Limits of elastic models of converging Riemannian manifolds
aa r X i v : . [ m a t h . A P ] M a r Limits of elastic models of convergingRiemannian manifolds
Raz Kupferman and Cy Maor
Abstract
In non-linear incompatible elasticity, the configurations are mapsfrom a non-Euclidean body manifold into the ambient Euclideanspace, R k . We prove the Γ -convergence of elastic energies for con-figurations of a converging sequence, M n → M , of body manifolds.This convergence result has several implications: (i) It can be viewedas a general structural stability property of the elastic model. (ii)It applies to certain classes of bodies with defects, and in particu-lar, to the limit of bodies with increasingly dense edge-dislocations.(iii) It applies to approximation of elastic bodies by piecewise-a ffi nemanifolds. In the context of continuously-distributed dislocations,it reveals that the torsion field, which has been used traditionallyto quantify the density of dislocations, is immaterial in the limitingelastic model. Contents Γ -convergence of elastic energies of converging manifolds 22 Γ -convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Convergence of minimizers . . . . . . . . . . . . . . . . . . . 27 One of the central notions in geometric theories of continuum mechanics,is that of a body manifold , M , whose points represent material elements.Mathematically, a body manifold is a topological, or di ff erentiable mani-fold. Di ff erent types of continuum systems are characterized by di ff erentgeometric structures imposed on the body manifolds. Body manifolds ofelastic solids are commonly smooth manifolds endowed with a Rieman-nian metric, i.e., Riemannian manifolds. A configuration of a body is an em-bedding of the body manifold into the ambient k -dimensional Euclideanspace. In hyper-elastic materials, both static and dynamics properties ofthe material are dictated by an elastic energy, which is an integral measureof local distortions of the configurations.In classical elasticity, the body manifold is assumed to be Euclidean, im-plying that it can be identified with a subset Ω of Euclidean k -dimensionalspace. The natural inclusion ι : Ω ֒ → R k is called a rest, or a reference config-uration , and it is a state of zero elastic energy. In the last several years, therehas been a growing interest in bodies that are pre-stressed . Pre-stressed bod-ies are modeled as Riemannian manifolds ( M , g ), where the reference metric g is non-flat, i.e., has a non-zero Riemann curvature tensor. Thus, it cannotbe embedded isometrically into Euclidean space. In particular, there is nonotion of reference configuration.The elastic theory of pre-stressed bodies is commonly known as the theoryof non-Euclidean , or incompatible elasticity. In its simplest versions, assum-2ng material isotropy, the elastic energy associated with a configuration u : M → R k is a distance of that configuration from being an isometricembedding (see e.g. [ESK09, KS14, LP10]). A prototypical energy densityis dist p ( du , SO( g , e )) for some p >
1, where e is the Euclidean metric in R k and SO( g , e ) is the set of orientation and inner-product preserving maps T M → R k [KS14, LP10]. A precise definition of this distance is given in thenext section.The theory of incompatible elasticity has numerous applications. It wasproposed originally in the 1950s in the context of crystalline defects (seee.g. Kondo [Kon55] and Bilby and co-workers [BBS55, BS56]). Then, thenon-Euclidean metric structure associated with the defects is singular. Inrecent years, incompatible elasticity is motivated by studies of growingtissues, thermal expansion, and other mechanisms involving di ff erentialexpansion of shrinkage [ESK09, AESK11, AAE +
12, OY09, KES07]; in thesesystems the intrinsic geometry is typically smooth.In the context of crystalline defects, an important field of interest concernsdistributed defects. Models of continuously-distributed defects were de-veloped during the 1950s and 1960s. Body manifolds of bodies with dis-tributed defects are endowed with structure additional to a metric. Forexample, bodies with continuously-distributed dislocations are modeledby
Weitzenb¨ock manifolds ( M , g , ∇ ), where ∇ is a flat connection consistentwith g (that is, a metric connection), whose torsion tensor represents thedistribution of the dislocations [Nye53, BBS55, Kr ¨o81]; see also the morerecent literature [MR02, YG12].The modeling of a body with distributed dislocations by a Weitzenb ¨ockmanifold is phenomenological, rather than mechanical. In particular, it isnot associated with a class of constitutive relations (or elastic energies), andit is not clear how does ∇ manifest (if at all) in the response to deformationand loading (as pointed out in Section 1a in [Wan67]). This is in contrast tobodies with finitely many dislocations, which can be modeled as (singular)Riemannian manifolds with no additional structure (no torsion field), andfor which standard elastic energies are applicable [MLA + ff erent phenomenological models fordescribing the dislocations (see e.g. [CGO15]), or assume general classes ofelastic energies that may or may not relate to the connection (e.g. [CK13],3hich does not use Weitzenb ¨ock manifolds explicitly, however their choiceof crystalline structure is equivalent to a choice of a flat connection), oronly rely on the Riemannian part when considering mechanical response([YG12]).In [KM15] and [KM] we showed that Weitzenb ¨ock manifolds (with non-zero torsion) can be obtained as rigorous limits of (torsion-free) singularRiemannian manifolds. Thus, the phenomenological model of a body withcontinuously distributed dislocations is a limit of bodies with finitely-many singular dislocations, as the density of the dislocations tends toinfinity. This new notion of converging manifolds calls for a rigorousderivation of a mechanical model for bodies with continuously-distributeddislocations: Assuming a mechanical model for bodies with finitely-manysingular dislocations, is there a limiting mechanical model for the limitingWeitzenb ¨ock manifolds?The main question addressed in this paper is the following: Given a se-quence of converging manifolds endowed with elastic energies dependingcontinuously (in a precise sense) on the metric structure, what can be saidabout the Γ -limit of these energies? To minimize technicalities, we considersystems free of external forces or constraints (note that the non-Euclideanstructure renders such systems non-trivial). Body forces and boundaryconditions can be included, if needed, in a standard way (see the Discus-sion section).Our main result (Theorem 4.1) can be summarized as follows: Let ( M n , g n ) be a sequence of body manifolds, with correspondingelastic energy densities W ( M n , g n ) satisfying boundedness and coerciv-ity conditions, and depending continuously on the metric g n (seeSection 3 for a precise definition). If ( M n , g n ) → ( M , g ) uniformly(see Definition 2.4), then the elastic energies Γ -converge to the re-laxation of an energy with density W ( M , g ) ; if ( M n , g n ) → ( M , g ) in aweaker sense (see Definition 2.5), then the relaxation of W ( M , g ) is anupper-bound to every Γ -convergent subsequence. As mentioned above, it is shown in [KM15, KM] that any 2D Weitzenb ¨ockmanifold ( M , g , ∇ ) can be obtained as a limit of bodies with finitely manydislocations ( M n , g n , ∇ n ), where ∇ n is the Levi-Civita connection. The con-vergence of the Riemannian part, ( M n , g n ) → ( M , g ), is with respect to the4eaker notion of convergence. Yet, a slight modification of our construc-tion yields uniform convergence.For a Weitzenb ¨ock manifold ( M , g , ∇ ) to constitute an adequate elasticmodel for a body with distributed dislocations, one would expect to havean elastic energy E ( M , g , ∇ ) associated with it. Since ( M , g , ∇ ) is an e ff ectivelimit model of bodies with finitely many defects, E ( M , g , ∇ ) should be a limit ofthe energies associated with these bodies. In the case where ∇ = ∇ LC is theLevi-Civita connection, the body has no continuously-distributed disloca-tions, so it is natural to choose E ( M , g , ∇ LC ) = E ( M , g ) , where E ( M , g ) is a standardnon-Euclidean elastic energy (say, with density dist p ( du , SO( g , e ))). Ouranalysis shows that in this case E ( M , g , ∇ ) (or more accurately, its relaxation)would be independent of ∇ even if it is not the Levi-Civita connection (andthus contains torsion).This paper is concerned with isotropic materials, in which the elastic energyis derived from the Riemannian metric of a body manifold (the referencemetric), which is fixed. In other models, involving anisotropy or defectdynamics, the connection ∇ (or equivalently its torsion field) can still playa role in a limit energy functional. This lies outside the scope of this paper,and it is a natural topic for further research.In addition, our main theorem implies the structural stability of non-Euclideanelasticity under certain perturbations of the reference metric, as well asthe convergence of certain approximation methods, based on locally-Euclidean approximations of body manifolds. These applications are elab-orated in the discussion (Section 5).The structure of the paper is as follows: In Section 2, we define notions ofconvergence for body manifolds, and define an L p -topology for functionsdefined on converging manifolds. In Section 3, we define a class of elasticenergy functionals for configurations of body manifolds, and prove, inparticular, that the energy densities dist p ( · , SO( g , e )) belong to this class.While the definitions in this section are straightforward, it is the first time(to the best of our knowledge) that a convergence analysis relies on a precisequantitative relation between the metric structure and the elastic energydensity. In Section 4 we state and prove the main Γ -convergence result,and in Section 5 we discuss applications and limitations of our results, aswell as some open questions. 5 Settings
Let ( V , g ) and ( W , h ) be two oriented k -dimensional inner-product spaces.For a linear map A : V → W we denote by | A | ∞ the operator norm of A ,that is | A | ∞ = sup , v ∈ V | A ( v ) | h | v | g , and by | A | = tr( A T A ) the inner-product (Frobenius) norm induced by g and h . Note that | A | ∞ ≤ | A | ≤ k | A | ∞ . (2.1)When the exact norm is irrelevant or clear from the context, we simplywrite | A | .We denote by dist g , h (resp. dist ∞ g , h ) the distance function on L ( V , W ) withrespect to the inner-product (resp. operator) norm induced by g and h . Weextend it to subsets of L ( V , W ) as a Hausdor ff distance.We denote by SO( g , h ) the set of inner-product and orientation-preservingisomorphisms ( V , g ) → ( W , h ). The distortion of a map A ∈ L ( V , W ) is definedas Dis A = dist g , h ( A , SO( g , h )) . (2.2)All the above is extended to vector bundles equipped with inner-productsin the standard way. If A is orientation preserving, and σ , . . . , σ k are thesingular values of A , then Dis A = p ( σ − + . . . + ( σ k − .Throughout the paper, we consider derivatives of maps F : M → N in thefollowing way: Pointwise, for every p ∈ M , we consider ( dF ) p : T p M → T F ( p ) N as a map between vector spaces. Globally, dF is considered as a map T M → F ∗ T N , where F ∗ T N is a vector bundle over M , with the fiber ( F ∗ T N ) p identified with the fiber T F ( p ) N . This way dF is a bundle map over M , thusseparating its linear part from its nonlinear part (the projection of dF onthe base space). Likewise, we denote by F ∗ the pullback of tensor fields(such as Riemannian metrics), considered as sections of tensor products of T N and T ∗ N . This should not be confused with the closely related pullbackinvolving composition with dF , which we denote by F ⋆ . For example, if h is a Riemannian metric on N , then F ∗ h is an inner product on the vector6undle F ∗ T N , whereas F ⋆ h is an inner product on T M (hence a Riemannianmetric on M , unlike F ∗ h ), which is defined by F ⋆ h ( v , w ) = F ∗ h ( dF ( v ) , dF ( w )) , for every two vector fields v , w ∈ Γ ( T M ), whereas for every p ∈ M we have, F ∗ h p ( dF p ( v p ) , d p F ( w p )) = h F ( p ) ( dF p ( v p ) , d p F ( w p )) . Body manifolds are a general notion in mechanics, whose precise definitiondepends on the specific context. In this section we define the class ofmanifolds to which our results refer. Since we are interested in bodieswith defects, our concept of body manifold allows for singularities, whichimplies that we cannot require a smooth structure on the entire manifold.
Definition 2.1 A body manifold is a quadruple ( M , d , ˜ M , g ) , where M is a k-dimensional compact, oriented, connected topological manifold with corners andd is a distance function on M . ˜ M ⊂ M is an open smooth submanifold, suchthat M \ ˜ M has a k-dimensional Hausdor ff measure zero with respect to d. g is aRiemannian metric on ˜ M , consistent with the distance d in the following sense:for every p , q ∈ M , d ( p , q ) is the infimum over the lengths Len( γ ) = Z I p g ( ˙ γ ( t ) , ˙ γ ( t )) dt . of continuous paths γ : I → M that are a.e. smooth. In particular, γ ( t ) ∈ ˜ M forall t except perhaps for a set of measure zero. The consistency between g and d ensures that there are no “shortcuts”through the non-smooth parts of the body, i.e. that the Riemannian metricinduces the distance. Note also that the Riemannian metric induces ameasure on ˜ M —the volume form. This measure can be extended into ameasure µ on M by setting µ ( M \ ˜ M ) =
0. Since d and g are consistent, thenull sets of µ coincide with the null sets of the k -dimensional Hausdor ff measure. Examples : 7. The trivial example: Every compact, oriented, connected Riemannianmanifold with corners ( M , g ) is a body manifold with ˜ M = M and d induced by the Riemannian metric.2. A cone is a body manifold: it is a two-dimensional topological man-ifold hemeomorphic to a disc, endowed with a locally Euclideanmetric everywhere but at one point—the tip of the cone. In the me-chanical context, a cone is a disclination-type defect.3. Every piecewise-a ffi ne manifold is a body manifold. The smoothcomponent ˜ M may be disconnected. Piecewise-a ffi ne manifolds areprevalent in mechanics in the context of numerical approximations. NNN
We now define morphisms between body manifolds: these are bi-Lipschitzhomeomorphisms that are local di ff eomorphisms whenever the di ff erentialis defined (the smooth parts need not be di ff eomorphic). Definition 2.2
Let ( M , d M , ˜ M , g M ) and ( N , d N , ˜ N , g N ) be body manifolds. A mor-phism between those manifolds is a bi-Lipschitz homeomorphism F : M → N ,such that the restriction of F to ˜ M ∩ F − ( ˜ N ) (which is a set of full measure, sinceF − is Lipschitz) is a smooth embedding. Examples :1. Every di ff eomorphism between Riemannian manifolds is a bodymanifold morphism.2. A cone can be parametrized by polar coordinates, ( r , θ ), with a metricwhose components g ( r , θ ) = α r ! , < α , , are defined for every r >
0. The identity map into a Euclidean disc isa body manifold morphism. Note that the smooth parts of the coneand the disc are not di ff eomorphic.3. Maps from smooth Riemannian manifolds to piecewise-a ffi ne ap-proximations are body manifold morphisms.8 NN Elasticity is concerned with material response to distortions. In our con-text, where a body has a two metric structure—a distance function and aRiemannian metric—we distinguish between local and global distortionsof body manifold morphisms:
Definition 2.3
Let ( M , d M , ˜ M , g M ) and ( N , d N , ˜ N , g N ) be body manifolds and letF : M → N be a morphism. The local distortion of F is the distortion of thelinear map dF as defined in (2.2) , i.e., it is the map Dis dF : ˜ M ∩ F − ( ˜ N ) → [0 , ∞ ) , Dis dF = dist g M , F ∗ g N ( dF , SO ( g M , F ∗ g N )) . The global distortion of F is a non-negative number defined as
Dis F = sup p , q ∈ M | d M ( p , q ) − d N ( F ( p ) , F ( q )) | . In this section we define two modes of convergence for body manifolds,which, loosely speaking, correspond to uniform and non-uniform conver-gence of the Riemannian metrics.
Definition 2.4 (Uniform convergence of body manifolds)
Let ( M n , d n , ˜ M n , g n ) bea sequence of body manifolds and let ( M , d , , ˜ M , g ) be a body manifold. We saythat the sequence M n converges uniformly to M , if there exists a sequence ofbody manifold morphisms F n : M → M n , such that the local distortion vanishesuniformly, lim n →∞ k Dis dF n k ∞ = . (2.3) Definition 2.5 (Mean convergence of body manifolds)
Let ( M n , d n , ˜ M n , g n ) be asequence of body manifolds and let ( M , d , ˜ M , g ) be a body manifold. We say thatthe sequence M n converges in the mean to M , if there exists a sequence of bodymanifold morphisms F n : M → M n , such that1. F n are uniformly bi-Lipschitz, i.e. there exists a constant C > , independentof n, such that | ( dF n ) p | , | (( dF n ) p ) − | < C , (2.4) for every p ∈ M where dF n is defined. (Note that ( dF n ) − = F ∗ n ( d ( F − n )) .) . F n are approximate distance-preserving as maps between metric spaces: theglobal distortion vanishes asymptotically, lim n →∞ Dis F n = . (2.5)
3. F n are asymptotically rigid in the mean: lim n →∞ Z M Dis dF n dVol g = . (2.6)
4. The volume forms converge uniformly:dVol F ⋆ n g n dVol g → in L ∞ . (2.7)To simplify notations, we will denote the body manifolds ( M n , d n , ˜ M n , g n )and ( M , d , ˜ M , g ) by M n and M , whenever no confusion should arise.These definitions, and especially the definition of mean convergence, mayseem a bit convoluted, so we first provide the rationale behind them.As our main motivation for this work is the convergence of bodies withdislocations, we consider notions of convergence that (i) are satisfied byconverging bodies with dislocations considered in [KM15, KM] (furtherdetails are given in the examples section below); and (ii) are strong enoughto imply the Γ -convergence of associated elastic energies.The crux in each type of convergence is the way Dis F n converges to zero.When the convergence is in L ∞ (uniform convergence), it follows automati-cally that F n are uniformly bi-Lipschitz and that the volume forms convergeuniformly; these properties are needed for our Γ -convergence proof. When Dis F n → L (mean convergence) both the uniform by-Lipschitzproperty and volume convergence are not guaranteed, hence have to beimposed explicitly, as Conditions (2.4) and (2.7) (which are satisfied by ourmain examples, see below). Future improvements of the Γ -convergenceproof may allow to relax these conditions.Condition (2.5) is of “global” nature, and unlike the other conditions,does not involve the di ff erentials dF n explicitly. Furthermore, it plays noexplicit role in the Γ -convergence proof; its role is to “enforce” Gromov-Hausdor ff convergence (see below), and as a result, the uniqueness of the10imit (a limit body independent of the mappings F n ). It is possible thatthe other conditions in Definition 2.5 su ffi ce for a unique limit, in whichcase Condition (2.5) can be omitted. This is, however, a pure question ofgeometric rigidity, and it is beyond the scope of this paper. It is furtherdiscussed in the open questions part of Section 5.In the rest of this subsection we prove some properties of convergentsequences, and give some examples. Lemma 2.6 If M n converges to M in the mean, and F n : M → M n are maps thatrealize the convergence, then for every p ∈ [1 , ∞ )lim n →∞ Z M ( Dis dF n ) p dVol g = , (2.8) and lim n →∞ Z M n ( Dis dF − n ) p dVol g n = . (2.9) Proof : Since
Dis dF n ≤ | dF n | + k , and since | dF n | is uniformly bounded by (2.4), it follows from the BoundedConvergence Theorem that L -convergence (2.6) implies L p -convergence(2.8).Similarly, it is enough to prove (2.9) for p =
1. It follows from (2.6) that forevery ε > A n ⊂ M whose complements have asymptoti-cally vanishing volume, Vol( M \ A n ) →
0, in which
Dis dF n < ε . It followsthat the singular values of dF n with respect to the frame are in the interval(1 − ε, + ε ), hence all the singular values of ( dF n ) − are in the interval((1 + ε ) − , (1 − ε ) − ) ⊂ (1 − ε, + ε ), from which follows that for everypoint in A n , F ∗ n Dis dF − n < ε · √ k . (2.10)From (2.10) and the uniform bound (2.4), it follows that Z M F ∗ n Dis dF − n d Vol g ≤ √ k ε Vol( A n ) + C Vol( M \ A n ) ≤ √ k Vol( M ) ε + o (1) as n → ∞ .11ince ε is arbitrary, lim n →∞ Z M F ∗ n Dis dF − n d Vol g = . Using the uniform convergence of the volume (2.7),lim n →∞ Z M F ∗ n Dis dF − n d Vol F ⋆ n g n = , from which (2.9) for p = ■
1. Uniform convergence is stronger than mean convergence. Indeed,(2.3) implies (2.4) and (2.6). Uniform convergence of volumes (2.7)follows from the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Vol F ⋆ n g n d Vol g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( Dis dF n + k − , see Lemma 4.5 in [KM15] for details. Finally, let γ be a curve whoselengths ℓ g ( γ ) and ℓ F ⋆ n g n ( γ ) with respect to g and F ⋆ n g n are well-defined.The uniform convergence (2.3) implies that | ℓ g ( γ ) − ℓ F ⋆ n g n ( γ ) | → R >
0, this convergence is uni-form over all curves of length less or equal R . This implies uniformconvergence of the distances d F ⋆ n g n → d g (the distance functions in-duced on M by the Riemannian metrics F ⋆ n g n and g ). Since d g = d and d F ⋆ n g n is the pullback by F n of d n , this implies the asymptotic vanishing(2.5) of the global distortion.2. Both types of convergence are weaker than ( m , α )-H ¨older conver-gence of smooth manifolds, for any m ≥ α ∈ (0 , M n , g n ) → ( M , g ) in the C m ,α -topology if there exists dif-feomorphisms F n : M → M n such that F ⋆ n g n → g in the C m ,α -topology,i.e., the components of the metric converge in the C m ,α -topology inany coordinate chart (see [Pet06, Chapter 10] for details). This implies(2.3), hence uniform convergence.Thus, all the results presented in this paper apply a fortiori to H ¨older-converging manifolds. 12. Mean convergence of M n to M implies measured Gromov-Hausdor ff convergence of the measured metric spaces ( M n , d g n , Vol g n ) to ( M , d g , Vol g )(see [Pet06, Chapter 10] for details). Indeed, (2.5) implies Gromov-Hausdor ff convergence, and (2.7) implies weak convergence of themeasures Vol F ⋆ n g n to Vol g .
1. The convergence defined in [KM15, KM] in the context of distributededge-dislocations is weaker than mean convergence, but on the otherhand, it also embodies the convergence of an additional structure—a flat metric connection. In the terminology of the present paper,[KM15, KM] deal with the convergence of quintuples ( M , d , ˜ M , g , ∇ ),where ∇ is a flat metric connection on ˜ M . The explicit sequencesof manifolds constructed in [KM15, KM] exhibit mean convergence(see Propositions 1 and 3 in [KM15]). Therefore, the main theoremin [KM] implies that generic smooth, 2-dimensional surfaces can beobtained as mean convergence limits of locally-Euclidean surfaceswith distributed edge-dislocations.2. The constructions in [KM15, KM] are composed from building blocks,each containing a pair of disclinations of opposite charge, as illus-trated in Figure 1. In the n -th stage, the two disclinations in eachbuilding block have angles ± θ , i.e., independent of n and identicalin all blocks, whereas the distance d between the disclinations is oforder 1 / n . This construction yields in each block a dislocation ofmagnitude 2 d sin θ ∼ / n .These constructions yield sequences of body manifolds that do notconverge uniformly, but do converge in the mean. The lack of uni-form convergence stems from the fact that the disclination angles donot vanish as n tends to infinity. When mapping the manifolds M n into the limit manifold M , one has to map curves such as xp − p + y in Figure 1 to smooth curves. This always results in asymptoticallysmall areas where dF n is bounded away from being a rigid transfor-mation.A slight modification of the constructions in [KM15, KM] yields a se-quence of locally-Euclidean surfaces with distributed edge-dislocations13 yp − p + d θ x yp − p + d θ Figure 1: A single edge-dislocation realized as a dipole of disclinations at p − and p + , by gluing the segments [ x , p − ], [ p − , p + ] and [ p + , y ] in the upperpolygon to the matching segments in the lower polygon. The disclinationangle is 2 θ and the distance between the dislocations is | [ p − , p + ] | = d ,yielding a dislocation magnitude (identified with the size of the Burgersvector) 2 d sin( θ ). 14hat converges uniformly to a smooth two-dimensional surface. Forthat, one has to take the angle θ in each building block to be of order1 / n ε for some small ε , and set the distance d between the disclina-tions such that the dislocation magnitude is the same as in the originalconstruction (hence d is of order 1 / n − ε ). This construction yields thesame limit as the original construction.While it can be argued that vanishing disclination angles are “lessphysical” than fixed ones (especially in the context of crystallinesolids), this shows that any smooth surface with a continuous dis-tribution of dislocations ( M , d , M , g , ∇ ) (since M is smooth ˜ M = M )can be approximated uniformly by surfaces with finitely many dislo-cations ( M n , d n , ˜ M n , g n , ∇ n ), where ∇ n is the Levi-Civita connection.3. Another example of uniform convergence is the convergence of ap-proximations of a surface via Euclidean triangulations: Any givensurface can be triangulated by geodesic triangles whose edge-lengthsare of order 1 / n and whose angles are bounded away from 0 and π .For n large enough, each such triangle can be replaced by a Eu-clidean triangle of the same edge lengths. This yields a surface hav-ing disclination-type singularities at the vertices, while being locallyEuclidean everywhere else. As n tends to infinity, these singular, lo-cally Euclidean surfaces converge uniformly to the original surface.Higher dimensional analogues to this construction are also possible.4. As an example of a sequence of manifolds converging to a smoothmanifold in a weak sense, but not in the mean (and therefore neitheruniformly), one can take any sequence of Riemannian manifolds( M n , g n ) that converges to ( M , g ) while lim n Vol( M n ) , Vol( M ); thereare many such examples in the literature (see e.g. [Iva98]).An example relevant to the homogenization of defects is the conver-gence of bodies with increasingly dense point-defects, as in [KMR15].There, lim n Vol( M n ) > Vol( M ). In this example the maps F n : M → M n are far from being rigid, as Dis dF n is uniformly bounded away fromzero almost everywhere. Thus, the homogenization of point-defectsdoes not fall under the framework of this paper.15 .4 Convergence of maps on converging manifolds Having two notions of convergence for body manifolds, we proceed todefine a topology for maps f n : M n → R k . Definition 2.7
Let M n be a sequence of body manifolds converging to a bodymanifold M (either uniformly or in the mean), and let F n : M → M n be bodymanifold morphisms that realize the convergence. We say that a sequence u n ∈ L p ( M n ; R k ) converges to u ∈ L p ( M ; R k ) in L p (relative to F n ) if k u n ◦ F n − u k L p ( M ; R k ) → . Note that this convergence depends on the maps F n , which means that wedo not have a general notion of convergence of sequences in L p ( M n ; R k ) toa limit in L p ( M ; R k ). This convergence induces a natural topology on thedisjoint union ( ⊔ n L p ( M n ; R k ) ⊔ L p ( M ; R k ); see [KS08] for details. In the termi-nology of [KS08], we defined an asymptotic relation between L p ( M n ; R k ) and L p ( M ; R k ), since the sequence F n also realizes measured Gromov-Hausdor ff convergence, as stated in the third item in Section 2.3.1.The following lemma establishes standard properties of L p -convergence,adapted to converging manifolds: Lemma 2.8
1. If u n → u in L p , then u n is a bounded sequence in L p ( M n ; R k ) , namely, k u n k L p ( M n ; R k ) is bounded.2. If u n is bounded in W , p ( M n ; R k ) (i.e. k u n k W , p ( M n ; R k ) is bounded), then u n ◦ F n is uniformly bounded in W , p ( M ; R k ) , and in particular admits a weaklyW , p -convergent subsequence.Proof : It is enough to prove the lemma under the assumption that M n → M in the mean. Let u n → u in L p . By the triangle inequality, k u n ◦ F n k L p ( M ; R k ) ≤ k u n ◦ F n − u k L p ( M ; R k ) + k u k L p ( M ; R k ) . k u n k pL p ( M n ; R k ) = Z M n | u n | p d Vol g n = Z M | u n ◦ F n | p d Vol F ⋆ n g n = Z M | u n ◦ F n | p d Vol F ⋆ n g n d Vol g d Vol g ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d Vol F ⋆ n g n d Vol g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ k u n ◦ F n k pL p ( M ; R k ) , hence lim sup n →∞ k u n k L p ( M n ; R k ) ≤ k u k L p ( M ; R k ) , which proves the first part.For the second part, assume that u n is bounded in W , p ( M n ; R k ). In particu-lar, u n is bounded in L p ( M n ; R k ). The same calculation as above yields that u n ◦ F n is bounded in L p ( M ; R k ). Moreover, k d ( u n ◦ F n ) k pp = Z M | d ( u n ◦ F n ) | p d Vol g ≤ Z M F ∗ n | du n | p · | dF n | p d Vol g ≤ C Z M F ∗ n | du n | p d Vol g = C Z M n | du n | p d Vol ( F n ) ⋆ g = C Z M n | du n | p d Vol ( F n ) ⋆ g d Vol g n d Vol g n ≤ C k du n k pL p ( M n ; R k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d Vol ( F n ) ⋆ g d Vol g n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ C ′ k du n k pL p ( M n ; R k ) , (2.11)where the norms of du n , dF n and d ( u n ◦ F n ) at a point p are in the space L ( T p M n , R k ), as described in Section ?? . Between the first and the secondline we used the uniform Lipschitz continuity of F n . In the passage to thelast line we used the uniform Lipschitz continuity of F − n , and the fact thatif G : ( N , h ) → ( N ′ , h ′ ) is a smooth map between k -dimensional Riemannianmanifolds ( G = F − n in our case), then Hadamard’s inequality (see Lemma4.5 in [KM15] for details) implies d Vol G ⋆ h ′ d Vol h ≤ | dG | d . Together with the boundedness of u n ◦ F n in L p ( M ; R k ), (2.11) implies that u n ◦ F n is bounded in W , p ( M ; R k ), which completes the proof. ■ Energy functionals on families of manifolds
Definition 3.1
Let M be a class of body manifolds.1. An energy density on M is a functionW : G ( M , d , ˜ M , g ) ∈ M T ∗ ˜ M ⊗ R k → R . We denote the restriction of W to ( M , d , ˜ M , g ) by W ( M , g ) .2. An energy density on M is called p -regular for p ∈ (1 , ∞ ) , if the followingholds:(a) Regularity: For every ( M , d , ˜ M , g ) ∈ M , W ( M , g ) is a Carath´eodoryfunction; see Appendix A in [KM14] for the definition of Carath´eodoryfunctions in Riemannian settings.(b) Uniform coercivity: There exist α, β > such that for every ( M , d , ˜ M , g ) ∈ M , W ( M , g ) ( A ) ≥ α | A | p − β, ∀ A ∈ T ∗ ˜ M ⊗ R k . (3.1) (c) Uniform boundedness: There exists a γ > such that for every ( M , d , ˜ M , g ) ∈ M ,W ( M , g ) ( A ) ≤ γ ( | A | p + , ∀ A ∈ T ∗ ˜ M ⊗ R k . (3.2) (d) Lipschitz continuity in the metric: There exists a C > such that forevery ( M , d , ˜ M , g ) , ( N , d ′ , ˜ N , h ) ∈ M , linear isomorphism L : T ˜ M → T ˜ N , and A ∈ T ∗ ˜ N ⊗ R k | W ( M , g ) ( A ◦ L ) / p − L ∗ W ( N , h ) ( A ) / p | < C (1 + L ∗ | A | ) Dis L . (3.3)
3. Let W be a p-regular energy density on M . Its associated energy func-tional is the functionE : G ( M , d , ˜ M , g ) ∈ M L p ( M ; R k ) → R ∪ { + ∞} defined byE ( M , g ) [ u ] = R M W ( du ) dVol g u ∈ W , p ( M ; R k ) + ∞ u ∈ L p ( M ; R k ) \ W , p ( M ; R k ) . a )–( c ) are standardregularity conditions. Condition 2( d ) is the one that involves dependenceon the metric. In particular, when reduced to a single manifold, it implieshomogeneity and isotropy. Indeed, let ( M , g ) be a Riemannian manifold, x , y ∈ M , A ∈ T ∗ x M × R k and B ∈ T ∗ y M × R k . If A and B are related by anisometry, namely, A = B ◦ L for some L ∈ SO( g x , g y ), then W ( M , g ) ( A ) = W ( M , g ) ( B ) . The motivation for the Lipschitz continuity (3.3) is that it is a key propertysatisfied by the prototypical energy density
Dis du , as proved in the nextproposition: Proposition 3.2
Let ( M , d , ˜ M , g ) be a k-dimensional body manifold. For everyp ∈ (1 , ∞ ) , the energy densityW ( M , g ) ( du ) = ( Dis du ) p , (3.4) is p-regular, where u is considered as a map ( M , d , ˜ M , g ) → ( R k , e ) .Proof : The regularity property (a) holds since W ( M , g ) is continuous on everyfiber and we have the smoothness of the manifold on its base.The coercivity (b) and boundedness (c) are immediate, hence it remainsto prove (d). Let ( M , d , ˜ M , g ) and ( N , d ′ , ˜ N , h ) be body manifolds. Let L : T ˜ M → T ˜ N be a linear isomorphism, and let A ∈ T ∗ ˜ N ⊗ R k . We need toprove (3.3).The energy density (3.4) depends on g in two ways: via the metric with re-spect to which the distortion is measured, and via the set of local isometriesSO( g , e ) whose distance from is being measured. We treat each dependenceseparately: | W ( M , g ) ( A ◦ L ) / p − L ∗ W ( N , h ) ( A ) / p | = | Dis ( A ◦ L ) − L ∗ Dis A | = | dist g , e ( A ◦ L , SO( g , e )) − L ∗ dist h , e ( A , SO( h , e )) |≤ | dist g , e ( A ◦ L , SO( g , e )) − dist g , e ( A ◦ L , SO( L ⋆ h , e )) | + | dist g , e ( A ◦ L , SO( L ⋆ h , e )) − L ∗ dist h , e ( A , SO( h , e )) |≤ dist g , e (SO( g , e ) , SO( L ⋆ h , e )) + | dist g , e ( A ◦ L , SO( L ⋆ h , e )) − L ∗ dist h , e ( A , SO( h , e )) | .
19n the passage to the last inequality we used the fact that in any metricspace ( X , d ), with x ∈ X and A , B ⊂ X , | d ( x , A ) − d ( x , B ) | ≤ d ( A , B ) , where the distance on the right-hand side is the hausdor ff distance.In Lemma 3.3 below we prove thatdist ∞ g , e (SO( g , e ) , SO( L ⋆ h , e )) = dist ∞ g , h ( L , SO( g , h )) . Together with the norm inequality (2.1), we getdist g , e (SO( g , e ) , SO( L ⋆ h , e )) ≤ k dist g , h ( L , SO( g , h )) = k Dis L . In Lemma 3.4 below we prove that (cid:12)(cid:12)(cid:12) dist g , e ( A ◦ L , SO( L ⋆ h , e )) − L ∗ dist h , e ( A , SO( h , e )) (cid:12)(cid:12)(cid:12) ≤ ( L ∗ | A | + k ) Dis L . Putting everything together, | W ( M , g ) ( A ◦ L ) / p − L ∗ W ( N , h ) ( A ) / p | ≤ ( L ∗ | A | + k ) Dis L , which conclude the proof. ■ Lemma 3.3
Let ( V , g ) and ( W , h ) be two oriented k-dimensional inner-productspaces, and let L : V → W be an isomorphism. Then, for any metric r on V, dist ∞ r , e ( SO ( g , e ) , SO ( L ⋆ h , e )) = dist ∞ r , h ( L , SO ( g , h )) . Proof : Let R ∈ SO( g , e ) and Q ∈ SO( L ⋆ h , e ); both are isomorphisms V → R k .The (operator norm) distance between R and Q is d ∞ ( R , Q ) = sup k v k r = k ( R − Q ) v k e = sup k v k r = ( h Rv , Rv i e + h Qv , Qv i e − h Rv , Qv i e ) = sup k v k r = (cid:16) h v , v i g + h v , v i L ⋆ h − h QQ − Rv , Qv i e (cid:17) = sup k v k r = (cid:16) k v k g + k Lv k h − h LQ − Rv , Lv i h (cid:17) , where in the last step we used the fact that for every u , v ∈ V , h Qu , Qv i e = h u , v i L ⋆ h = h Lu , Lv i h . S = LQ − R : V → W , and observe that S ∈ SO( g , h ) as h Sv , Su i h = h LQ − Rv , LQ − Ru i h = h Q − Rv , Q − Ru i L ⋆ h = h Rv , Ru i e = h v , u i g . Also, d ∞ ( L , S ) = sup k v k r = k ( L − S ) v k h = sup k v k r = (cid:16) k Lv k h + k v k g − h Lv , Sv i h (cid:17) = d ∞ ( R , Q ) . It follows that for every R ∈ SO( g , e ) and Q ∈ SO( L ⋆ h , e ),dist ∞ r , e ( R , Q ) ≥ dist ∞ r , h ( L , SO( g , h )) , which implies thatdist ∞ r , e (SO( g , e ) , SO( L ⋆ h , e )) ≥ dist ∞ r , h ( L , SO( g , h )) . For the reverse inequality, the same arguments imply that for every S ∈ SO( g , h ) and Q ∈ SO( L ⋆ h , e ), R = Q L − S ∈ SO( g , e ) satisfies d ∞ ( R , Q ) = d ∞ ( L , S ) . Taking S to be a minimizer for the right-hand side, we obtain that for every Q ∈ SO( L ⋆ h , e ), dist ∞ r , e (SO( g , e ) , Q ) ≤ dist ∞ r , h ( L , SO( g , h )) . Similarly, since for every S ∈ SO( g , h ) and R ∈ SO( g , e ), Q = RS − L ∈ SO( L ⋆ h , e ) satisfies d ( R , Q ) = d ( L , S ) we obtain that for every R ∈ SO( g , e ),dist ∞ r , e ( R , SO( L ⋆ h , e )) ≤ dist ∞ r , h ( L , SO( g , h )) . From the definition of Hausdor ff distance, these two inequalities implythat dist ∞ r , e (SO( g , e ) , SO( L ⋆ h , e )) ≤ dist ∞ r , h ( L , SO( g , h )) . ■ Lemma 3.4
Let ( V , g ) and ( W , h ) be two oriented k-dimensional inner-productspaces, and let L : V → W be an isomorphism and A : W → R k . Then (cid:12)(cid:12)(cid:12) dist g , e ( A ◦ L , SO ( L ⋆ h , e )) − dist h , e ( A , SO ( h , e )) (cid:12)(cid:12)(cid:12) ≤ ( | A | + k ) Dis L . roof : Let B : W → R k . Then, for every Q ∈ SO( g , h ), | | B | h , e − | B ◦ L | g , e | = | | B ◦ Q | g , e − | B ◦ L | g , e | ≤ | B ◦ ( Q − L ) | g , e ≤ | B | h , e | Q − L | g , h , where we used the sub-multiplicativity of the Frobenius norm. Hence, | | B | h , e − | B ◦ L | g , e | ≤ | B | h , e dist g , h ( L , SO( g , h )) . Take B = A − R with R ∈ SO( h , e ). Then, | B | h , e ≤ | A | h , e + k , and( | A | h , e + k ) dist g , h ( L , SO( g , h )) ≥ | B | h , e dist g , h ( L , SO( g , h )) ≥ | B | h , e − | B ◦ L | g , e ≥ dist h , e ( A , SO( h , e )) − | B ◦ L | g , e = dist h , e ( A , SO( h , e )) − | A ◦ L − R ◦ L | g , e . Since R ◦ L ∈ SO( L ⋆ h , e ) and this holds for all R ∈ SO( h , e ) we obtain( | A | h , e + k ) dist g , h ( L , SO( g , h )) ≥ dist h , e ( A , SO( h , e )) − dist g , e ( A ◦ L , SO( L ⋆ h , e )) . Repeating the same argument the other way around we obtain an absolutevalue in the second line. ■ Γ -convergence of elastic energies of convergingmanifolds Let M be a class of k -dimensional body manifolds. Fix p ∈ (1 , ∞ ), andlet W be a p -regular energy density on M , with E the associated energyfunctional. For ( M , d , ˜ M , g ) ∈ M , denote Γ E M = R M QW ( M , g ) ( du ) d Vol g u ∈ W , p ( M ; R k ) , + ∞ u ∈ L p ( M ; R k ) \ W , p ( M ; R k ) , where QW ( M , g ) is the quasi-convex envelope of W ( M , g ) (see Section 3.4 in[KM14] for a discussion on quasi-convexity in Riemannian settings).In this section we prove Γ -convergent results for a sequence E M n , where M n ∈ M is a convergent sequence of body manifolds. In Section 4.1 weprove Γ -convergence, or establish an upper bound to Γ -convergent subse-quences, depending on whether M n → M uniformly or in the mean. InSection 4.2 we adapt to our setting the standard convergence of minimizersfor Γ -convergent (sub)sequences. 22 .1 Γ -convergence Theorem 4.1
Let M n , M ∈ M , then the following holds:1. If M n → M uniformly, then E M n Γ -converges to Γ E M .2. If M n → M in the mean, then the Γ -limit of every Γ -convergent subsequenceof E M n is bounded from above by Γ E M .The Γ -convergence is with respect to the L p -topology induced by some realizationF n : M → M n of the convergence. Note that although the topology depends on the choice of realizations F n ,neither the Γ -limit (in the first case) or the bound on the Γ -limit (in thesecond case) depends on this choice. Proof : For succinctness, we will write E n = E M n , E = E M and Γ E = Γ E M .Similarly we will write W n = W ( M n , g n ) and W ∞ = W ( M , g ) .Let E ∞ be the Γ -limit of a (not-relabeled) subsequence of E n . Such a subse-quence always exists by the general compactness theorem of Γ -convergence(see Theorem 8.5 in [DM93] for the classical result, or Theorem 4.7 in [KS08]for the case where each functional is defined on a di ff erent space).Part 2, which only assumes convergence in the mean, states that E ∞ ≤ Γ E .This upper bound follows from Propositions 4.2 and 4.3.To prove Part 1, which assumes uniform convergence, it is enough to provethat E ∞ = Γ E . Indeed, since by the compactness theorem, every sequencehas a Γ -converging subsequence, the Urysohn property of Γ -convergence(see Proposition 8.3 in [DM93]) implies that if all converging subsequencesconverge to the same limit, then the entire sequence converges to that limit.Proposition 4.4 establishes the lower bound E ∞ ≥ Γ E , which together withthe upper bound concludes the proof. ■ Proposition 4.2 (Infinity case)
Assume M n → M in the mean, and let u ∈ L p ( M ; R k ) \ W , p ( M ; R k ) . Then E ∞ [ u ] = ∞ = Γ E [ u ] .Proof : Suppose, for contradiction, that E ∞ [ u ] < ∞ . Let u n → u be a recoverysequence, namely, lim n →∞ E n [ u n ] = E ∞ [ u ] < ∞ . E n [ u n ] < ∞ for all n , and in particular, u n ∈ W , p ( M n , R k ). The coercivity of W n implies thatsup n Z M n | du n | p g n , e d Vol g n < ∞ . Thus, u n is uniformly bounded in W , p , and by Lemma 2.8, u n ◦ F n weaklyconverges (modulo a subsequence) in W , p ( M ; R k ). By the uniqueness ofthe limit, this limit is u , hence u ∈ W , p ( M ; R k ), which is a contradiction. ■ Proposition 4.3 (Upper bound)
Assume M n → M in the mean. Then, for everyu ∈ W , p ( M ; R k ) , E ∞ [ u ] ≤ Γ E [ u ] . Proof : Let u ∈ W , p ( M ; R k ). Define u n = u ◦ F − n ∈ L p ( M n ; R k ). Trivially, u n → u in L p , and by the definition of Γ -limit, E ∞ [ u ] ≤ lim inf n E n [ u n ] . We now show that lim n E n [ u n ] = E [ u ] . (4.1)Since | dF − n | is uniformly bounded, u n ∈ W , p ( M n ; R k ). Therefore, (4.1) readslim n Z M n W n ( d ( u ◦ F − n )) d Vol g n = Z M W ∞ ( du ) d Vol g . First,lim n Z M n W n ( d ( u ◦ F − n )) d Vol g n = lim n Z M F ∗ n W n ( d ( u ◦ F − n )) d Vol F ⋆ n g n = lim n Z M F ∗ n W n ( d ( u ◦ F − n )) d Vol g + Z M F ∗ n W n ( d ( u ◦ F − n )) − d Vol g d Vol F ⋆ n g n ! d Vol F ⋆ n g n = lim n Z M F ∗ n W n ( d ( u ◦ F − n )) d Vol g . (4.2)In the passage from the second to the third line we used the boundedness(3.2) of W and the uniform convergence (2.7) of the volume forms.24econd, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z M F ∗ n W n ( d ( u ◦ F − n )) d Vol g ! / p − Z M W ∞ ( du ) d Vol g ! / p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z M (cid:12)(cid:12)(cid:12) F ∗ n W n ( d ( u ◦ F − n )) / p − W ∞ ( du ) / p (cid:12)(cid:12)(cid:12) p d Vol g ! / p ≤ C Z M (1 + | du | ) p dist pF ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) d Vol g ! / p . (4.3)In the passage from the first to the second line we used the reverse trian-gle inequality, and in the passage to the third line we used the Lipschitzcontinuity (3.3) of W , with L = d ( F − n ) and A = du .Since by (2.9), dist F ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) → L p , we can assume bymoving to a subsequence that this sequence converges almost everywhere.Let ε >
0. By Egorov’s theorem, there exists an A ⊂ M such that Vol g ( M \ A ) < ε and dist F ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) → A . Since | du | is in L p and dist F ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) is bounded uniformly by some constant C ′ , we obtain thatlim sup n Z M (1 + | du | ) p dist pF ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) d Vol g ≤ lim sup n Z M \ A (1 + | du | ) p dist pF ∗ n g n , g ( dF − n , SO( F ∗ n g n , g )) d Vol g ≤ lim sup n C ′ Z M \ A (1 + | du | ) p d Vol g . (4.4)Since M \ A is arbitrary small and | du | ∈ L p , the righthand side is arbitrarysmall, hence (4.3) and (4.4) imply thatlim n Z M F ∗ n W n ( d ( u ◦ F − n )) d Vol g = Z M W ∞ ( du ) d Vol g . (4.5)Together with (4.2), (4.1) follows.We therefore obtain that for every u ∈ W , p ( M ; R k ) E ∞ [ u ] ≤ E [ u ] . (4.6)25ogether with Proposition 4.2, we obtain that (4.6) holds for every u ∈ L p ( M ; R k ). Since E ∞ is a Γ -limit with respect to the L p topology, it is lower-semicontinuous (see Proposition 6.8 in [DM93] or Lemma 4.6 in [KS08]),and E ∞ ≤ ˜ E , (4.7)where ˜ E is the lower semicontinuous envelope of E with respect to thestrong L p topology. We complete the proof by showing that ˜ E = Γ E .The argument is essentially the same as in the proof of Proposition 4.3 in[KM14], using Lemma 5 in [LDR95] and the results of [AF84] (see AppendixB in [KM14] for the relevant generalization of [AF84] to manifolds). ■ Proposition 4.4 (Lower bound)
Assume M n → M uniformly. Then, for everyu ∈ W , p ( M ; R k ) , E ∞ [ u ] ≥ Γ E [ u ] . Proof : Let u ∈ W , p ( M ; R k ), and let u n → u be a recovery sequence. If E ∞ [ u ] = ∞ , then the claim is trivial. Otherwise, we may assume that u n ∈ W , p ( M n ; R k ) for all n . By the coercivity of W n , u n is bounded in W , p , hence u n ◦ F n is bounded in W , p ( M ; R k ) and weakly W , p -converges(modulo a subsequence) to u .We will show that E ∞ [ u ] = lim n E n [ u n ] = lim n Z M n W n ( du n ) d Vol g n = lim n Z M W ∞ ( d ( u n ◦ F n )) d Vol g ≥ lim n Z M QW ∞ ( d ( u n ◦ F n )) d Vol g ≥ Z M QW ∞ ( du ) d Vol g = Γ E [ u ] . (4.8)The passage from the second to the third line follows from the definitionof the quasi-convex envelope. The passage from the third to the fourthline follows from the weak lower-semicontinuity of an integral functionalwith a Carath´eodory quasiconvex integrand (see Section 3.4 in [KM14] fordetails). The rest of the proof derives the equality between the first and thesecond line. 26irst, by the same arguments as in (4.2),lim n Z M n W n ( du n ) d Vol g n = lim n Z M F ∗ n W n ( du n ) d Vol g . (4.9)Second, we use the Lipschitz continuity (3.3) of W , with L = dF n and A = du n , and obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z M F ∗ n W n ( du n ) d Vol g ! / p − Z M W ∞ ( d ( u n ◦ F n )) d Vol g ! / p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z M (cid:12)(cid:12)(cid:12) F ∗ n W n ( du n ) / p − W ∞ ( d ( u n ◦ F n )) / p (cid:12)(cid:12)(cid:12) p d Vol g ! / p ≤ C Z M (1 + F ∗ n | du n | ) p dist p g , F ∗ n g n ( dF n , SO( g , F ∗ n g n )) d Vol g ! / p ≤ C (cid:13)(cid:13)(cid:13) dist g , F ∗ n g n ( dF n , SO( g , F ∗ n g n )) (cid:13)(cid:13)(cid:13) ∞ Z M (1 + F ∗ n | du n | ) p d Vol g ! / p → F ∗ n | du n | is uniformlybounded in L p ( M , g ).Using (4.9) and (4.10) we obtain thatlim n E n [ u n ] = lim n Z M n W n ( du n ) d Vol g n = lim n Z M W ∞ ( d ( u n ◦ F n )) d Vol g (4.11)which completes the proof. ■ The following proposition is a standard convergence of minimizers resultthat typically accompanies Γ -convergence results. Proposition 4.5
Assume that M n → M either uniformly or in the mean, andthat E M n Γ -converges to E ∞ (in the case of uniform convergence we always haveE ∞ = Γ E M ). Let u n ∈ W , p ( M n ; R k ) be a sequence of (approximate) minimizersof E M n , and denote by u n the mean of u n . Then the translated sequence u n − u n s relatively compact (with respect to the L p topology defined by F n ), and all itslimits points are minimizers of E ∞ . Moreover, lim n →∞ inf L p ( M n ; R k ) E M n = min L p ( M ; R k ) E ∞ . Proof : Once again, we write E n = E M n and Γ E = Γ E M . Let u n be a sequenceof approximate minimizers of E n . Since E n is invariant to translations (inthe sense that E n [ u n ] = E n [ u n + x ] for every x ∈ R k ), we will assume w.l.o.g.that u n =
0. We first prove that it is relatively compact, i.e. that everysubsequence (not relabeled) of u n has a subsequence converging in L p .Let w ∈ W , p ( M n ; R k ) be arbitrary and let w n ∈ L p ( M n ; R k ) be a recoverysequence for w . Then, by Theorem 4.1,inf L p E n [ · ] ≤ E n [ w n ] −→ n →∞ E ∞ [ w ] ≤ Γ E [ w ] < ∞ . This shows that inf L p E n [ · ] is a bounded sequence.It follows that E n [ u n ] is bounded, hence, by coercivity, du n is uniformlybounded in L p . Together with the Poincar´e inequality, we obtain that u n ◦ F n is uniformly bounded in W , p ( M ; R k ). This implies the existence of a(not relabeled) subsequence u n → u in L p , proving the relative compactnessof u n .We now prove that u is a minimizer of E ∞ . Let w ∈ L p ( M ; R k ) be an arbitraryfunction, and let w n ∈ L p ( M n ; R k ) be a recovery sequence for w . Then, E ∞ [ w ] = lim n →∞ E n [ w n ] ≥ lim n →∞ inf L p E n [ · ] = lim n →∞ E n [ u n ] ≥ E ∞ [ u ] , where the last inequality follows from the lower-semicontinuity propertyof Γ -limits. Since w is arbitrary, u is a minimizer of E ∞ . Moreover, bychoosing w = u we conclude that E ∞ [ u ] = lim n →∞ inf L p E n [ · ] . ■ In this paper we proved a Γ -convergence result for elastic models of uni-formly converging manifolds. This result is intrinsic, in the following28enses: first, it does not depend on the parametrizations of the manifolds M n and M . Second, while the L p -topology described in Section 2.4 de-pends on the maps F n , the limiting functional Γ E M itself is independentof these maps. That is, Γ E M [ u ] is defined independently of the choice ofmaps, even though recovery sequences converging to u depend on them.The intrinsic nature of the limit model highlights the geometric natureof non-Euclidean elasticity, which is sometimes obscured by choices ofcoordinates and maps.For manifolds that converge in the mean, we do not obtain a Γ -convergenceresult, but, similarly, the Γ -upper bound is independent of parametrizationand of the maps F n . Boundary conditions and external forces
For the sake of clarity, we lim-ited our analysis to unconstrained systems, i.e., systems without externalforces and without boundary constraints. Forces and boundary conditionscan be included in a standard way. Note, however, that boundary condi-tions should be specified for configurations of each of the manifolds M n and M , and the maps F n : M → M n that realize the convergence shouldmap admissible configurations to admissible configurations. That is, themaps F n must satisfy the condition that u n ∈ W , p ( M n ; R k ) is M n -admissibleif and only if u n ◦ F n is M -admissible. Applications of the main theorem to dislocation theory
As discussedin the Introduction, bodies with continuously-distributed dislocations arecommonly modeled as smooth Weitzenb ¨ock manifolds ( M , g , ∇ ), where thea ffi ne connection ∇ is metrically consistent with g and flat (hence, uniquelydetermined by its torsion tensor). In the terminology of this paper, thiscorresponds to a body manifold ( M , d , M , g , ∇ ) (since M is smooth ˜ M = M ).This model is, naturally, viewed as a limit of bodies with finitely manydislocations ( M n , d n , ˜ M n , g n , ∇ n ), were ∇ n is the Levi-Civita connection, i.e.,torsionless.In this paper we associate with each body manifold ( M , d , ˜ M , g ) an elasticenergy functional E ( M , g ) . In the case of continuously-distributed disloca-tions, we would expect to have an energy functional that depends onthe connection, namely, E ( M , g , ∇ ) . When there are no distributed disloca-tions, ∇ is the Levi-Civita connection, ∇ LC , hence it is natural to assume29 ( M , g , ∇ LC ) = E ( M , g ) , so that E ( M , g , ∇ ) extends E ( M , g ) . Since a body with a con-tinuous distribution of dislocations ( M , d , M , g , ∇ ) is an e ff ective modelfor bodies with finitely many dislocations ( M n , d n , ˜ M n , g n , ∇ LCn ), we expect E ( M , g , ∇ ) to be a limit of the elastic energies E ( M n , g n ) (up to relaxation).Two questions arise in this context: First, is E ( M , g , ∇ ) well-defined as a limit ofenergies E ( M n , g n ) , independently of the converging sequence of manifolds?Second, how does E ( M , g , ∇ ) depend on ∇ ?The first part of Theorem 4.1 implies that the limiting elastic energy doesnot depend on the limiting process as long as the sequence of manifoldswith finitely-many dislocations converges uniformly (like the variation ofthe constructions in [KM15, KM] presented in Example 2 in Section 2.3.2).In this case, the limiting energy does not depend on the connection ∇ (orequivalently on the torsion). In other words, the limiting elastic model isonly sensitive to the metric structure of the limit manifold .If one rather considers a larger class of body manifolds that converges tothe limit ( M , d , M , g , ∇ ), including the original constructions in [KM15, KM](which converges only in the mean, see Example 1 in Section 2.3.2), thenour results do not guarantee the existence of a Γ -limit independent of theconverging sequence. However, one would still expect that if a specificsequence of bodies with dislocations has a Γ -limit energy, then the e ff ectof the torsion would be an additional compatibility constrain, and hencewould increase the energy compared to the torsion-free case. In otherwords, the inequality E ( M , g , ∇ ) ≥ E ( M , g ) is expected. The second part ofTheorem 4.1 shows that it is not the case, as the limit energy is boundedfrom above by that determined by the metric. In particular, if ( M , g ) canbe isometrically embedded in R k (e.g. as in the main example of [KM15]),then ( M , d , M , g , ∇ ) has a zero energy embedding in R k , regardless of ∇ andthe converging sequence. Other applications of the main theorem
The first part of Theorem 4.1also holds for approximations of a surface M by Euclidean triangles M n , asdescribed in Example 3 in Section 2.3.2. The Γ -convergence still holds if oneconsiders only maps M n → R k which are a ffi ne on every triangle. Indeed,the only change in the proof is to replace the recovery sequence u n : M n → R k in Proposition 4.3 with a piecewise a ffi ne sequence that L p -convergesto the same limit u : M → R k ; this is always possible. This implies the30onsistency of finite element approximations based on triangulations ofsurfaces (or on simplices in higher dimensions) and their piecewise a ffi neembeddings into Euclidean space.Finally, our results establish the structural stability of elastic models that are p -regular according to Definition 3.1: if two metrics are arbitrarily closeto each other (with respect to the sup-norm), then their elastic energiesare arbitrarily close. This observation validates experimental estimatesof reference metrics via interpolations based on finite sets of measureddistances (e.g., [SRS07]). Other rigidity criteria
The distortion of a linear map
Dis A defined in(2.2) plays a role both in the definition of convergence of body manifoldsin Section 2.3 and in the Lipschitz continuity property of p -regular energydensities in Definition 3.1.In principle, one can choose other measures for the distortion of a lin-ear map, for which other energy densities may be p -regular according toDefinition 3.1. If we change the definition of Dis A accordingly also inthe definitions of converging body manifolds (Definitions 2.3–2.5), thenour results do not change, providing that for uniformly converging bodymanifolds, the uniform convergence of the new distortion criterion in Def-inition 2.4 continues to imply uniform bi-Lipschitzness (2.4) and uniformvolume convergence (2.7). Even if this is not true for the new distortioncriterion, (2.4) and (2.7) can be assumed in addition to (2.3) in the definitionof uniform convergence 2.4 (as in the definition of mean convergence 2.5),and the proof will still hold. Open questions
Outside the context of dimension reduction, this is, tothe best of our knowledge, the first paper to consider Γ -convergence of elas-tic energies of converging manifolds. Unlike dimension reduction, wherethe converging manifolds are ordered by an inclusion relation, here thenotion of convergence allows for varying topologies and metric structures.Naturally, there remain numerous open questions, among which are:1. Do the elastic energies Γ -converge in the case of manifolds convergingin the mean? Even if such a result does not hold in general, it isof interest to determine whether it holds for the specific sequence31f manifolds with dislocations considered in [KM15, KM] (see alsoExample 1 in Section 2.3.2).2. It would also be interesting to relax some of the assumptions on theelastic energy densities. In particular, for a more physical model onemay want to modify the growth condition in Definition 3.1, such toinclude densities tending to infinity when deformations tend to besingular (the relaxation of the growth condition is of interest in manyother contexts as well, see [CD15] for details).3. This paper considers “bulk” elasticity—the embedding of a k -dimensionalmanifold in the k -dimensional Euclidean space. Another main themein elasticity theory (both classical and non-Euclidean) is the deriva-tion of dimensionally reduced models for bodies with one or moreslender dimensions (see e.g. [LDR95, FJM02, KS14, KM14]). Aninteresting question concerns the two-parameter limit of changingmetrics and dimension reduction. A result in this direction wouldalso relate to the von-K´arm´an limits of slender bodies whose metricstend to a Euclidean metric; such a situation was treated in [LMP11].4. Another question, which unlike the previous ones is of geometricnature rather than analytic, concerns the role of global distortion inmanifolds that converge in the mean. An asymptotically vanishingglobal distortion is part of our mean convergence definition (Con-dition 2 in Definition 2.5). Its only role in the present paper is toguarantee the uniqueness of the limit (as vanishing distortion im-plies Gromov-Hausdor ff convergence); it doesn’t play any role in thesubsequent analysis.It would be interesting to understand whether this condition can beomitted, that is to say, whether asymptotic vanishing of the meanlocal distortions (Condition 3 in Definition 2.5) su ffi ces to define anotion of convergence (i.e., that the limit does not depend on thechoice of morphisms). Such a result would require rigidity estimatesfor Riemannian manifolds, analogous to Reshetnyak’s generalizationof Liouville’s rigidity theorem (which is in a Euclidean setting, see[Res67] for the original paper and [FJM02] for a modern restatement).32 cknowledgements This research was supported by the Israel-US Bina-tional Foundation (Grant No. 2010129), by the Israel Science Foundation(Grant No. 661 /
13) and by a grant from the Ministry of Science, Technol-ogy and Space, Israel and the Russian Foundation for Basic Research, theRussian Federation.
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