-Euclidean elasticity Lemma (L-Pakzad 2009) inf u 2 W 1 , 2 E ( u ) > 0 , Riem ( G ) 6⌘ 0 . Thin non-Euclidean plates: Ω = Ω h = ω ⇥ ( � h / 2 , h / 2 ) , ω ⇢ R 2 Scaling of: inf E h ⇠ h β ? argmin E h ! argmin I β ? As h ! 0: Hierarchy of theories I β , where β depends on Riem ( G h ) Bhattacharya, Li, L., Mahadevan, Pakzad, Raoult, Schaffner When G = Id : dimension reduction in nonlinear elasticity seminal analysis by LeDret-Raoult 1995, Friesecke-James-Muller 2006 Manufacturing residually-strained thin films: Shaping of elastic sheets by prescription of Non-Euclidean metrics (Klein, Efrati, Sharon) Science, 2007 Half-tone gel lithography (Kim, Hanna, Byun, Santangelo, Hayward) Science, 2012 Defect-activated liquid crystal elastomers (Ware, McConney, Wie, Tondiglia, White) Science, 2015 5 / 23

The Monge-Amp` ere constrained energy E h ( u h ) = 1 Z � � ( ∇ u h )( G h ) � 1 / 2 ( x ) Energy Ω h W d x h Theorem (L-Ochoa-Pakzad 2014) Let: G h ( x 0 , x 3 ) = Id 3 + 2 hS ( x 0 ) . Then: inf E h  Ch 3 , 9 v 2 W 2 , 2 ( ω ) , det ∇ 2 v = � curl curl S 2 ⇥ 2 Γ 1 h 3 E h � ! I, where I is the 2-d energy: ω | ∇ 2 v | 2 for v 2 W 2 , 2 ( ω ) , det ∇ 2 v = � curl curl S 2 ⇥ 2 I ( v ) = R [More general result for G h ( x 0 , x 3 ) = Id 3 + 2 h γ S ( x 0 ) and γ 2 ( 0 , 2 ) . When γ � 2 then higher order models.] u h ( x 0 , 0 ) = x 0 + h 1 / 2 ve 3 Structure of minimizers to E h : κ ( ∇ ( id + h 1 / 2 ve 3 ) T ∇ ( id + h 1 / 2 ve 3 )) = κ ( Id 2 + h ∇ v ⌦ ∇ v ) = � 1 2 h curl curl ( ∇ v ⌦ ∇ v )+ O ( h 2 ) = h det ∇ 2 v + O ( h 2 ) Gauss curvature: κ ( Id 2 + 2 hS 2 ⇥ 2 ) = � h curl curl S 2 ⇥ 2 + O ( h 2 ) 6 / 23

Weak formulation of the Monge-Amp` ere equation • existence of W 2 , 2 solutions is not guaranteed det ∇ 2 v = f � � v 2 W 1 , 2 ( ω ) Det ∇ 2 v = � 1 ∇ v ⌦ ∇ v 2 curl curl curl curl ( ∇ v ⌦ ∇ v ) = curl curl S 2 ⇥ 2 Need to solve: where S 2 ⇥ 2 = λ Id 2 ∆ λ = � 2 f in ω . with 3 eqns in 3 unknowns ∇ v ⌦ ∇ v + sym ∇ w = S 2 ⇥ 2 Equivalently: on a 2d domain ( ∇ u ) T ∇ u = G 2 ⇥ 2 , where u : ω ! R 3 Similar problem 1: isometric immersion of 2d metric in R 3 . Nirenberg (1953): 8 G 2 ⇥ 2 , κ > 0 9 smooth isometr. embed. in R 3 Poznyak-Shikin (1995): Same true for κ < 0 on bounded ω ⇢ R 2 Nash-Kuiper (1956): 8 n -dim G 9 C 1 , α isometr. embed. in R n + 1 Case G 2 ⇥ 2 : Borisov (2004), Conti-Delellis-Szekelyhidi (2010) α < 1 7 Delellis-Inauen-Szekelyhidi (2015) α < 1 5 . C 1 , 2 3 + solutions are rigid – convex case: Borisov (2004). 7 / 23

Weak formulation of the Monge-Amp` ere equation • existence of W 2 , 2 solutions is not guaranteed det ∇ 2 v = f � � v 2 W 1 , 2 ( ω ) Det ∇ 2 v = � 1 ∇ v ⌦ ∇ v 2 curl curl curl curl ( ∇ v ⌦ ∇ v ) = curl curl S 2 ⇥ 2 Need to solve: where S 2 ⇥ 2 = λ Id 2 ∆ λ = � 2 f in ω . with 3 eqns in 3 unknowns ∇ v ⌦ ∇ v + sym ∇ w = S 2 ⇥ 2 Equivalently: on a 2d domain ∂ t u + div ( u ⌦ u )+ ∇ p = 0 , div u = 0 Similar problem 2: 3d incompressible Euler equations, ( u , p ) : T 4 ⇥ [ 0 , T ] ! R 4 . Onsager’s conjecture: rigidity/flexibility treshold = 1 3 . Constantin-E-Titi, Eyink (1994): Every L ∞ ( 0 , T ; C α ( T 3 )) solution, α > 1 3 , is energy conserving. Delellis, Szekelyhidi, Buckmaster, Isett: existence of non-energy- conserving solutions, for every α < 1 3 : compactly supported in time; arbitrary temporal kinetic energy profile. 8 / 23

Rigidity for the Monge-Amp` ere equation Theorem (L-Pakzad 2015) Let v 2 C 1 , 2 3 + . If Det ∇ 2 v = 0 , then v is developable, i.e. 8 x 2 ω : • either v is affine in B ε ( x ) • or ∇ v is constant on a segment through x, joining ∂ω on both ends. If Det ∇ 2 v � c > 0 is Dini continuous, then v is locally convex and an Alexandrov solution in ω . Theorem (Pakzad 2004, Sverak 1991, L-Mahadevan-Pakzad 2013) Let v 2 W 2 , 2 . If det ∇ 2 v = 0 then v is developable and v 2 C 1 . If det ∇ 2 v � c > 0 then v 2 C 1 and v (or � v) is locally convex. 9 / 23

Flexibility for the Monge-Amp` ere equation Theorem (L-Pakzad 2015) Let ( v 0 , w 0 ) : ω ! R ⇥ R 2 be a smooth short infinitesimal, i.e.: ∇ v 0 ⌦ ∇ v 0 + sym ∇ w 0 < S 2 ⇥ 2 . 7 � ( v n , w n ) Then 9 ( v n , w n ) 2 C 1 , 1 uniformly � ! ( v 0 , w 0 ) and ∇ v n ⌦ ∇ v n + sym ∇ w n = S 2 ⇥ 2 . 1 5 � . Extension [L-Pakzad-Inauen 2017]: flexibility at C Corollary (“Ultimate flexibility”) 7 7 . The set of C 1 , α (¯ 6 ( ω ) and α < 1 ω ) solutions to the Monge - Let f 2 L ere equation: Det ∇ 2 v = f is dense in the space C 0 (¯ ω ) . Amp` For f 2 L p ( ω ) and p 2 ( 1 , 7 6 ) , the density holds for any α < 1 � 1 p . Det ∇ 2 is weakly discontinuous everywhere in W 1 , 2 ( ω ) . 7 � ) inf E h  Ch Consequences for energy scaling: flexibility at C 1 , 1 9 4 . 3 � , optimal for Nash-Kuiper, ) inf E h  Ch (If flexibility at C 1 , 1 10 4 ). Energy gap: Ch 3 �  inf E h  Ch 10 4 + : residual energy? fine crumpling? 10 / 23

Rigidity for the Monge-Amp` ere equation Tool: Degree formula via the commutator estimate. Commutator estimate argument in the Euler rigidity: ∂ t u + div ( u ⌦ u )+ ∇ p = 0 , div u = 0. Mollify on scale ε : ∂ t ( u ε )+ div ( u ⌦ u ) ε + ∇ p ε = 0 , div u ε = 0 � R 1 2 | u ε | 2 � R ⌦ ↵ d � ( u ⌦ u ) ε : ∇ u ε = 0 Integrate by parts with u ε : dt Add new trilinear term for free: � R 1 2 | u ε | 2 � R ⌦ ↵ d = ( u ⌦ u ) ε � ( u ε ⌦ u ε ) : ∇ u ε . dt Use commutator estimate: k ( fg ) ε � f ε g ε k C k  C ε 2 α � k k f k C 0 , α k g k C 0 , α � � R ⌦ ↵� �  C ε 2 α ε α � 1 = C ε 3 α � 1 ( u ⌦ u ) ε � ( u ⌦ u ) ε : ∇ u ε to bound: ! 0 when α > 1 3 . For the Monge-Amp` ere equation, use similar argument in the geometrical context. 11 / 23

Rigidity for the Monge-Amp` ere equation Lemma (L-Pakzad 2015) Let v 2 C 1 , 2 3 + , f 2 L 1 + satisfy: Det ∇ 2 v = f. Then: Z Z 8 φ 2 L ∞ ( U b ω ) U ( φ � ∇ v ) f = R 2 φ ( y ) deg ( ∇ v , U , y ) d y supp φ ⇢ R 2 \ ( ∇ v )( ∂ U ) . 1 2 ∇ v ⌦ ∇ v + sym ∇ w = A , f = � curlcurl A . Proof. 1 Mollify on scale ε : 2 ( ∇ v ⌦ ∇ v ) ε + sym ∇ w ε = A ε Apply degree formula to smooth v ε : U ( φ � ∇ v ε ) det ∇ 2 v ε = R R R 2 φ ( y ) deg ( ∇ v ε , U , y ) d y ! RHS U ( φ � ∇ v ε ) det ∇ 2 v ε � ( φ � ∇ v ) f R Error in LHS: R ( φ � ∇ v ε � φ � ∇ v ) f � R ( φ � ∇ v ε ) curlcurl � 1 � = 2 ∇ v ε ⌦ ∇ v ε � A R ( φ � ∇ v ε ) curlcurl � 1 � first term ! 0 , last term: 2 ∇ v ε ⌦ ∇ v ε � A ε R ( φ � ∇ v ε ) curlcurl � � = 1 ∇ v ε ⌦ ∇ v ε � ( ∇ v ⌦ ∇ v ) ε 2 R ⌦ � �↵ ∇ ? ( φ � ∇ v ε ) , curl = � 1 ∇ v ε ⌦ ∇ v ε � ( ∇ v ⌦ ∇ v ) ε 2 bounded by: C ε α � 1 ε 2 α � 1 = C ε 3 α � 2 ! 0 when α > 2 3 . 12 / 23