Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models
aa r X i v : . [ m a t h . A P ] D ec QUANTITATIVE IMMERSABILITY OF RIEMANN METRICSAND THE INFINITE HIERARCHY OF PRESTRAINED SHELL MODELS
MARTA LEWICKA
Abstract.
This paper concerns the variational description of prestrained materials, in the contextof dimension reduction for thin films Ω h = ω × ( − h , h ). Given a Riemann metric G on Ω , we studythe question of what is the infimum of the averaged pointwise deficit of a given immersion from beingan orientation-preserving isometric immersion of G | Ω h on Ω h , over all weakly regular immersions.This deficit is measured by the non-Euclidean energies E h , which can be seen as modifications of theclassical nonlinear three-dimensional elasticity.Building on our previous results, we complete the scaling analysis of E h and the derivation ofΓ-limits of the scaled energies h − n E h , for all n ≥
1. We show the energy quantisation, in thesense that the even powers 2 n of h are indeed the only possible ones (all of them are also attained).For each n , we identify the equivalent conditions for the validity of the corresponding scaling, interms of the vanishing of appropriate Riemann curvatures of G up to certain orders, and in termsof the matched isometry expansions. We also establish the asymptotic behaviour of the minimizingimmersions as h → Introduction
In this paper, we propose results that address and relate the following two contexts:(i) Quantitative analysis of immersability of Riemann metrics.(ii) Dimension reduction in non-Euclidean elasticity of prestrained thin films.It is a well-known fact that a three-dimensional Riemann metric G has a smooth isometricimmersion in R , if an only if its curvature tensor R ( G ) = { R ab,cd } a,b,c,d =1 ... vanishes identi-cally. The smoothness requirement may be replaced by the orientation-preservation of a Lips-chitz continuous immersion; then the condition R ( G ) = 0 automatically yields smoothness anduniqueness, up to rigid motions. When R ( G ) = 0, one may pose the question of what is theinfimum of the average pointwise deficit from being an orientation-preserving isometric immer-sion, over all, weakly regular, immersions. We study this question on a family of thin films (cid:8) Ω h = ω × ( − h , h ) (cid:9) h → around a given two-dimensional midplate ω , where the said deficit ismeasured by the energy: E h ( u ) = ffl Ω h dist (( ∇ u ) G − / , SO (3)). Our first goal is to determine thepossible scalings: inf E h ∼ h β , as h →
0, in terms of powers β of the thickness h . We are theninterested in identifying properties of G , that correspond to each scaling range, in function of thecurvature components and their derivatives. Finally, we want to predict the asymptotics of theminimizing immersions as h → G , which is induced by mechanisms such as growth, plasticity, thermal expansion etc. Thebody wants to realize the distances between its constitutive cell elements, which are set by G , bydeforming its shape. Since this realization is taking place in the flat three-dimensional space, it isimpossible unless R ( G ) = 0. This condition is precisely equivalent to having the stored non-Euclidean Date : December 27, 2018. energy of deformations infimize to zero. In the variational description of thin prestrained films Ω h ,we thus study the nonlinear energies: (cid:8) E h ( u ) = ffl Ω h W (( ∇ u ) G − / ) } h → and, as above, want todetermine the viable scalings of their infima, their singular limits as h →
0, and the asymptoticbehaviour of the three-dimensional minimizing shapes.In our previous works [28, 6] we analyzed the scenario: inf E h ∼ h , whereas in [29, 30] we showedthat the next limiting energy level beyond h is: inf E h ∼ h , arising when { R ,ab } a,b =1 ... = 0on ω . Then we observed that the further scaling level is: inf E h ∼ h and that it corresponds to R ( G ) = 0 on ω . In the present paper, we complete this analysis and provide the derivation of theΓ-limits I n to scaled energies h − n E h , for all n ≥
1. We prove the previously conjectured energyquantisation so that h n are indeed the only possible scalings, all of them attained (by G = e x n Id .The structure of {I n } n ≥ should be compared with the hierarchy of plate models in the classicalnonlinear elasticity [9], as follows. The energy I consists of pure bending, quantifying the curvatureunder the midplate isometric immersion constraint. This is a Kirchhoff-like model, relative to theambient metric G . The next energy I consists of linearised first order bending and second orderstretching; this is a von Karman-like model, augmented by terms carrying the relevant componentsof the Riemann tensor R ( G ). Each higher order energy I n consists of linearised bending augmentedby the the order-related covariant derivatives of R ( G ) on the midplate. This is a linear elasticity-likemodel, in the present context valid in the quantized scaling regimes n ≥
3, whereas in the classicalcase appearing in the regimes h β for all β > E h ∼ h β with β ∈ (0 , β ≥ G × on ω , admits a W , isometric immersion in R . While the systematic description of the singular limits at scalings β < h β in non-even regimes of β > The set-up of the problem.
Let ω ⊂ R be an open, bounded, connected set with Lipschitzboundary. We consider a family of thin hyperelastic sheets occupying the reference domains:Ω h = ω × (cid:16) − h , h (cid:17) ⊂ R , < h ≪ . A typical point in Ω h is denoted by x = ( x , x , x ) = ( x ′ , x ). We often use the unit-thickness plateΩ as the referential rescaling of each Ω h via: Ω h ∋ ( x ′ , x ) ( x ′ , x /h ) ∈ Ω .The films Ω h are characterized by the given smooth incompatibility (Riemann metric) tensor: G ∈ C ∞ ( ¯Ω , R × , pos )and we want to study the singular limit behaviour, as h →
0, of the following energy functionals:(1.1) E h ( u h ) = 1 h ˆ Ω h W (cid:0) ∇ u h ( x ) G ( x ) − / (cid:1) d x = ˆ Ω W (cid:0) ∇ u h ( x ′ , hx ) G ( x ′ , hx ) − / (cid:1) d x, IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 3 defined on vector fields u h ∈ W , (Ω h , R ) interpreted as deformations of Ω h . Above, G ( x ) − / standsfor the inverse of G ( x ). When G = Id , the functionals E h are the classical Hookean nonlinear elasticenergies of deformations, with the density W obeying the properties listed below.In the present general setting, E h ( u h ) is designed to measure the deviation of u h from beingan (equidimensional) isometric immersion of G on Ω h . Indeed, by polar decomposition theorem, F G − / ∈ SO (3) if and only if F T F = G and det F >
0. The Borel-regular, homogeneous density W : R × → [0 , ∞ ] is thus assumed to satisfy:(i) W ( RF ) = W ( F ) for all R ∈ SO (3) and F ∈ R × ,(ii) W ( F ) = 0 for all F ∈ SO (3),(iii) W ( F ) ≥ C dist (cid:0) F, SO (3) (cid:1) for all F ∈ R × , with some uniform constant C > U of SO (3) such that W is finite and C regular on U .By a more refined analysis [28] one can prove the global counterpart of the above pointwise statement,namely that: inf W , E h = 0 if an only if all the components of the Riemann curvature tensor of G vanish identically: { R ab,cd } a,b,c,d =1 ... = 0 on Ω h .In this paper, we determine the possible energy scalings: inf E h ∼ h β in the limit of vanishingthickness h →
0, and the corresponding variational limits (Γ-limits) I β of h − β E h , in the regime β > β ∈ [2 , I β are typically given by energies of the form I = k T ensor ( y ) k E defined on the appropriateset of limiting deformations/displacements y of the midplate ω . They quantify the resulting effectivecurvatures in T ensor ( y ) relative to G at the level induced by β , and in the weighted L norm on ω :(1.2) E . = (cid:0) L ( ω, R × ) , k · k Q (cid:1) , k F k Q = (cid:16) ˆ ω Q ( x ′ , F ( x ′ )) d x ′ (cid:17) / . Above, the quadratic form Q carries the two-dimensional reduction of the first nonzero term in theTaylor expansion of W close to its energy well SO (3). More precisely, we define: Q ( F ) = D W ( Id )( F, F ) Q ( x ′ , F × ) = min n Q (cid:0) G ( x ′ , − / ˜ F G ( x ′ , − / (cid:1) ; ˜ F ∈ R × with ˜ F × = F × o . (1.3)The form Q is defined for all F ∈ R × , while each Q ( x ′ , · ) is defined on F × ∈ R × . Both Q and all Q are nonnegative definite and depend only on the symmetric parts of their arguments,in view of the assumptions on W . The quadratic minimization problem in (1.3) has thus a uniquesolution among symmetric matrices ˜ F , which for each x ′ ∈ ω is given via the linear function:(1.4) F × c ( x ′ , F × ) ∈ R with: Q ( x ′ , F × ) = Q (cid:16) G ( x ′ , − / (cid:0) F ∗ × + c ⊗ e (cid:1) G ( x ′ , − / (cid:17) . Description of the main results of this paper.
As already pointed out, we will be con-cerned with the regimes of curvatures of G , yielding the incompatibility rate, quantified by inf E h ,of order higher than h in the thickness h . We first recall the following result from [30]:(1.5) lim h → h inf E h = 0 ⇔ R ab,cd ( x ′ ,
0) = 0 for all x ′ ∈ ω, for all a, b, c, d : 1 . . . . The above conditions are further equivalent to existence of smooth vector fields y ,~b ,~b : ¯ ω → R ,defined uniquely up to rigid motions, such that for the following smooth R × matrix fields on ¯ ω : B = (cid:2) ∂ y , ∂ y , ~b (cid:3) , B = (cid:2) ∂ ~b , ∂ ~b , ~b (cid:3) , MARTA LEWICKA there holds: B T B = G ( x ′ ,
0) with det B > B T B ) sym = 12 ∂ G ( x ′ , (cid:0) ( ∇ y ) T ∇ ~b (cid:1) sym + ( ∇ ~b ) T ∇ ~b = 12 ∂ G ( x ′ , × . (1.6)Note that the last equality above implies that we can uniquely define a new smooth vector andmatrix fields: ~b : ¯ ω → R and B = (cid:2) ∂ ~b , ∂ ~b , ~b (cid:3) , so that: ( B T B ) sym + B T B = ∂ G ( x ′ , (cid:16) X k =0 x k k ! B k (cid:17) T (cid:16) X k =0 x k k ! B k (cid:17) = G ( x ′ , x ) + O ( h ) on Ω h , as h → . In conclusion, the following three conditions: the two conditions in (1.5) and the one in (1.7), areequivalent. Our first main result generalizes this statement to all even order powers 2( n + 1) inthe infimum energy scaling, for any n ≥
2. Moreover, these scalings exhaust all possibilities in theremaining regime: inf E h ∼ h β with β > Theorem 1.1.
The following three statements are equivalent, for each fixed integer n ≥ : (i) R , ( x ′ ,
0) = R , ( x ′ ,
0) = R , ( x ′ ,
0) = 0 for all x ′ ∈ ω , and ∂ ( k )3 R i ,j ( x ′ ,
0) = 0 for all x ′ ∈ ω , all k = 0 . . . n − and all i, j = 1 . . . . (ii) inf E h ≤ Ch n +1) . (iii) There exist smooth fields y , { ~b k } n +1 k =1 : ¯ ω → R such that calling (cid:8) B k = (cid:2) ∂ ~b k , ∂ ~b k , ~b k +1 (cid:3)(cid:9) nk =1 ,in addition to B = (cid:2) ∂ y , ∂ y , ~b (cid:3) satisfying det B > , we have: (1.8) (cid:16) n X k =0 x k k ! B k (cid:17) T (cid:16) n X k =0 x k k ! B k (cid:17) = G ( x ′ , x ) + O ( h n +1 ) on Ω h , as h → , or in other words: m X k =0 (cid:18) mk (cid:19) B T k B m − k − ∂ ( m )3 G ( x ′ ,
0) = 0 for all m = 0 . . . n , for all x ′ ∈ ω . We further prove compactness and the lower bound, at any of the new viable scaling levelsinf E h ∼ h n +1) , completing thus the analysis done for n = 0 in [28, 6] and for n = 1 in [29, 30]: Theorem 1.2.
Fix n ≥ and assume that any of the equivalent conditions in Theorem 1.1 holds.Let the sequence of deformations { u h ∈ W , (Ω h , R ) } h → satisfy: E h ( u h ) ≤ Ch n +1) . Then: (i) There exists ¯ R h ∈ SO (3) , c h ∈ R such that the displacements { V h ∈ W , ( ω, R ) } h → in: V h ( x ′ ) = 1 h n h/ − h/ ( ¯ R h ) T (cid:0) u h ( x ′ , x ) − c h (cid:1) − (cid:16) y ( x ′ ) + n X k =1 x k k ! ~b k ( x ′ ) (cid:17) d x converge as h → , strongly in W , ( ω, R ) , to the limiting displacement: (1.9) V ∈ V y = n V ∈ W , ( ω, R ); (cid:0) ( ∇ y ) T ∇ V (cid:1) sym = 0 a.e. in ω o . (ii) The above condition V ∈ V y automatically defines ~p ∈ W , ( ω, R ) such that: (cid:0) B T (cid:2) ∇ V, ~p (cid:3)(cid:1) sym = 0 a.e. in ω, IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 5 and then we have: lim inf h → h n +1) E h ( u h ) ≥ I n +1) ( V ) , where: I n +1) ( V ) = 124 · (cid:13)(cid:13)(cid:13) ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b + α n (cid:2) ∂ ( n − R i ,j (cid:3) i,j =1 ... (cid:13)(cid:13)(cid:13) Q + β n · (cid:13)(cid:13)(cid:13) P S ⊥ y (cid:0)(cid:2) ∂ ( n − R i ,j (cid:3) i,j =1 ... (cid:1)(cid:13)(cid:13)(cid:13) Q + γ n · (cid:13)(cid:13)(cid:13) P S y (cid:0)(cid:2) ∂ ( n − R i ,j (cid:3) i,j =1 ... (cid:1)(cid:13)(cid:13)(cid:13) Q . (1.10) Above, S y is the following closed subspace of the Hilbert space E in (1.2): S y = closure E n(cid:0) ( ∇ y ) T ∇ w (cid:1) sym ; w ∈ W , ( ω, R ) o , whereas P S y ( F ) and P S ⊥ y ( F ) denote, respectively, the orthogonal projections of F onto thespace S y and its orthogonal complement S ⊥ y in E . The coefficients in (1.10) are: α n = for n odd n ( n + 3)( n + 1)! for n even ,β n = 12 n +3 (2 n + 3) (cid:0) ( n + 1)! (cid:1) · for n odd n ( n + 3) for n even ,γ n = 12 n +3 (2 n + 3) (cid:0) ( n + 1)! (cid:1) · ( n + 1) ( n + 2) for n odd n ( n + 3) for n even . (1.11)(iii) There holds on ω : · (cid:2) ∂ ( n − R i ,j ( · , (cid:3) i,j =1 ... = 2 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( · , × . (1.12)We point out a few related observations:(i) When G = Id , then each functional in (1.10) reduces to the classical linear elasticity. Wehave: y = id , ~b = e and V = (cid:8) ( αx ⊥ + ~β, v ); α ∈ R , ~β ∈ R , v ∈ W , ( ω ) (cid:9) , and for V ∈ V ,there holds: ~p = ( −∇ v, I n +1) ( V ) = 124 ˆ ω Q (cid:0) x ′ , ∇ v (cid:1) d x ′ , in functionof the out-of-plane scalar displacement v .(ii) In the present geometric context, the bending term is: ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b . It is of order h n x and it interacts with the curvature term (cid:2) ∂ ( n − R i ,j ( · , (cid:3) i,j =1 ... , which is of order x n +13 . The interaction occurs only when the two terms have same parity, which happens ateven n , so α n = 0 for n odd. The two remaining terms in (1.10) measure the L norm of (cid:2) ∂ ( n − R i ,j ( · , (cid:3) i,j =1 ... , with distinct weights assigned to the S y and (cid:0) S y (cid:1) ⊥ projections,again according to the parity of n .(iii) The formula in (1.12) relates the quantities appearing in conditions (i) and (iii) of The-orem 1.1. The curvature (cid:2) ∂ ( n − R i ,j ( · , (cid:3) i,j =1 ... is thus precisely the coefficient of thediscrepancy of the order h n +1 in (1.8) at the 2 × n + 1)! / MARTA LEWICKA (iv) The finite strain space S y can be identtified in the the following two cases.When y = id ,then S y = { S ∈ L ( ω, R × ); curl T curl S = 0 } . When the Gauss curvature κ (( ∇ y ) T ∇ y ) = κ (cid:0) G × ) > ω , then S y = L ( ω, R × ), as shown in [26].Our next result proves the upper bound that is consistent with Theorem 1.1 and yields the Γ-convergence of the rescaled energies h − n +1) E h to the dimensionally reduced limits I n +1) in (1.10): Theorem 1.3.
Fix n ≥ and assume that any of the equivalent conditions in Theorem 1.1 hold.Then for every V ∈ V y as defined in (1.9), there exists a sequence { u h ∈ W , (Ω h , R ) } h → so that: (1.13) 1 h n h/ − h/ u h ( x ′ , x ) − (cid:16) y + n X k =1 x k k ! ~b k (cid:17) d x → V as h → , strongly in W , ( ω, R ) , and that: lim inf h → h n +1) E h ( u h ) = I n +1) ( V ) , where the limiting energy functional is as in (1.10). It is worth noting the following self-evident application of Theorems 1.1, 1.2 and 1.3:
Corollary 1.4.
Under either of the equivalent conditions in (1.5), assume that for some n ≥ thereholds: ∂ ( m )3 (cid:2) R i j ( · , (cid:3) i,j =1 ... = 0 on ω , for all m = 0 . . . n − , but ∂ ( n − (cid:2) R i j ( · , (cid:3) i,j =1 ... .Then there exist c, C > such that: (1.14) ch n +1) ≤ inf E h ≤ Ch n +1) . Moreover, the scaled energies h n − E h , Γ -converge to the limiting functional I n +1) in (1.10),effectively defined on the space V y of first order infinitesimal isometries in (1.9). For completeness, we note that the conformal metrics of the form: G ( x ′ , x ) = e φ ( x ) Id providea class of examples for the viability of all scalings in (1.14). Indeed, the trace midplate metric e φ (0) Id has a smooth isometric immersion y = e φ (0) id : ω → R , and the only possibly nonzeroRiemann curvatures of G are given by: R = − φ ′ ( x ) e φ ( x ) , R = R = − φ ′′ ( x ) e φ ( x ) .By Corollary 1.4 we see that inf E h ∼ h n if and only if φ ( k ) (0) = 0 for k = 1 . . . n − φ ( n ) (0) = 0.1.3. The structure of the paper.
In sections 2 and 3 we work under the assumption (iii) ofTheorem 1.1. First, in Lemma 2.1, we give an easy proof of the implication ( iii ) ⇒ ( ii ). Theparticular energy-consistent deformation field can be further used as the local change of variablesallowing for the application of the nonlinear rigidity estimate [8] in the present context. This isdone in Lemma 2.2 and Corollary 2.3, providing an approximation of an arbitrary energy-consistentdeformation gradient ∇ u h , by a non-symmetric square root of the n -th order Taylor expansion ofthe metric G , derived from the expansion guaranteed in (iii). Both the approximation error andthe L norm of the gradient of the rotation field are energy-controlled. In Lemma 2.4 we prove thecompactness part of Theorem 1.2. In Lemma 2.5 we conclude a preliminary lower bound estimate,involving a version of the functional I n +1) , whose curvature terms are still expressed in terms ofthe expansion fields in (iii), as suggested in the right hand side of (1.12).In section 3, we develop a geometric line of arguments, serving to prove (in Corollary 3.6) theidentity (1.12) under assumption (iii). In Lemmas 3.1 and 3.2, we partially reprove the equivalentconditions valid at the previously analyzed scalings h and h . These statements are then generalizedin Lemma 3.5, where we show the implication (iii) ⇒ (i), resulting also in the existence of a oneorder higher approximate field ~b n +1 , that is given solely through the Christoffel symbols of G on ω .In section 4 we finally prove Theorem 1.1, showing equivalence of the stated three conditions, byinduction on n ≥
2. We also finish the proof of Theorem 1.2 by: improving the lower bound from
IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 7 section 2, identifying its curvature components via (1.12), and separating the bending and the excessterms. In section 5 we prove Theorem 1.3, constructing a energy-consistent recovery sequence.1.4.
Notation.
Given a matrix F ∈ R n × n , we denote its transpose by F T and its symmetric partby F sym = ( F + F T ). The space of symmetric n × n matrices is denoted by R n × n sym , whereas R n × n sym , pos stands for the space of symmetric, positive definite n × n matrices. By SO ( n ) = { R ∈ R n × n ; R T = R − and det R = 1 } we mean the group of special rotations; its tangent space at Id n consists ofskew-symmetric matrices: T Id n SO ( n ) = so ( n ) = { F ∈ R n × n ; F sym = 0 } . We use the matrix norm | F | = (trace( F T F )) / , which is induced by the inner product h F : F i = trace( F T F ). The 2 × F ∈ R × is denoted by F × . Conversely, for a given F × ∈ R × , the 3 × F × and all other entries equal to 0, is denoted by F ∗ × . Unlessspecified otherwise, all limits are taken as the thickness parameter h vanishes: h →
0. By C wedenote any universal positive constant, independent of h .1.5. Acknowledgments.
M.L. is grateful to Annie Raoult and Shankar Venkataramani for interestin the project and discussions. Support by the NSF grant DMS-1613153 is acknowledged.2.
A proof of Theorem 1.2: compactness and a preliminary lower bound
In this section, assuming condition (iii) of Theorem 1.1, we derive the compactness and (a versionof) the lower bound in Theorem 1.2. We first observe the implication ( iii ) ⇒ ( ii ) in Theorem 1.1: Lemma 2.1.
Assume that condition (iii) in Theorem 1.1 holds, for some n ≥ . Then we have: inf E h ≤ Ch n +1) . Proof.
Define u h ( x ′ , x ) = y + n +1 X k =1 x k k ! ~b k , so that: ∇ u h ( x ′ , x ) = n X k =0 x k k ! B k + x n +13 ( n + 1)! (cid:2) ∂ ~b n +1 , ∂ ~b n +1 , (cid:3) . Consequently, ( ∇ u h ) G − / is positive definite for all small h , and modulo a rotation field it equalsthe following matrix field on Ω h , where we used the assumption (1.8): q(cid:0) ( ∇ u h ) G − / (cid:1) T (cid:0) ( ∇ u h ) G − / (cid:1) = q Id + G − / (cid:0) ( ∇ u h ) T ∇ u h − G (cid:1) G − / = p Id + O ( h n +1 ) = Id + O ( h n +1 ) . This implies: E h ( u h ) = 1 h ˆ Ω h W (cid:0) Id + O ( h n +1 ) (cid:1) d x ≤ Ch n +1) , as claimed.Recalling results (1.5) and (1.7) quoted from [30], we already see that lim h → h inf E h = 0 auto-matically implies: inf E h ≤ Ch . Before addressing compactness at h n with h ≥
3, we develop thenonlinear rigidity estimates applicable in the present context.
Lemma 2.2.
Assume that condition (iii) in Theorem 1.1 holds, for some n ≥ . Let V ⊂ ω be anopen, Lipschitz subdomain such that y is injective on V . Denote V h = V × ( − h , h ) . Then for every u h ∈ W , ( V h , R ) there exists ¯ R h ∈ SO (3) such that: h ˆ V h (cid:12)(cid:12) ∇ u h − ¯ R h n X k =0 x k k ! B k (cid:12)(cid:12) d x ≤ C (cid:16) h ˆ V h W (cid:0) ( ∇ u h ) G − / (cid:1) d x + h n +1 | V h | (cid:17) . MARTA LEWICKA
The constant C is uniform for all V h ⊂ Ω that are bi-Lipschitz equivalent with controlled Lipschitzconstants.Proof. Define Y = y + n +1 X k =1 x k k ! ~b k , and observe that for h sufficiently small, Y is a smooth diffeomor-phism of V h onto its image U h ⊂ R . Consider the change of variables v h = u h ◦ Y − ∈ W , ( U h , R )and apply the fundamental geometric rigidity estimate [8], yielding existence of ¯ R h ∈ SO (3) with: ˆ U h |∇ v h − ¯ R h | ≤ C ˆ U h dist (cid:0) ∇ v h , SO (3) (cid:1) . Changing variable in the left hand side gives: ˆ U h |∇ v h − ¯ R h | = ˆ V h (det ∇ Y ) · (cid:12)(cid:12) ( ∇ u h )( ∇ Y ) − − ¯ R h (cid:12)(cid:12) ≥ C ˆ V h (cid:12)(cid:12) ∇ u h − ¯ R h ∇ Y (cid:12)(cid:12) = C ˆ V h (cid:12)(cid:12) ∇ u h − ¯ R h (cid:0) n X k =0 x k k ! B k (cid:1)(cid:12)(cid:12) + C ˆ V h O ( h n +1) ) . Changing now variable in the right hand side and using ( ∇ Y ) G − / ∈ SO (3) (cid:0) Id + O ( h n +1 ) (cid:1) , asestablished in Lemma 2.1, results in: ˆ U h dist (cid:0) ∇ v h , SO (3) (cid:1) = ˆ V h (det ∇ Y ) · dist (cid:0) ( ∇ u h )( ∇ Y ) − , SO (3) (cid:1) ≤ C ˆ V h dist (cid:0) ( ∇ v h ) G − / , SO (3)( ∇ Y ) G − / (cid:1) ≤ C ˆ V h dist (cid:0) ( ∇ v h ) G − / , SO (3) (cid:1) + C ˆ V h O ( h n +1) ) . Combining the three displayed inequalities above proves the result.The well-known approximation technique [9] together with the arguments in [29, Corollary 2.3],yield the following estimate, whose proof we leave to the reader:
Corollary 2.3.
Assume condition (iii) in Theorem 1.1, for some n ≥ . Then, given a sequence { u h ∈ W , (Ω h , R ) } h → such that E h ( u h ) ≤ Ch n +1) , there exists { R h ∈ W , ( ω, SO (3)) } h → with: h ˆ Ω h (cid:12)(cid:12) ∇ u h − R h n X k =0 x k k ! B k (cid:12)(cid:12) d x ≤ Ch n +1) and ˆ ω |∇ R h ( x ′ ) | d x ′ ≤ Ch n . We now show the compactness part of Theorem 1.1:
Lemma 2.4.
Assume condition (iii) in Theorem 1.1, for some n ≥ . Let the sequence of deforma-tions { u h ∈ W , (Ω h , R ) } h → satisfy: E h ( u h ) ≤ Ch n +1) . Then: (i) The averaged displacements V h converge, up to a subsequence, to the first order isometry V as in Theorem 1.2 (i). (ii) The scaled strains h (cid:0) ( ∇ y ) T ∇ V h (cid:1) sym converge, up to a subsequence, weakly in L ( ω, R × ) to some S ∈ S y . IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 9
Proof. Define the following rotation: ¯ R h = P SO (3) Ω h ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − d x . In order toobserve that the above definition is legitimate, we write:dist (cid:16) Ω h ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − d x, SO (3) (cid:17) ≤ (cid:12)(cid:12)(cid:12) Ω h ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − d x − R h ( x ′ ) (cid:12)(cid:12)(cid:12) ≤ Ω h (cid:12)(cid:12) ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − − R h (cid:12)(cid:12) d x + 2 (cid:12)(cid:12)(cid:0) ω R h d x ′ (cid:1) − R h ( x ′ ) (cid:12)(cid:12) , and upon integrating d x ′ on the domain ω while noting Corollary 2.3, obtain:dist (cid:16) Ω h ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − d x, SO (3) (cid:17) ≤ Ch n +1) + Ch n ≤ Ch n . Consequently, there also follows:(2.1) (cid:12)(cid:12)(cid:12) Ω h ( ∇ u h ) (cid:16) n X k =0 x k k ! B k (cid:17) − d x − ¯ R h (cid:12)(cid:12)(cid:12) ≤ Ch n , ω | R h − ¯ R h | d x ′ ≤ Ch n . Set now c h ∈ R so that ´ ω V h d x ′ = 0. We get: ∇ V h = 1 h n h/ − h/ ( ¯ R h ) T (cid:2) ∂ u h , ∂ u h (cid:3) − ( ¯ R h ) T R h (cid:16) n X k =0 x k k ! B k (cid:17) × d x + S h h/ − h/ (cid:16) n X k =0 x k k ! B k (cid:17) × d x , (2.2)where we define the following matrix fields whose convergence (up to a subsequence) results fromthe second bound in (2.1) and from Corollary 2.3:(2.3) S h = 1 h n (cid:16) ( ¯ R h ) T R h − Id (cid:17) ⇀ S weakly in W , ( ω, R × ) . We also note that S ∈ so (3) a.e. in ω . Since the first term in the right hand side of (2.2) convergesto 0 in L ( ω ), in virtue of Corollary 2.3, we conclude the following convergence, up to a subsequence: ∇ V h → ( SB ) × = S ∇ y strongly in L ( ω, R × ) . It also follows that the limit S ∇ y ∈ W , ( ω, R × ). A further application of the Poincare in-equality to the mean-zero displacements V h , yields their strong convergence (up to a subsequencein W , ( ω, R )) to some V ∈ W , ( ω, R ) satisfying ∇ V = ( SB ) × . By skew-symmetry of S , itfollows that ( ∇ y ) T ∇ V is skew a.e. in ω , proving (i). We observe that the first term in the right hand side of (2.2) has its L ( ω ) norm boundedby Ch , in view of the first estimate in Corollary 2.3. Consequently, in the decomposition of1 h (cid:0) ( ∇ y ) T ∇ V h (cid:1) sym , parallel to that in (2.2), the corresponding first term has a weakly converg-ing subsequence. The remaining second term equals:1 h (cid:16) ( ∇ y ) T S h (cid:0) ∇ y + O ( h ) (cid:1)(cid:17) sym = 1 h ( ∇ y ) T S h sym ∇ y + O ( h | S h | ) . The L ( ω ) norm of the second term above clearly converges to 0, whereas the first term obeys:(2.4) 1 h S h sym = − h n − S h ) T S h → L ( ω, R × ) . This ends the proof of the claim.We are now ready to derive the lower bound on the scaled energies h − n +1) E h ( u h ), in terms ofthe expansion fields y , { ~b k } n +1 k =1 in condition (iii) of Theorem 1.1: Lemma 2.5.
In the context of Lemma 2.4, there holds: lim inf h → h n +1) E h ( u h ) ≥ ˆ Ω Q (cid:18) x ′ , S − δ n +1 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + x (cid:0) ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b (cid:1) + x n +13 n + 1)! (cid:16) (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( x ′ , (cid:17)(cid:19) d x, with the coefficient δ n +1 given by: (2.5) δ n +1 = n + 2)!2 n +1 for n odd for n even . Proof. By Corollary 2.3, the following matrix fields {Z h ∈ L (Ω , R × ) } h → have a convergingsubsequence, weakly in L (Ω , R × ):(2.6) Z h ( x ′ , x ) = 1 h n +1 (cid:16) ∇ u h ( x ′ , hx ) − R h ( x ′ ) n X k =0 h k x k k ! B k ( x ′ ) (cid:17) ⇀ Z . We write: ( R h ) T ∇ u h ( x ′ , hx ) = n X k =0 h k x k k ! B k + h n +1 ( R h ) T Z h and observe that: E h ( u h ) = ˆ Ω W (cid:0) ( R h ) T ∇ u h ( x ′ , hx ) G ( x ′ , hx ) − / (cid:1) d x ≥ ˆ {|Z h | ≤ h } W (cid:0)q Id + G ( x ′ , hx ) − / J h G ( x ′ , hx ) − / (cid:1) d x, (2.7)where the intermediary field J h has the following expansion, on the set {|Z h | ≤ h } ⊂ Ω : J h ( x ′ , x ) = (cid:16) n X k =0 h k x k k ! B k (cid:17) T (cid:16) n X k =0 h k x k k ! B k (cid:17) − G ( x ′ , hx )+ 2 h n +1 (cid:18)(cid:16) n X k =0 h k x k k ! B k (cid:17) T ( R h ) T Z h (cid:19) sym + h n +1) ( Z h ) T Z h = h n +1 x n +13 ( n + 1)! (cid:16) n X k =1 (cid:18) n + 1 k (cid:19) ( B k ) T B n +1 − k − ∂ ( n +1)3 G ( x ′ , (cid:17) + 2 h n +1 (cid:0) B T ( R h ) T Z h (cid:1) sym + o ( h n +1 )Consequently, we get from (2.7) and Taylor expanding W at Id :1 h n +1) E h ( u h ) ≥ ˆ {|Z h | ≤ h } Q (cid:16) G ( x ′ , hx ) − / (cid:0) h n +1 J h + o (1) (cid:1) G ( x ′ , hx ) − / (cid:17) d x. IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 11
Since B T ( R h ) T Z h converges weakly in L (Ω , R × ), up to a subsequence, to B T ¯ R T Z , for some¯ R ∈ SO (3) (which is an accumulation point of ¯ R h in the proof of Lemma 2.4), the above results in:lim inf h → h n +1) E h ( u h ) ≥ ˆ Ω Q (cid:18) x n +13 n + 1)! G ( x ′ , − / (cid:16) n X k =1 (cid:18) n + 1 k (cid:19) ( B k ) T B n +1 − k − ∂ ( n +1)3 G ( x ′ , (cid:17) G ( x ′ , − / + G ( x ′ , − / (cid:0) B T ¯ R T Z (cid:1) sym G ( x ′ , − / (cid:19) d x (2.8) We need to identify the relevant 2 × (cid:0) B T ¯ R T Z (cid:1) sym in (2.8). We applythe finite difference technique [9] and consider the following fields { f s,h ∈ W , (Ω , R ) } s> ,h → : f s,h ( x ) = s h ( ¯ R h ) T Z h ( x ′ , x + t ) + S h n X k =0 h k ( x + t ) k k ! B k ( x ′ ) d t e = 1 h n +1 s ( ¯ R h ) T (cid:0) u h ( x ′ , h ( x + s )) − u h ( x ′ , hx ) (cid:1) − h n s n X k =0 h k ( x + t ) k k ! ~b k +1 d t. where S h is defined in (2.3). Recall that, as proved in Lemma 2.4, ∇ V = ( SB ) × and that S is a.e.in so (3). It follows that the vector ~p defined in Theorem 1.2 (ii) must coincide with SB e = S~b .Consequently, using the first definition above it now easily follows that:(2.9) f s,h → S~b = ~p strongly in L (Ω , R ) . Using the second definition, we further compute the in-plane derivatives of f s,h for j = 1 . . . ∂ j f s,h ( x ) = 1 sh n +1 ( ¯ R h ) T (cid:0) ∂ j u h ( x ′ , h ( x + s )) − ∂ j u h ( x ′ , hx ) (cid:1) − h n s n X k =0 h k ( x + t ) k k ! ∂ k ~b k +1 d t = 1 s ( ¯ R h ) T (cid:16) Z h ( x ′ , x + s ) − Z h ( x ′ , x ) (cid:17) e j + 1 sh n +1 ( Id + h n S h ) n X k =1 h k k ! (cid:0) ( x + s ) k − x k (cid:1) B k e j − h n s n X k =0 h k ( x + t ) k k ! ∂ j ~b k +1 d t. The first term in the right hand side above converges to 1 s ¯ R T (cid:0) Z ( x ′ , x + s ) − Z ( x ′ , x ) (cid:1) e j , weakly in L (Ω , R ), whereas the last two terms may be rewritten as:1 sh n +1 ( Id + h n S h ) n X k =1 h k k ! (cid:0) ( x + s ) k − x k (cid:1) ∂ j ~b k − sh n +1 n +1 X k =1 h k k ! (cid:0) ( x + s ) k − x k (cid:1) ∂ j ~b k = 1 sh S h n X k =1 h k k ! (cid:0) ( x + s ) k − x k (cid:1) ∂ j ~b k − s ( n + 1)! (cid:0) ( x + s ) n +1 − x n +13 (cid:1) ∂ j ~b n +1 ⇀ S∂ j ~b − s ( n + 1)! (cid:0) ( x + s ) n +1 − x n +13 (cid:1) ∂ j ~b n +1 weakly in W , (Ω , R ) . In conclusion, and recalling (2.9), we obtain the following convergence, weakly in W , ( ω, R ): ∂ j f s,h ( x ) ⇀ s ¯ R T (cid:0) Z ( x ′ , x + s ) − Z ( x ′ , x ) (cid:1) e j + S∂ j ~b − s ( n + 1)! (cid:0) ( x + s ) n +1 − x n +13 (cid:1) ∂ j ~b n +1 = ∂ j ~p. We thus see that:¯ R T (cid:16) Z ( x ′ , x ) − Z ( x ′ , (cid:17) e j = x (cid:0) ∂ j ~p − S∂ j ~b (cid:1) + 1( n + 1)! x n +13 ∂ j ~b n +1 for j = 1 . . . , which finally yields: (cid:16) B T ¯ R T Z ( x ′ , x ) (cid:17) × = (cid:16) B T ¯ R T Z ( x ′ , (cid:17) × + x (cid:16) ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b (cid:17) + 1( n + 1)! x n +13 ( ∇ y ) T ∇ ~b n +1 . (2.10) We now compute the symmetric part of the trace term (cid:0) B T ¯ R T Z ( x ′ , (cid:1) × , sym and concludethe proof of the Lemma. It follows from (2.2) and the definition of Z h in (2.6) that: ∇ V h = h ˆ / − / ( ¯ R h ) T Z h × d x + S h (cid:0) ∇ y + O ( h ) (cid:1) In virtue of (2.6), (2.10) and (2.4), we obtain convergence, weakly in E :1 h (cid:0) ( ∇ y ) T ∇ V h (cid:1) sym ⇀ (cid:16) ( ∇ y ) T ¯ R T Z ( x ′ , × (cid:17) sym + ˆ / − / x n +13 ( n + 1)! d x (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym , which allows to conclude, by Lemma 2.4 (ii):(2.11) (cid:0) B T ¯ R T Z ( x ′ , (cid:1) × , sym = S − δ n +1 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym . This ends the proof of Lemma, in virtue of (2.8), (2.10), (2.11) and recalling definitions (1.3).3.
Relations between (i) and (iii) of Theorem 1.1 and a proof of Theorem 1.2 (iii)
In this section we show the relation between the defining quantities appearing in conditions (i)and (iii) of Theorem 1.1. Equivalence of (i) and (iii) at n = 2 has been shown in [30], building onthe previous results in [6, 29], while the proof of the general case will be carried out by induction on n ≥
2. We start by introducing some notation that allows for a systematic approach.Define the smooth matrix fields { Γ a : ¯Ω → R × } a =1 ... by setting their coefficients (Γ a ) bc = Γ bac to be the usual Christoffel symbols Γ bac = 12 X m =1 G bm (cid:0) ∂ b G mc + ∂ c G mb − ∂ m G bc (cid:1) of the metric G .Recall the standard notation for the coefficients of the inverse: ( G − ) ab = G ab . Since the Levi-Civitaconnection is torsion-free, it follows that Γ a e b = Γ b e a for all a, b = 1 . . . c, d = 1 . . . (cid:2) R ab,cd (cid:3) a,b =1 ... = (cid:0) ∂ c Γ d + Γ c Γ d (cid:1) − (cid:0) ∂ d Γ c + Γ d Γ c (cid:1) , (cid:2) R ab,cd (cid:3) a,b =1 ... = G (cid:2) R ab,cd (cid:3) a,b =1 ... . Given a matrix field F : Ω → R × , we define: ∇ a F = ∂ a F + Γ a F for each a = 1 . . .
3, so that( ∇ a F ) e b coincides with the usual covariant derivative of vector fields: ∇ a ( F e b ). It also follows that: ∇ c ∇ d F − ∇ d ∇ c F = (cid:2) R ab,cd (cid:3) a,b =1 ... F and ∇ a ( F F ) = ( ∇ a F ) F + F ∂ a F . We now partially reprove the mentioned statements at n = 1 , IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 13
Lemma 3.1.
Assume that there exist smooth fields y ,~b : ¯ ω → R such that the matrix field: B = (cid:2) ∂ y , ∂ y ,~b (cid:3) has positive determinant and such that: B T B = G ( x ′ , and (cid:0) ( ∇ y ) T ∇ ~b (cid:1) sym = 12 ∂ G ( x ′ , × for all x ′ ∈ ω. Then: (i) ∂ i B = B Γ i for all i = 1 . . . , and in particular: ∂ i ~b = B Γ e i . (ii) R ab,ij ( x ′ ,
0) = R ab,ij ( x ′ ,
0) = 0 for all x ′ ∈ ω and all a, b = 1 . . . , i, j = 1 . . . . (iii) There exists a unique smooth field ~b : ¯ ω → R such that defining the matrix field B = (cid:2) ∂ ~b , ∂ ~b , ~b (cid:3) , there holds: (cid:0) B T B (cid:1) sym = 12 ∂ G ( x ′ , for all x ′ ∈ ω . Moreover: B = B Γ and ∂ i ~b = B ∇ i Γ e for all i = 1 . . . . Proof. One easily calculates, by a repeated use of the assumed identities, that: h ∂ i y , ∂ j ~b i = ∂ j G i − h ∂ ij y ,~b i = ∂ j G i − ∂ i G j + h ∂ j y , ∂ i ~b i and thus: ∂ G ij = h ∂ i y , ∂ j ~b i + h ∂ j y , ∂ i ~b i = ∂ j G i − ∂ i G j + 2 h ∂ j y , ∂ i ~b i , for all i, j = 1 . . .
2, where all the identities are taken on ω × { } . Thus:(3.1) h ∂ j y , ∂ i ~b i = 12 (cid:0) ∂ G ij + ∂ i G j − ∂ j G i = (cid:0) G Γ (cid:1) ji for all i, j = 1 . . . . Secondly: h ∂ j y , ∂ ik y i = ∂ i G jk − h ∂ k y , ∂ ij y i = ∂ i G jk − ∂ j G ik + h ∂ i y , ∂ jk y i = ∂ i G jk − ∂ j G ik + ∂ k G ij − h ∂ j y , ∂ ik y i , which results in: h ∂ j y , ∂ ik y i = 12 (cid:0) ∂ i G jk + ∂ k G ij − ∂ j G ik (cid:1) = (cid:0) G Γ i (cid:1) jk for all i, j, k = 1 . . . . Thirdly, from (3.1) we obtain: h ~b , ∂ ik y i = ∂ i G k − h ∂ i ~b , ∂ k y i = 12 (cid:0) ∂ i G k + ∂ k G i − ∂ G ik (cid:1) = (cid:0) G Γ i (cid:1) k for all i, k = 1 . . . . Finally: h ~b , ∂ i ~b i = ∂ i G = (cid:0) G Γ i (cid:1) , so that the last two identities yield: B T ∂ i B = G Γ i for all i = 1 . . . , on ω × { } . This proves (i) and further: ∂ i ~b = B Γ i e = B Γ e i , as claimed. Using (i) we compute:0 = ∂ ij B − ∂ ji B = ∂ i (cid:0) B Γ j (cid:1) − ∂ j (cid:0) B Γ i (cid:1) = B Γ i Γ j + B ∂ i Γ j − (cid:0) B Γ j Γ i + B ∂ j Γ i (cid:1) = − B (cid:2) R ks,ij ( · , (cid:3) k,s =1 ... for all i, j = 1 . . . . which implies (ii). For (iii), uniqueness of ~b is obvious, while ~b = B Γ e follows from the requesteddefining identity, in view of (3.1). The covariant derivative formula is a consequence of (i). Lemma 3.2.
Assume that there exist smooth fields y ,~b ,~b : ¯ ω → R such that the matrix field: B = (cid:2) ∂ y , ∂ y , ~b (cid:3) has positive determinant and that together with B = (cid:2) ∂ ~b , ∂ ~b , ~b (cid:3) itsatisfies: B T B = G ( x ′ , and (cid:0) B T B (cid:1) sym = 12 ∂ G ( x ′ , (cid:0) ( ∇ y ) T ∇ ~b (cid:1) sym + ( ∇ ~b ) T ∇ ~b = 12 ∂ G ( x ′ , × for all x ′ ∈ ω. Then: (i) R ab,cd ( x ′ ,
0) = 0 for all x ′ ∈ ω and all a, b, c, d = 1 . . . . (ii) There exists a unique smooth field ~b : ¯ ω → R such that defining the matrix field B = (cid:2) ∂ ~b , ∂ ~b , ~b (cid:3) , there holds: (cid:0) B T B (cid:1) sym + B T B = 12 ∂ G ( x ′ , for all x ′ ∈ ω . Moreover: B = B ∇ Γ and ∂ i ~b = B ∇ i ∇ Γ e for all i = 1 . . . . Proof.
Observe first that for all a, b = 1 . . . h ∂ Ge a , e b i = ∂ (cid:0) h G Γ e a , e b i + h G Γ e b , e a i (cid:1) = h∇ Γ a e , Ge b i + h∇ Γ b e , Ge a i + 2 h G Γ e a , Γ e b i . (3.2)Consequently, and using Lemma 3.1 (iii), the last assumed condition is equivalent to:0 = h B e i , ∂ j ~b i + 2 h ∂ i ~b , ∂ j ~b i + h B e j , ∂ i ~b i − h ∂ Ge i , e j i = h Ge i , ∇ j Γ e i + 2 h G Γ e i , Γ e j i + h Ge j , ∇ i Γ e i− (cid:16) h∇ Γ j e , Ge i i + h∇ Γ i e , Ge j i + 2 h G Γ e j , Γ e j i (cid:17) = h Ge i , (cid:2) R a j ( · , (cid:3) a =1 ... i + h Ge j , (cid:2) R a j ( · , (cid:3) a =1 ... i = 2 R i ,j ( · ,
0) on ω, for all i, j = 1 . . . . The above proves (i), in virtue of Lemma 3.1 (ii) that guarantees R ab,ij ( · ,
0) = 0 for all a, b = 1 . . . i, j = 1 . . .
2. To show (ii), we observe that by Lemma 3.1 and by (i): B e i = ∂ i ~b = B ∇ i Γ e = B ∇ Γ i e = B ∇ Γ e i for all i, j = 1 . . . B T B ) sym + B T B − ∂ G ( x ′ ,
0) = ( B T B ) sym + Γ T G Γ − (cid:16) ( G ∇ Γ ) sym + Γ T G Γ (cid:17) , inview of (3.2), so B = B ∇ Γ satisfies the defining relation. Finally, ∂ i ~b = ∂ i (cid:0) B ∇ Γ e (cid:1) = B ∇ i ∇ Γ e results from Lemma 3.1 (i).We state the following two useful observations: Lemma 3.3.
For all n ≥ there holds: ∂ ( n +1)3 G ( x ′ ,
0) = 2 (cid:16) G ∇ ( n )3 Γ (cid:17) sym + n X k =1 (cid:18) n + 1 k (cid:19)(cid:0) ∇ ( k − Γ (cid:1) T G ∇ ( n − k )3 Γ for all x ′ ∈ ω. Proof.
The proof follows by induction. For n = 0, the statement is obviously true. Assume that itis true for some n −
1, then: ∂ ( n +1)3 G ( x ′ ,
0) = ∂ (cid:18) (cid:0) G ∇ ( n − Γ (cid:1) sym + n − X k =1 (cid:18) nk (cid:19)(cid:0) ∇ ( k − Γ (cid:1) T G ∇ ( n − − k )3 Γ (cid:19) = G ∇ ( n )3 Γ + (cid:0) ∇ ( n )3 Γ (cid:1) T G + Γ T G ∇ ( n − Γ + (cid:0) ∇ ( n − Γ (cid:1) T G Γ + n − X k =1 (cid:18) nk (cid:19)(cid:16)(cid:0) ∇ ( k − Γ (cid:1) T G ∇ ( n − k )3 Γ + (cid:0) ∇ ( k )3 Γ (cid:1) T G ∇ ( n − k − Γ (cid:17) = G ∇ ( n )3 Γ + (cid:0) ∇ ( n )3 Γ (cid:1) T G + Γ T G ∇ ( n − Γ + (cid:0) ∇ ( n − Γ (cid:1) T G Γ + n − X k =1 (cid:16)(cid:18) nk (cid:19) + (cid:18) nk − (cid:19)(cid:17)(cid:0) ∇ ( k − Γ (cid:1) T G ∇ ( n − k )3 Γ − (cid:18) n (cid:19) Γ T G ∇ ( n − Γ + (cid:18) nn − (cid:19)(cid:0) ∇ ( n − Γ (cid:1) T G Γ . IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 15
Collecting all the terms and recalling that (cid:0) nk (cid:1) + (cid:0) nk − (cid:1) = (cid:0) n +1 k (cid:1) implies the result. Lemma 3.4.
Assume that R , ( x ′ ,
0) = R , ( x ′ ,
0) = R , ( x ′ ,
0) = 0 for all x ′ ∈ ω and alsothat ∂ ( k )3 R i ,j ( x ′ ,
0) = 0 for all k = 0 . . . n , all i, j = 1 . . . and all x ′ ∈ ω . Then all the mixed partialderivatives of both R ab,cd and R ab,cd , of any order up to n , are zero on ω , for all a, b, c, d = 1 . . . .Proof. The proof proceeds by induction on n . For n = 0 the result is obviously true. Assume thatit is true for some n ≥ n + 1 hold. Then: ∂ ( k ) R ab,cd ( x ′ ,
0) = ∂ ( k ) R ab,cd ( x ′ ,
0) = 0 for all k = 0 . . . n, a, b, c, d = 1 . . . ,∂ ( n +1)3 R i ,j ( x ′ ,
0) = 0 for all i, j = 1 . . . , x ′ ∈ ω, and we need to show that any partial derivatives of order n + 1, of the Riemann tensor’s componentsis zero on ω . This is certainly true for partial derivatives containing ∂ i for some i = 1 . . .
2, so itsuffices to prove the claim for ∂ ( n +1)3 . Below, we consider various combinations of indices i, j = 1 . . . a, b = 1 . . .
3. Firstly:(3.3) ∂ ( n +1)3 R ab,ij = ∂ ( n )3 ∇ R ab,ij = ∂ ( n )3 (cid:0) − ∇ i R ab,j − ∇ j R ab, (cid:1) = ∂ ( n )3 (cid:0) − ∂ i R ab,j − ∂ j R ab, (cid:1) = 0 , where we used the induction assumption in the first and the third equalities and the second Bianchiidentity in the second one. Secondly:(3.4) ∂ ( n +1)3 R ab,ij = ∂ ( n +1)3 h (cid:2) G ap (cid:3) p =1 ... , (cid:2) R pb,ij (cid:3) p =1 ... i = h (cid:2) G ap (cid:3) p =1 ... , (cid:2) ∂ ( n +1)3 R pb,ij (cid:3) p =1 ... i = 0 , where we used the induction assumption and (3.3) in the last equality. Thirdly:(3.5) ∂ ( n +1)3 R ab,i = ∂ ( n +1)3 h (cid:2) G ap (cid:3) p =1 ... , (cid:2) R pb,i (cid:3) p =1 ... i = h (cid:2) G ap (cid:3) p =1 ... , (cid:2) ∂ ( n +1)3 R pb,i (cid:3) p =1 ... i = 0 , by using (3.4) and the result assumption at n + 1, in the last equality. Finally: ∂ ( n +1)3 R ab,cd = 0 by(3.4) and the result assumption.The following is the main result of this section: Lemma 3.5.
Fix n ≥ . Assume that there exist smooth y , { ~b k } nk =1 : ¯ ω → R such that the matrixfields: B = (cid:2) ∂ y , ∂ y , ~b (cid:3) with positive determinant and { B k = (cid:2) ∂ ~b k , ∂ ~b k , ~b k +1 (cid:3) } n − k =1 , satisfy: m X k =0 (cid:18) mk (cid:19) B T k B m − k − ∂ ( m )3 G ( x ′ ,
0) = 0 for all m = 0 . . . n − , (cid:0) ( ∇ y ) T ∇ ~b n (cid:1) sym + n − X k =1 (cid:18) nk (cid:19) ( ∇ ~b k ) T ∇ ~b n − k = ∂ ( n )3 G ( x ′ , × for all x ′ ∈ ω. Then: (i)
Condition in Theorem 1.1 (i) holds. (ii)
There exists a unique smooth field ~b n +1 : ¯ ω → R such that defining the matrix field B n = (cid:2) ∂ ~b n , ∂ ~b n , ~b n +1 (cid:3) , there holds: n X k =0 (cid:18) nk (cid:19) B T k B n − k = ∂ ( n )3 G ( x ′ , for all x ′ ∈ ω . Moreover: B n = B ∇ ( n − Γ and ∂ i ~b n +1 = B ∇ i ∇ ( n − Γ e for all i = 1 . . . . Proof. The proof proceeds by induction. The statement at n = 2 has been shown in Lemma 3.2.We now assume it to be true for some n ≥
2. By Lemma 3.4, we get:(3.6) All mixed partial derivatives up to order n −
2, of all components of theRiemann curvature tensor, are 0 at ω × { } .Since B k = (cid:2) ∂ ~b n , ∂ ~b n , ~b n +1 (cid:3) with ~b n +1 as in (ii), and recalling Lemma 3.3, we obtain for all x ′ ∈ ω :2 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( x ′ , × = 2 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:16) B T k B n +1 − k (cid:17) × − (cid:18) (cid:0) G ∇ ( n )3 Γ (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19)(cid:0) ∇ ( k − Γ (cid:1) T G ∇ ( n − k )3 Γ (cid:19) × = 2 (cid:16) ( ∇ y ) T ∇ ~b n +1 − G ∇ ( n )3 Γ (cid:17) sym = 2 h h Ge i , ∇ j ∇ ( n − Γ e − ∇ ( n )3 Γ e j i i i,j =1 ... , sym = 2 h h Ge i , ∇ j ∇ ( n − Γ e − ∇ ( n )3 Γ j e i i i,j =1 ... , sym . (3.7)By (3.6) we can consecutively swap the order of all the covariant derivatives on ω × { } in: ∇ j ∇ ( n − Γ = ∇ ∇ j ∇ ( n − Γ = ∇ (2)3 ∇ j ∇ ( n − Γ = ∇ (3)3 ∇ j ∇ ( n − Γ = ( . . . ) = ∇ ( n − ∇ j Γ , so that:(3.8) ∇ j ∇ ( n − Γ − ∇ ( n )3 Γ j = ∇ ( n − (cid:0) ∇ j Γ − ∇ Γ j (cid:1) = ∇ ( n − (cid:2) R ab,j ( x ′ , (cid:3) a,b =1 ... . In conclusion, using (3.6) again, the formula in (3.7) becomes:2 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( x ′ , × = h h Ge i , (cid:2) ∂ ( n − R a ,j ( x ′ , (cid:3) a =1 ... i + h Ge j , (cid:2) ∂ ( n − R a ,i ( x ′ , (cid:3) a =1 ... i i i,j =1 ... = 2 h ∂ ( n − R i ,j ( x ′ , i i,j =1 ... for all x ′ ∈ ω, (3.9)proving (i) in view of the second assumption at n + 1. For (ii), observe that B n +1 is indeed uniquely defined, by choosing ~b n +2 = B n +1 e such that: n +1 X k =0 (cid:18) n + 1 k (cid:19) B T k B n +1 − k = ∂ ( n +1)3 G ( x ′ ,
0) for all x ′ ∈ ω, since the principal 2 × n + 1, we get: B n +1 e i = ∂ i ~b n +1 = ∂ i (cid:16) B ∇ ( n − Γ e (cid:17) = B ∇ i ∇ ( n − Γ e = B ∇ ( n )3 Γ i e + ∇ ( n − (cid:2) R a ,i ( x ′ , (cid:3) a =1 ... = B ∇ ( n )3 Γ i e = B ∇ ( n )3 Γ e i for all i = 1 . . . x ′ ∈ ω. Hence, there must be ~b n +1 = B ∇ ( n )3 Γ , as claimed. This ends the proof of the Lemma. IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 17
We note that the argument in the proof above leading to (3.9), automatically gives:
Corollary 3.6.
For any n ≥ , condition (iii) in Theorem 1.1 implies the formula (1.12). The end of proof of Theorem 1.2 and a proof of Theorem 1.1
The following statement concludes the proof of Theorem 1.2, assuming (iii) of Theorem 1.1:
Lemma 4.1.
In the context of Lemma 2.4, there holds: lim inf h → h n +1) E h ( u n ) ≥ I n +1) ( V ) .Proof. By Lemma 2.4 and Corollary 3.6, we get:lim inf h → h n +1) E h ( u h ) ≥ ˆ Ω Q (cid:18) x ′ , S − δ n +1 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + x (cid:0) ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b (cid:1) + x n +13 ( n + 1)! (cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:19) d x. Denoting the x ′ -dependent tensor terms at different powers of x in the integrand above by I, II and
III , and recalling the definition of δ n +1 in (2.5), the right hand side becomes:12 ˆ Ω Q (cid:0) x ′ , I + x II + x n +13 III (cid:1) d x = 12 ˆ ω Q (cid:16) x ′ , I + (cid:0) ˆ / − / x n +13 d x (cid:1) III (cid:17) + 124 Q (cid:16) x ′ , II + 12 (cid:0) ˆ / − / x n +23 d x (cid:1) III (cid:17) + 12 (cid:18)(cid:0) ˆ / − / x n +23 d x (cid:1) − (cid:0) ˆ / − / x n +13 d x (cid:1) − (cid:0) ˆ / − / x n +23 d x (cid:1) (cid:19) Q (cid:0) x ′ , III (cid:1) d x ′ = 12 ˆ ω Q (cid:18) x ′ , S − δ n +1 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + δ n +1 (cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:19) d x ′ + 124 ˆ ω Q (cid:18) x ′ , ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b + α n (cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:19) d x ′ + γ n ˆ ω Q (cid:18) x ′ , (cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:19) d x ′ , where by a direct calculation one easily checks that the numerical coefficients α n and γ n have theform (1.11). Further, since S − δ n +1 (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym ∈ S y , the first term in the right hand sideabove is bounded from below by:12 dist Q (cid:16) δ n +1 (cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... , S y (cid:17) = δ n +1 Q (cid:16)(cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... , S y (cid:17) = δ n +1 (cid:13)(cid:13)(cid:13) P S ⊥ y (cid:0)(cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:1)(cid:13)(cid:13)(cid:13) Q . Decomposing the third term into: γ n (cid:13)(cid:13)(cid:13) P S ⊥ y (cid:0)(cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:1)(cid:13)(cid:13)(cid:13) Q + γ n (cid:13)(cid:13)(cid:13) P S y (cid:0)(cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:1)(cid:13)(cid:13)(cid:13) Q , the claim follows by checking that: δ n +1 γ n = β n in (1.11).We are now ready to give: A proof of Theorem 1.1.
The proof is carried out by induction on n ≥
2. When n = 2, then (i)is equivalent with (iii) by facts recalled in the preliminary discussion in section 1.2. Condition (iii)implies (ii) by Lemma 2.1, whereas (ii) implies (i) again in view of (1.5).Assume now the equivalence of the three conditions at some n ≥
2. We want to show theequivalence at n + 1. Condition (i) implies (iii) by Corollary 3.6. Condition (iii) implies (ii) byLemma 2.1. Finally, assuming (ii) at n + 1 allows to write:0 = lim h → h n +1) inf E h = lim h → h n +1) E h ( u h ) ≥ I n +1) ( V ) ≥ γ n · (cid:13)(cid:13)(cid:13)(cid:2) ∂ ( n − R i ,j ( x ′ , (cid:3) i,j =1 ... (cid:13)(cid:13)(cid:13) Q , for some infimizing sequence { u h ∈ W , (Ω h , R ) } h → and a resulting V from Theorem 1.2. Thisestablishes (i) at n + 1, in view of the inductive assumption.For completeness, we state the following auxiliary observations: Lemma 4.2.
In the context of Theorem 1.2, we have: (i)
The bending term ( ∇ y ) T ∇ ~p + ( ∇ V ) T ∇ ~b is symmetric and it equals: h(cid:10) Γ j e i , (cid:20) ( ∇ V ) T ~b (cid:21) (cid:11) − h ∂ ij V,~b i i i,j =1 ... . (ii) Under any of the equivalent conditions in Theorem 1.1 at n + 1 , we have: Ker I n +1) = (cid:8) Sy + c ; S ∈ so (3) , c ∈ R (cid:9) , and the following coercivity estimate holds: dist W , ( ω, R ) (cid:0) V, Ker I n +1) (cid:1) ≤ C I n +1) ( V ) for all V ∈ V y y with a constant C > that depends on G, ω and W but is independent of V .Proof. The symmetry of the bending term in (i) follows from: h ∂ i y , ∂ j ~p i + h ∂ i V, ∂ j ~b i = ∂ j (cid:0) h ∂ i y , ~p i + h ∂ i V,~b i (cid:1) − (cid:0) h ∂ ij y , ~p i + h ∂ ij V,~b i (cid:1) = −h ∂ ij y , ~p i − h ∂ ij V,~b i for all i, j = 1 . . . . The coercivity statement in (ii) has been proved in [30, Theorems 8.2, 8.3].5.
A proof of Theorem 1.3
In this section, we prove the upper bound result of Theorem 1.3. In view of the already establishedTheorem 1.1, it suffices to show:
Lemma 5.1.
Fix n ≥ and assume condition (iii) in Theorem 1.1. Let V ∈ V y be a first or-der isometry displacement as in (1.9). Then, there exists a sequence { u h ∈ W , (Ω h , R ) } h → ofdeformations satisfying (1.13), and such that: lim inf h → h n +1) E h ( u h ) = I n +1) ( V ) .Proof. Denote Y ( x ′ , x ) = y + n +1 X k =1 x k k ! ~b k and define:(5.1) u h ( x ′ , x ) = Y ( x ′ , x ) + h n v h ( x ′ ) + h n +1 w h ( x ′ ) + h n x ~p h ( x ′ ) + h n +1 x ~q h ( x ′ )+ x n +23 ( n + 2)! ~k ( x ′ ) + h n x ~r h ( x ′ ) for all ( x ′ , x ) ∈ Ω h . IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 19
We now introduce terms in the above expansion. For a fixed small ε >
0, the truncated sequence { v h ∈ W , ∞ ( ω, R ) } h → is chosen according to the standard construction in [8] (see also referencestherein), in a way that: v h → V strongly in W , ( ω, R ) as h → ,h n k v h k W , ∞ ( ω, R ) ≤ ε and lim h → h n (cid:12)(cid:12) { x ′ ∈ ω ; v h ( x ′ ) = V ( x ′ ) } (cid:12)(cid:12) = 0 . (5.2)The sequence { ~p h ∈ W , ∞ ( ω, R ) } h → is defined by:(5.3) B T ~p h = (cid:20) − ( ∇ v h ) T ~b (cid:21) so that (cid:16) B T (cid:2) ∇ v h , ~p h (cid:3)(cid:17) sym = (cid:16) ( ∇ y ) T ∇ v h (cid:17) ∗ sym . The sequence { w h ∈ C ∞ (¯ ω, R ) } h → is such that, recalling (2.5): (cid:16) ( ∇ y ) T ∇ w h (cid:17) sym → − δ n +1 P S y (cid:16)(cid:2) ∂ ( n − R i j (cid:3) i,j =1 ... (cid:17) strongly in E = L ( ω, R × ) as h → , lim h → h / k w h k W , ∞ ( ω, R ) = 0 . (5.4)Finally, ~k ∈ C ∞ (¯ ω, R ) and { ~q h ∈ C ∞ (¯ ω, R ) } h → , { ˜ r h ∈ L ∞ ( ω, R ) } h → are defined by:(5.5) 2 B T ~k = c (cid:16) x ′ , (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( x ′ , × (cid:17) − n X k =0 (cid:18) n + 1 k (cid:19) ( ∇ ~b n +1 − k ) T ∇ ~b k +1 n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k +1 ) T ∇ ~b n +2 − k + " ∂ ( n +1)3 G ( x ′ , , ∂ ( n +1)3 G ( x ′ , ,B T ~q h = c (cid:16) x ′ , (cid:0) ( ∇ y ) T ∇ w h (cid:1) sym (cid:17) − " ( ∇ w h ) T ~b ,B T ˜ r h = c (cid:16) x ′ , ( ∇ y ) T ∇ ~p h + ( ∇ v h ) T ∇ ~b (cid:17) − " ( ∇ v h ) T ~b h ~p h ,~b i . Further, we choose { ~r h ∈ C ∞ (¯ ω, R ) } h → to satisfy, in view of (5.2):(5.6) lim h → k ~r h − ˜ r h k L ( ω, R ) = 0 and lim h → h / k ~r h k W , ∞ ( ω, R ) = 0 . By (5.4) and (5.6) we easily deduce (1.13). Compute now, for all rescaled variables ( x ′ , x ) ∈ Ω : ∇ u h ( x ′ , hx ) = h n (cid:2) ∇ v h , ~p h (cid:3) + n X k =0 h k x k k ! B k + h n +1 x n +13 ( n + 1)! (cid:2) ∂ ~b n +1 , ∂ ~b n +1 , ~k (cid:3) + h n +1 x (cid:2) ∇ ~p h , ~r h (cid:3) + h n +1 (cid:2) ∇ w h , ~q h (cid:3) + O ( h n +2 ) (cid:0) |∇ ~q h | + |∇ ~r h | (cid:1) . Consequently, it follows that for h small enough we have:dist (cid:0) ( ∇ u h ) G − / , SO (3) (cid:1) ≤ C (cid:0) |∇ u h − B | + h (cid:1) ≤ Cǫ, which justifies writing, by Taylor’s expansion of W and taking ǫ ≪ W (cid:0) ( ∇ u h ) G − / (cid:1) = W (cid:16)q Id + G − / (cid:0) ( ∇ u h ) T ∇ u h − G (cid:1) G − / (cid:17) = W (cid:16) Id + 12 G − / (cid:0) ( ∇ u h ) T ∇ u h − G (cid:1) G − / + O (cid:0) | ( ∇ u h ) T ∇ u h − G | (cid:1)(cid:17) = W (cid:16) Id + 12 G ( x ′ , − / (cid:0) ( ∇ u h ) T ∇ u h − G (cid:1) G ( x ′ , − / + O (cid:0) h | ( ∇ u h ) T ∇ u h − G | (cid:1) + O (cid:0) | ( ∇ u h ) T ∇ u h − G | (cid:1)(cid:17) = 18 Q (cid:16) G ( x ′ , − / (cid:0) ( ∇ u h ) T ∇ u h − G (cid:1) G ( x ′ , − / (cid:17) + O (cid:0) h | ( ∇ u h ) T ∇ u h − G | (cid:1) + O (cid:0) | ( ∇ u h ) T ∇ u h − G | (cid:1) . This implies that:1 h n +2 E h ( u h )= 18 ˆ Ω Q (cid:16) h n +1 G ( x ′ , − / (cid:0) ( ∇ u h ) T ∇ u h ( x ′ , hx ) − G ( x ′ , hx ) (cid:1) G ( x ′ , − / (cid:17) d x + ˆ Ω h n +2 O (cid:0) h | ( ∇ u h ) T ∇ u h − G | (cid:1) + 1 h n +2 O (cid:0) | ( ∇ u h ) T ∇ u h − G | (cid:1) d x. (5.7)We thus compute, for all ( x ′ , x ) ∈ Ω :( ∇ u h ) T ∇ u h ( x ′ , hx ) − G ( x ′ , hx ) = 2 h n (cid:0) ( ∇ y ) T ∇ v h (cid:1) ∗ sym + h n +1 x n +13 ( n + 1)! (cid:18) n X k =1 (cid:18) n + 1 k (cid:19) B T k B n +1 − k + 2 (cid:0) B T (cid:2) ∂ ~b n +1 , ∂ ~b n +1 , ~k (cid:3)(cid:1) sym − ∂ ( n +1)3 G ( x ′ , (cid:19) + 2 h n +1 x (cid:0) B T (cid:2) ∇ ~p h , ~r h (cid:3)(cid:1) sym + 2 h n +1 x (cid:0) B T (cid:2) ∇ ~v h , ~p h (cid:3)(cid:1) sym + 2 h n +1 (cid:0) B T (cid:2) ∇ w h , ~q h (cid:3)(cid:1) sym + R h , where: R h = o ( h n +1 ) + O ( h n +2 ) (cid:0) |∇ ~v h | + |∇ ~v h | (cid:1) + O ( h n ) |∇ ~v h | + O ( h n +2 ) |∇ ~v h | . We now estimate the two last (error) terms in the right hand side of (5.7). Observe that: | ( ∇ u h ) T ∇ u h − G | = O ( h n +1 ) (cid:0) |∇ v h | + |∇ w h | + | ~p h | + |∇ ~p h | + | ~q h | + | ~r h | (cid:1) + R h + O ( h n ) | (cid:0) ( ∇ y ) T ∇ v h (cid:1) sym | = O ( h n +1 ) (cid:0) |∇ v h | + |∇ v h | + h − / o (1) (cid:1) + O ( h n ) | (cid:0) ( ∇ y ) T ∇ v h (cid:1) sym | + O ( h n ) |∇ v h | + O ( h n +2 ) |∇ v h | , where we have repeatedly used (5.3), (5.4), (5.5) and (5.6), Consequently:1 h n +2 O (cid:0) | ( ∇ u h ) T ∇ u h − G | (cid:1) = O ( h n +1 ) (cid:0) |∇ v h | + |∇ v h | + h − / o (1) (cid:1) + O ( h n +4 ) |∇ v h | + O ( h n − ) |∇ v h | + O ( h n − ) | (cid:0) ( ∇ y ) T ∇ v h (cid:1) sym | . The first two terms in the right hand side above converge to 0 in L ( ω ) by (5.2) and (5.4). The L norm of the third term is bounded by Ch n − k∇ v h k W , and thus converges to 0 as well. The final IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 21 fourth term is bounded, in virtue of (5.2) by:1 h ˆ ω (cid:12)(cid:12)(cid:0) ( ∇ y ) T ∇ v h (cid:1) sym (cid:12)(cid:12) d x ≤ Ch (cid:0) k∇ v h k L ∞ + k∇ v h k L ∞ (cid:1) ˆ { v h = V } dist ( x ′ , { v h = V } ) d x ′ ≤ Cǫ h n +2 ˆ { v h = V } dist ( x ′ , { v h = V } ) d x ′ ≤ Cǫ h n +2 (cid:12)(cid:12) { v h = V } (cid:12)(cid:12) ≤ Cǫ h n +2 h n · o (1) → h → . (5.8)This completes the convergence analysis of the first error term in (5.7). For the second term, we get:1 h n +2 O (cid:0) h | ( ∇ u h ) T ∇ u h − G | (cid:1) = O ( h ) (cid:0) |∇ v h | + |∇ v h | + h − o (1) (cid:1) + O ( h n − ) |∇ v h | + O ( h n +3 ) |∇ v h | + 1 h O (cid:0) | (cid:0) ( ∇ y ) T ∇ v h (cid:1) sym | (cid:1) . As before, the first three terms converge to 0 in L ( ω ), whereas convergence of the last term followsby (5.8). Concluding, and since h n +1 R h converges to 0 in L (Ω ), the limit in (5.7) becomes:lim h → h n +2 E h ( u h )= lim h → ˆ Ω Q (cid:16) h n +1 G ( x ′ , − / (cid:0) ( ∇ u h ) T ∇ u h ( x ′ , hx ) − G ( x ′ , hx ) (cid:1) G ( x ′ , − / (cid:17) d x = lim h → ˆ Ω Q (cid:16) G ( x ′ , − / K h ( x ′ , x ) G ( x ′ , − / (cid:17) d x, (5.9)where for a.e. ( x ′ , x ) ∈ Ω we define: K h ( x ′ , x ) = 2 h (cid:0) ( ∇ y ) T ∇ v h (cid:1) ∗ sym + x n +13 ( n + 1)! (cid:18) n X k =1 (cid:18) n + 1 k (cid:19) B T k B n +1 − k + 2 (cid:0) B T (cid:2) ∂ ~b n +1 , ∂ ~b n +1 , ~k (cid:3)(cid:1) sym − ∂ ( n +1)3 G ( x ′ , (cid:19) + 2 x (cid:0) B T (cid:2) ∇ ~p h , ~r h (cid:3) + (cid:0) B T (cid:2) ∇ ~v h , ~p h (cid:3)(cid:1) sym + 2 (cid:0) B T (cid:2) ∇ w h , ~q h (cid:3)(cid:1) sym . In view of (5.8) and since k ~r h − ˜ r h k L converges to 0 as requested in (5.6), the compatibility in thedefinition (5.5) now yields from (5.9):lim h → h n +2 E h ( u h )= lim h → ˆ ω Q (cid:18) x ′ , x n +13 n + 1)! (cid:18) (cid:0) ( ∇ y ) T ∇ ~b n +1 (cid:1) sym + n X k =1 (cid:18) n + 1 k (cid:19) ( ∇ ~b k ) T ∇ ~b n +1 − k − ∂ ( n +1)3 G ( x ′ , × (cid:19) + 2 x (cid:16) ( ∇ y ) T ∇ ~p h + ( ∇ v h ) T ∇ ~b (cid:17) + (cid:0) ( ∇ y ) T ∇ w h (cid:1) sym (cid:19) d x ′ . (5.10)Now, decomposing the integrand above as in the proof of Lemma 4.1 and recalling convergences in(5.2) and (5.4), we conclude that the right hand side of (5.10) equals I n +1) ( V ), as claimed.It is worth observing that directly from Theorems 1.2 and 1.3 we obtain: Corollary 5.2.
Each functional I n +1) attains its infimum and there holds: lim h → h n +1) inf E h = min I n +1) . The infima in the left hand side are taken over W , (Ω , R ) deformations u h , whereas the minimumin the right hand side is taken over admissible displacements V ∈ V y . References [1]
P. Bella and R.V. Kohn , Metric-induced wrinkling of a thin elastic sheet , J. Nonlinear Sci. (2014), pp. 1147–1176.[2] P. Bella and R.V. Kohn , The coarsening of folds in hanging drapes , Comm Pure Appl Math (5) (2017),pp. 978–2012.[3] H. Ben Belgacem, S. Conti, A. DeSimone and S. Muller , Rigorous bounds for the Foppl–von K´arm´an theoryof isotropically compressed plates , J. Nonlinear Sci. , (2000), pp. 661– 683.[4] H. Ben Belgacem, S. Conti, A. DeSimone and S. Muller , Energy scaling of compressed elastic films—three-dimensional elasticity and reduced theories , Arch. Ration. Mech. Anal. (2002), no. 1, pp. 1–37.[5]
S. Conti and F. Maggi , Confining thin elastic sheets and folding paper
Archive for Rational Mechanics andAnalysis (2008), , Issue 1, pp. 1–48.[6]
K. Bhattacharya, M. Lewicka and M. Sch¨affner , Plates with incompatible prestrain , Archive for RationalMechanics and Analysis, 221 (2016), pp. 143–181.[7]
E. Efrati, E. Sharon and R. Kupferman , Elastic theory of unconstrained non-Euclidean plates , J. Mech.Phys. Solids, (2009), pp. 762–775.[8] G. Friesecke, R. D. James and S. M¨uller , A theorem on geometric rigidity and the derivation of nonlinearplate theory from three-dimensional elasticity , Comm. Pure Appl. Math., 55 (2002), pp. 1461–1506.[9]
G. Friesecke, R. D. James and S. M¨uller , A hierarchy of plate models derived from nonlinear elasticity bygamma-convergence , Arch. Ration. Mech. Anal., 180 (2006), no. 2, pp. 183–236.[10]
J. Gemmer and S. Venkataramani , Shape selection in non-Euclidean plates , Physica D: Nonlinear Phenomena(2011), pp. 1536–1552.[11]
J. Gemmer and S. Venkataramani , Shape transitions in hyperbolic non-Euclidean plates , Soft Matter (2013),pp. 8151–8161.[12]
J. Gemmer, E. Sharon, T. Shearman and S. Venkataramani , Isometric immersions, energy minimizationand self-similar buckling in non-Euclidean elastic sheets , Europhysics Letters (2016), 24003.[13]
A. Gladman, E. Matsumoto, R. Nuzzo, L. Mahadevan and J. Lewis , Biomimetic 4D printing , NatureMaterials , (2016) pp. 413–418.[14] G. Jones and L. Mahadevan , Optimal control of plates using incompatible strains , Nonlinearity (2015), 3153.[15] W. Jin and P. Sternberg , Energy estimates for the von K´arm´an model of thin-film blistering , Journal ofMathematical Physics , 192 (2001).[16] R. Kempaiah and Z. Nie , From nature to synthetic systems: shape transformation in soft materials
J. Mater.Chem. B (2014) , pp. 2357–2368.[17] J. Kim, J. Hanna, M. Byun, C. Santangelo, and R. Hayward , Designing responsive buckled surfaces byhalftone gel lithography , Science , (2012) pp. 1201–1205.[18]
Y. Klein, E. Efrati and E. Sharon , Shaping of elastic sheets by prescription of non-Euclidean metrics , Science (2007), pp. 1116–1120.[19]
R. Kupferman and C. Maor , A Riemannian approach to the membrane limit in non-Euclidean elasticity ,Comm. Contemp. Math. (2014), no.5, 1350052.[20] R. Kupferman and J.P. Solomon , A Riemannian approach to reduced plate, shell, and rod theories , Journal ofFunctional Analysis (2014), pp. 2989–3039.[21]
M. Dias, J. Hanna and C. Santangelo , Programmed buckling by controlled lateral swelling in a thin elasticsheet , Phys. Rev. E , (2011), 036603.[22] H. Le Dret and A. Raoult , The nonlinear membrane model as a variational limit of nonlinear three-dimensionalelasticity , J. Math. Pures Appl. (1995), pp. 549–578.[23] H. Le Dret and A. Raoult , The membrane shell model in nonlinear elasticity: a variational asymptotic deriva-tion , J. Nonlinear Sci. (1996), pp. 59–84.[24] M. Lewicka, L. Mahadevan and R. Pakzad , Models for elastic shells with incompatible strains , Proc. Roy.Soc. A (2014), 2165 20130604; pp. 1471–2946. IMENSION REDUCTION FOR NON-EUCLIDEAN ELASTICITY 23 [25]
M. Lewicka, L. Mahadevan and R. Pakzad , The Monge-Ampere constrained elastic theories of shallow shells ,Annales de l’Institut Henri Poincare (C) Non Linear Analysis , Issue 1, (2017), pp. 45–67.[26] M. Lewicka, M. Mora and R. Pakzad , The matching property of infinitesimal isometries on elliptic surfacesand elasticity of thin shells , Arch. Rational Mech. Anal. (3) (2011), pp. 1023–1050.[27]
M. Lewicka, P. Ochoa and R. Pakzad , Variational models for prestrained plates with Monge-Ampere con-straint , Diff. Integral Equations, , no 9-10 (2015), pp. 861–898.[28] M. Lewicka and R. Pakzad , Scaling laws for non-Euclidean plates and the W , isometric immersions ofRiemannian metrics , ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), pp. 1158–1173.[29] M. Lewicka, A. Raoult and D. Ricciotti , Plates with incompatible prestrain of high order , Annales del’Institut Henri Poincare (C) Non Linear Analysis, 34, Issue (2017), pp. 1883–1912.[30]
M. Lewicka and D. Lucic , Dimension reduction for thin films with transversally varying prestrain: the oscilla-tory and the non-oscillatory case , to appear (2018).[31]
H. Liang and L. Mahadevan , The shape of a long leaf , Proc. Nat. Acad. Sci. (2009).[32]
C. Maor and A. Shachar , On the role of curvature in the elastic energy of non-Euclidean thin bodies , preprint.[33]
S. Muller and H. Olbermann , Conical singularities in thin elastic sheets , Calculus of Variations and PartialDifferential Equations, (2014) , Issue 3–4, pp. 1177–1186.[34] H. Olbermann , Energy scaling law for the regular cone , Journal of Nonlinear Science (2016) , Issue 2, pp. 287–314.[35] H. Olbermann , On a boundary value problem for conically deformed thin elastic sheets , preprint arXiv:1710.01707.[36]
P.E.K. Rodriguez, A. Hoger and A. McCulloch , Stress-dependent finite growth in finite soft elatic tissues
J. Biomechanics (1994), pp. 455–467.[37] E. Sharon, B. Roman and H.L. Swinney , Geometrically driven wrinkling observed in free plastic sheets andleaves , Phys. Rev. E (2007), pp. 046211–046217.[38] I. Tobasco , Curvature-driven wrinkiling of thin elastic shells , preprint.[39]
S. Venkataramani , Lower bounds for the energy in a crumpled elastic sheet—a minimal ridge , Nonlinearity (2004), no. 1, pp. 301–312.[40] Z. Wei, J. Jia, J. Athas, C. Wang, S. Raghavan, T. Li and Z. Nie , Hybrid hydrogel sheets that undergopre-programmed shape transformations , Soft Matter , (2014), pp. 8157–8162. Marta Lewicka: University of Pittsburgh, Department of Mathematics, 139 University Place, Pitts-burgh, PA 15260
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