Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity
Marco Barchiesi, Duvan Henao, Carlos Mora-Corral, Rémy Rodiac
aa r X i v : . [ m a t h . A P ] F e b HARMONIC DIPOLES AND THE RELAXATION OF THENEO-HOOKEAN ENERGY IN 3D ELASTICITY
MARCO BARCHIESI, DUVAN HENAO, CARLOS MORA-CORRAL, AND R´EMY RODIAC
Abstract.
In a previous work, Henao & Rodiac (2018) proved an existence result forthe neo-Hookean energy in 3D for the essentially 2D problem of axisymmetric domainsaway from the symmetry axis. In the relaxation approach for the neo-Hookean energyin 3D, minimizers are found in the space of maps that can be obtained as the weak H -limit of a sequence of diffeomorphisms; see Mal´y (1993). Among the singular mapscontained in that H -weak closure, a particularly upsetting pathology, constructed uponthe dipoles found in harmonic map theory, was presented by Conti & De Lellis (2003).Their example involves a cavitation-type discontinuity around which the orientation isreversed, the cavity created there being furthermore filled with material coming fromother part of the body. Since physical minimizers should not exhibit such behaviour, thisraises the regularity question of proving that minimizers have Sobolev inverses (since theinverses of the harmonic dipoles have jumps across the created surface). The first naturalstep is to address the problem in the axisymmetric setting, without the assumption thatthe domain is hollow and at a distance apart of the symmetry axis. Attacking that problemwill presumably demand a thorough analysis of the fine properties of singular maps in the H sequential weak closure, as well as an explicit characterization of the relaxed energyfunctional. In this paper we introduce an explicit energy functional (which coincides withthe neo-Hookean energy for regular maps and expresses the cost of a singularity in terms ofthe jump and Cantor parts of the inverse) and an explicit admissible space (which containsall weak H limits of regular maps) for which we can prove the existence of minimizers.Chances to succeed in establishing that minimizers do not have dipoles are higher in thisexplicit alternative variational problem than when working with the abstract relaxationapproach in the abstract space of all weak limits.Our candidate for relaxed energy has many similarities with the relaxed energy intro-duced by Bethuel-Br´ezis-Coron for a problem with lack of compactness in harmonic mapstheory. The proof we present in this paper for the lower semicontinuity of the augmentedenergy functional, partly inspired by the prominent role played by conformality in thatcontext, further develops the connection between the minimization of the 3D neo-Hookeanenergy with the problem of finding a minimizing smooth harmonic map from B into S with zero degree boundary data. Introduction
A regularity problem for the well-posedness of the neo-Hookean model.
Weconsider the problem of the existence of minimizers for polyconvex energies of the form Z Ω W ( x , D u ( x )) d x , (1.1) as is customary in nonlinear elasticity (see, e.g. [3, 12]). Here Ω ⊂ R represents thereference configuration of an elastic body, u : Ω → R is the deformation map, and W : Ω × R × → R ∪ {∞} is the elastic stored-energy function of the material. In thecase of a neo-Hookean material (the most widely used model in physics, engineering, andmaterials science, as well as the only one derived from first principles) the energy is E ( u ) = Z Ω (cid:2) | D u | + H (det D u ) (cid:3) d x (1.2)where H : (0 , ∞ ) → [0 , ∞ ) is a convex function such thatlim t →∞ H ( t ) t = lim s → H ( s ) = ∞ . (1.3)Since interpenetration of matter is physically unrealistic, minimizers are sought in a suitablesubclass of A (Ω) := { u ∈ H (Ω , R ) : E ( u ) < ∞ , u is injective a.e. } , that is, a set of full Lebesgue measure in Ω is assumed to exist such that u restricted tothis set is injective.In his celebrated existence theory Ball [3] was able to apply the direct method of thecalculus of variations, in spite of the lack of convexity of the energy functional, based onthe identitydet D u = Det D u , h Det D u , ϕ i := − Z Ω u ( x ) · (cid:0) (cof D u ) Dϕ (cid:1) , ϕ ∈ C c (Ω) , whereby the Jacobian determinant can be written as a distributional divergence, and ananalogous identity for cof D u , the matrix of 2 × W ( x , F ) ≥ c | F | + c | cof F | / + H (det F ) − c , ( x , F ) ∈ R × . (1.4)However, this coercivity excludes the neo-Hookean materials. Moreover, for neo-Hookeanmaterials the hypothesis of finite energy alone is insufficient to ensure that det D u =Det D u , as shown by the solutions to the boundary value problems in the models forcavitation [4, 54, 50, 53]. The difficulty is not related to a specific method of proof butis of a more fundamental nature: from the example of cavitation it follows that the neo-Hookean energy is not H -quasiconvex, and this H -quasiconvexity is necessary for (1.1)to be H -weakly lower semicontinuous in a general space such as A (Ω), as proved by Ball& Murat [5].Essentially three different strategies have been proposed to overcome the aforementionedobstacle. The three implicitly suggest that the Lavrentiev phenomenon associated to cav-itation might be indicative of a missing ingredient in the model. If the working space A (Ω) ⊂ H contains singular maps capturing the physically relevant phenomenon of inter-nal rupture at the microscopic length scale (even more if these maps consistently appearas counterexamples blocking the mathematical analysis) then it is important to preciselydefine the energy to be assigned to this rupture singularities (as done in the variational ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 3 analysis of brittle fracture by Ambrosio & Braides [1]—in the static case—and by Franc-fort & Marigo [23, 8]—for quasistatic crack propagation). One such strategy goes in thisdirection of completing the model so that it includes (at least part of) the energy of a cav-itation, while the other two restrict their attention to working spaces A (Ω) that excludecavitation altogether.We begin by mentioning the strategy that allows for cavitation. It consists in accountingfor the energy of the stretch of the surfaces of the elastic body. This energy is usually muchsmaller than the bulk energy and can be neglected, but in the case of cavitation it showsthat Ball & Murat’s [5] counterexample to lower semicontinuity is unphysical . Giaquinta,Modica & Souˇcek [25, Sect. 3.3.1, p. 284], [26, Sect. 3.6.2, p. 264] used the mass of theboundary of the graph of u (regarded as a current) as an expression of that surface energy;M¨uller & Spector [50] used the perimeter Per u (Ω) of the deformed configuration. In [31, 32]the two approaches were shown to be conceptually equivalent in the Sobolev setting . Avariant of this strategy, analyzed by Sivaloganathan & Spector [53] in the static problemand by Mora-Corral [45] for quasistatic evolutions, is to assign a fixed quantized energy atany new cavitation point.Regarding the two strategies that exclude cavitation, the first starts with defining theneo-Hookean energy for a singular H map by relaxation, regarding formula (1.2) as validonly for smooth maps. As shown by Marcellini [42] and Dacorogna & Marcellini [15],classical cavitation requires then infinite energy since the function H that penalizes volumechanges blows up in any attempt to regularize a map of the form u ( x ) := r ( | x | ) x | x | , with r ∈ C (cid:0) [0 , ∞ ) , (0 , ∞ ) (cid:1) , opening a cavity with radius r (0) >
0. In particular, the relaxationof the neo-Hookean energy is not an approach aiming at modelling cavitation, but the goalis rather to provide an existence theory for more regular maps. This is consistent withthe interpretation that, in reality, an ε -neighbourhood of a cavitation point consists of adamaged or weakened material (which is far from opposing the usual resistance to increasesin volume), so the way it stretches to cover a non-infinitesimal volume cannot be assignedthe expensive energy H (det D u ε ) of a healthy elastic material. The second strategy amongthose not allowing for cavitation is to look for minimizers in the smaller class A r (Ω) ofmaps in A (Ω) for which the generalized divergence identitiesDiv (cid:0) (adj D u ) g ◦ u (cid:1) = (div g ) ◦ u det D u ∀ g ∈ C c ( R , R ) (1.5)(of which det D u = Det D u is a particular case) are satisfied. This was proposed byGiaquinta, Modica & Souˇcek [24], a simplified version of it then being provided also byM¨uller [48].The various strategies outlined in the previous paragraph have succesfully yielded ex-istence theories, at least for materials very similar to the neo-Hookean. In particular, aspecific case of the last strategy (working with a class of functions for which the divergence The domain is decomposed in ever smaller regions where the deformations are defined by suitablyrescaling a single cavitation map, and this results in the opening of cavities with an arbitrarily largesurface. Discrepancies between the two surface functionals other than those originated by two (and only two)fracture surfaces coming in contact with each other were ruled out.
MARCO BARCHIESI, DUVAN HENAO, CARLOS MORA-CORRAL, AND R´EMY RODIAC identities (1.5) hold) consist on imposing coercivity hypotheses of the form (1.4), while thecoercivity assumptions W ( x , F ) ≥ c | F | p + H (det F ) − c , ( x , F ) ∈ R × , for any p > W ( x , F ) ≥ c | F | + H ( | cof F | ) + H (det F ) − c , ( x , F ) ∈ R × , with H and H only superlinear at infinity [32, Th. 8.5] are enough to carry out existencetheories both in the settings of the last strategy (for which (1.5) holds) and of the firststrategy (for which an energy associated to cavitation is added). Nevertheless, these threeassumptions (and, in fact, most of the analyses available) exclude the borderline quadraticcase of the neo-Heokean energy, where a superlinear dependence on the cofactors is alsoabsent. The only results known by the authors for this case are that of Henao & Rodiac[36], who addresses the essentially 2D problem of axisymmetric domains away from thesymmetry axis, and the full 3D result by Mal´y [41], who proves, via relaxation, the existenceof minimizers in the H -weak closure of the space of diffeomorphisms. Among the singularmaps contained in that H -weak closure, a particularly upsetting pathology, constructedupon the dipoles found in harmonic map theory in [9], was presented by Conti & De Lellis[13]. In it, two cavitations with opposite signs (in the sense that the orientation is reversedin one part of the body—as a result of it being transported through a portal of the sizeof a single line segment) cancel each other (the volume opened by one cavity is filled withthe part of the body whose orientation is reversed). Since such flagrant interpenetration ofmatter can hardly be accepted as physical, an existence theory for the neo-Hookean energyis incomplete until the following regularity problem (the object of study of this paper) issolved: to prove that minimizers of the relaxed energy in the weak closure of some classof regular maps (where the divergence identities hold and cavitations—of either sign—areexcluded) are not only weak limits but enjoy the same regularity. Furthermore, since itwas proved in [41, Thm. 3.1] that the original neo-Hookean energy is lower semicontinuouswhen the limit map and the maps in the sequence are regular (see also Lemma 4.1), thenthe regularity of the minimizer of the relaxed energy would yield a full result of existencenot only for that functional but for the original neo-Hookean problem.The regularity problem (to which the construction of an existence theory has essentiallyreduced) is already very hard in the simplified setting of axisymmetric maps (which stillcontains the pathological dipoles by Conti & De Lellis). Hence, the natural next stepis to establish the existence of minimizers of the original (unrelaxed) neo-Hookean en-ergy among injective and axisymmetric maps enjoying enough regularity (no cavitations).Knowing only abstractly that the space is the weak closure of a regular class of maps, andthat the functional is the least of all inferior limits taken over all possible weakly convergentsequences, will probably prove insufficient for that research program. In this paper we pro-vide a complementary route: we define an explicit admissible space, which can be provedto be weakly closed as well as to contain all weak limits of regular maps; define a muchmore explicit energy functional (that coincides with (1.2) for regular maps); and prove thatminimizers are attained for that functional in that admissible space. We believe that the
ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 5 energy functional that we introduce is actually the relaxation of the neo-Hookean energy.However, even without having to wait for a proof of that relaxation result, studying the (par-tial) regularity of minimizers (just ruling out—classical or lower-dimensional—cavitationand dipoles) in the alternative explicit variational problem hereby presented already offersa new plan to address the well-posedness of the neo-Hookean model (and constitutes themost substantial step towards this main and long-standing open problem since the work ofMal´y and Conti & De Lellis). In particular, having a geometric handle of the singularitiesand an estimate on their energetic cost makes it more likely to build regular competitorsshowing that singularities are too expensive and cannot appear in a minimizer.Regarding boundary conditions, our attention is restricted to the pure displacementproblem. This gives a target domain that is fixed and open, making the Sobolev regularityfor the inverse easier to obtain. For the same reason, it is important to rule out theformation of cavities at the boundary, hence we assume that Ω ⋐ e Ω, where e Ω is a smoothbounded domain of R , and require the deformations u to coincide with a bounded C orientation-preserving diffeomorphism b : e Ω → R not only on ∂ Ω but on the whole of e Ω \ Ω, and to be injective a.e. not only in Ω but on the whole of e Ω (a device used beforein elasticity [54, 53, 35], which can nevertheless be avoided with the techniques of [34]).Set Ω b := b (Ω) , e Ω b := b ( e Ω) . We assume that Ω, e Ω and b are axisymmetric (see the definition (2.5) in Section 2.3).Define A s := { u ∈ H ( e Ω , R ) : u is injective a.e. and axisymmetric,det D u > , u = b in e Ω \ Ω , and E ( u ) ≤ E ( b ) } , (1.6) A rs := { u ∈ A s : the divergence identities (1.5) are satisfied } , (1.7)and B := { u ∈ A s : e Ω b = im G ( u , e Ω) a.e. and u − ∈ W , ( e Ω b , R ) × BV ( e Ω b ) } (1.8)(the geometric image im G ( u , e Ω) is that of Definition 2.5).
Theorem 1.1.
The energy F ( u ) := Z Ω (cid:2) | D u | + H (det D u ) (cid:3) d x + 2 k D s u − k M (˜Ω b , R ) attains its infimum in B . Moreover, if there exists a minimizer in B which is also in theregular subclass A rs , then the infimum of the original neo-Hookean energy E of (1.2) in A rs is attained. The proof is at the end of Section 6. Note that the energy F coincides with the neo-Hookean energy E for diffeormorphisms, in agreement with Marcelinni’s [42] and Mal´y’s[41] approaches. The explicit new term 2 k D s u − k M (˜Ω b , R ) is the main contribution of thiswork. MARCO BARCHIESI, DUVAN HENAO, CARLOS MORA-CORRAL, AND R´EMY RODIAC
We believe that, at least for certain generic boundary conditions, the minimizer of The-orem 1.1 is indeed regular (in the sense of at least satisfying the divergence identities).This leads us to conjecture that minimizers of the neo-Hookean energy cannot exhibit thepathological behaviour of the harmonic dipoles constructed by Conti & De Lellis.
Conjecture.
The neo-Hookean energy E ( u ) = R Ω | D u | + H (det D u ) d x attains its infi-mum in A rs . As mentioned above, the strategy of proof that we are proposing is to take advantage ofthe explicit energy functional defined in Theorem 1.1 and of the explicit characterizationof the functional space B in order to prove that harmonic dipoles (or any jump or Cantorpart in the vertical component of the inverse) are so expensive that they cannot be presentin an energy minimizer.1.2. The singular energy.
The axisymmetric minimization problem when the referenceconfiguration Ω does not contain its axis of symmetry was considered in [36]. In thispaper we pursue the more difficult case when Ω contains its axis of symmetry (wherethe singularities can appear). Away from the axis the divergence identities are satisfied(Lemma 3.1). Combining this with the convergence of the images established in [30] weobtain (Lemma 4.1) that weak limits of regular maps belong to B . In particular, theexample of Conti & De Lellis is generic in that any cavitation produced by a map in ourclass B must be filled with the image of some other part of the body. Also, the only wayin which the divergence identities do not pass to the limit is when the cofactors are notequiintegrable, so the example is again generic in that a ‘stack’ of surfaces with smallerand smaller diameters, orthogonal to the axis of symmetry, are necessarily being stretchedwithout control in any regular sequence weakly converging to a singular map in our class.Since the area vector (cof D u ) e d H over an ( x , x )-disk of radius δ is the wedge productof the tangent vectors ( D u ) e and ( D u ) e of the deformed surface, Cauchy’s inequalitygives Z C δ | D u j | d x ≥ Z C δ | (cof D u j ) e | d x = 2 Z u j ( C δ ) | D ( u − ) j | d y where the integrals are over small δ -cylinders around the symmetry axis. The sets u j ( C δ )collapse to a set with zero volume (thanks to the equiintegrability of the determinants);in the example by Conti & De Lellis, they collapse to a sphere, which can be thought ofexactly as the jump set of the (vertical component of the) inverse u − for the limit map u (since two different parts of the body end up being in contact through that sphere, whichis the fracture surface in the deformed configuration for the singular map). However, Z Ω | D u | d x = lim δ ց Z Ω \ C δ | D u | d x ≤ lim inf δ ց lim inf j →∞ Z Ω \ C δ | D u j | d x so the original formula for the neo-Hookean energy completely misses out the concentrationof the Dirichlet energy if applied directly to the singular map u .The above heuristic argument aims to show the need for the relaxation approach in orderto define the neo-Hookean energy for the singular maps in the weak closure of any regular ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 7 class. In addition, it aims to clarify why the singular part of the distributional derivative ofthe inverse has the potential to adequately describe the energy concentration: in the refer-ence configuration all evidence of the abnormal activity of the regular sequence is lost andhidden into the one-dimensional symmetry axis, but in the deformed configuration large(two-dimensional or fractal) structures remain and allow the singular map to rememberthe energy invested into their formation. All in all, the singular term 2 k D s u − k M (˜Ω b , R ) is not artificial or invented at all , but it emerges naturally from the Dirichlet energy. Atthe very least, we prove (equation (6.16)) that it is a lower bound of the abstract relaxedenergy functional.A final word is in order regarding the relaxation approach. As opposed to what occurs inthe modelling of cavitation (or of brittle fracture), where the relaxation is not appropriatesince the deformation of a broken and softened portion of the body should not be associatedwith the same constituive law for the energy per unit volume as the healthy material, in thestudy of the neo-Hookean energy in (the weak closure of) the class of regular maps we seemto be confronted not with singularities that are physically relevant but with pathologicaldeformations which we would prefer to exclude from our analysis. Instead of facing thehard modelling problem of assigning a correct and tractable energy of a Sobolev mapwith physical singularities, we are in need of a strategy of proof to solve a purely analyticregularity problem. The discussion is not about whether it was because the material hadsoftened that the map by Conti & De Lellis could stretch δ -disks into a -radius sphere viabubbling, case in which the full Dirichlet term | D u | would be disproportionate. It is notabout whether | D u | + H (det D u ) is the right physical energy for the harmonic dipoles.The discussion is about how to reach a contradiction from the assumption that a minimizingsequence of regular maps ends up forming those dipoles, and a successful argument couldprobably use that if that were the case then the regular maps in the sequence (for whichthe neo-Hookean model would be taken to be physically realistic) would produce an energyconcentration of (at least)2 k D s u − k M = 2 · (area of the bubble) · (the length of the dipole) , and that this is (presumably) more than what a minimizer can afford.1.3. Connection with harmonic map theory.
Bethuel, Br´ezis and Coron [7] (see also[26]) also derived a relaxed energy to treat a problem of lack of compactness in the theoryof harmonic maps between manifolds. The alternative expression for the energy that weobtain, when evaluated at the map by Conti & De Lellis (which seems to represent gener-ically one of the two types of singularities that can appear, the other type correspondingto Cantor parts in the inverse), gives exactly the same expression as the relaxed energy inthe context of harmonic maps.In a forthcoming article we will construct a recovery sequence for the map by Conti & DeLellis showing that, at least in that (generic) pathological deformation, the lower boundwe obtained coincides with the relaxed energy. Also, under the additional assumption E ( u ) < ∞ for the surface energy functional defined in Section 2.2, we will prove thatfor any weak limit of regular maps there exist a countable family of points ξ i , ξ ′ i lying MARCO BARCHIESI, DUVAN HENAO, CARLOS MORA-CORRAL, AND R´EMY RODIAC on the axis of symmetry and a countable family of sets of finite perimeter whose reducedboundaries Γ i satisfy k D s u − k M (Ω ′ , R ) = X i ∈ N | ξ i − ξ ′ i |H (Γ i ) . The right-hand side of the latter expression is the analogue of the “length of minimalconnection”, introduced in [9], connecting singularities of harmonic maps. Besides, thesupplementary term in the harmonic maps relaxed energy can be expressed in terms ofthis length of minimal connection in the case where the map considered has a finite num-ber of singularities. Furthermore, we are also able to express our candidate for the relaxedenergy in the framework of Cartesian currents. The expression we find is also analogous tothe relaxed energy for harmonic maps in this framework. This reveals a strong connectionbetween the problem of minimizing the neo-Hookean energy and finding a smooth mini-mizing harmonic maps from B into S with a smooth boundary data with zero degree.This problem was raised by Hardt and Lin in [28] and is still open nowadays. For the studyof partial regularity and prescribed singularities problem for harmonic maps from B to S in the axially symmetric setting we refer to [29] and [44].1.4. Organization of the paper.
The paper is organized as follows. In Section 2 weintroduce some notation and definitions that will be used in the sequel. More precisely wedefine there the geometric image of a map and the surface energy of a map. This notionquantifies how much the divergence identities (1.5) are not satisfied by a map. We thenmake precise what we mean by axisymmetry and specify in which sense the boundarycondition is satisfied. We also introduce the notion of family of ‘good open sets’ and showhow to relate the properties of a 3 D axisymmetric map u to the properties of its associated2 D map v . In section 3 we describe fine properties of maps in A s , i.e., axisymmetric mapswhich are injective a.e. and with a.e. positive Jacobian. We start with regularity propertiesof general axisymmetric maps, then we define the topological degree and topological imageof maps. We will need both the definition of the classical degree for continuous functionsand the definition of the Brezis–Nirenberg degree for Sobolev maps. A particular role isplayed by the topological image of the segment which is the intersection of the domain Ωand the symmetry axis. Then we focus on the invertibility property of maps in A s . Themain result of this section states that a suitably defined inverse of a map in A s has its twofirst components which are Sobolev functions. Section 4 is devoted to the proof of existenceof minimizers in the class A s . However we suspect that minimizers in this class could beirregular, in the sense that they could exhibit a pathology similar to the one of the exampleof Conti–De Lellis. That is why our main focus in this paper is to derive an explicit energyplaying the role of a relaxed energy for the neo-Hookean problem. In section 5, we focusfirst on regularity properties of weak limits in H of regular maps, i.e., maps in A rs . It isof crucial importance for the rest of the paper that we obtain that their geometric imageis equal to the entire target domain (up to a null- L -set) and that their inverses defined inall e Ω b are in BV ( e Ω b , R ), with the first two components in W , ( e Ω b ). These results rest onthe preliminary analysis of section 3. At last, in section 6, we give a lower semicontinuityresult for our candidate relaxed energy, hence proving a lower bound on the actual relaxed ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 9 energy. We also obtain various existence results thanks to the previous analysis and givea proof of Theorem 1.1. 2.
Notation and preliminaries
Geometric image and area formula.
In this section Ω is a bounded open set of R N , N ≥ A ⊂ R N at x ∈ R N : D ( A, x ) = lim r → | B ( x , r ) ∩ A || B ( x , r ) | . Here use |·| for the Lebesgue measure in R N . An alternative notation is L N . The Hausdorffmeasure of dimension d is denoted by H d . The abbreviation a.e. for almost everywhere or almost every will be intensively used. It refers to the Lebegue measure, unless otherwisestated. Given two sets A, B of R N , we write A ⊂ B a.e. if L N ( A \ B ) = 0, while A = B a.e. or A a.e. = B a.e. both mean A ⊂ B a.e. and B ⊂ A a.e. An analogous meaning is givento the expression H d -a.e.The definition of approximate differentiability can be found in many places (see, e.g.,[20, Sect. 3.1.2.], [50, Def. 2.3] or [32, Sect. 2.3]).We recall the area formula (or change of variable formula) of Federer. We will use thenotation N ( u , A, y ) for the number of preimages of a point y in the set A under u . Proposition 2.1.
Let u ∈ W , (Ω , R N ) , and denote the set of approximate differentiabilitypoints of u by Ω d . Then, for any measurable set A ⊂ Ω and any measurable function ϕ : R N → R , Z A ( ϕ ◦ u ) | det D u | d x = Z R N ϕ ( y ) N ( u , Ω d ∩ A, y ) d y whenever either integral exists. Moreover, if a map ψ : A → R is measurable and ¯ ψ : u (Ω d ∩ A ) → R is given by ¯ ψ ( y ) := X x ∈ Ω d ∩ A, u ( x )= y ψ ( x ) then ¯ ψ is measurable and Z A ψ ( ϕ ◦ u ) | det D u | d x = Z u (Ω d ∩ A ) ¯ ψϕ d y , y ∈ u (Ω d ∩ A ) , (2.1) whenever the integral on the left-hand side of (2.1) exists. We refer to [50, Prop. 2.6] and [20, Thm. 3.2.5 and Thm. 3.2.3] for the area formula.
Definition 2.2.
Let u ∈ W , (Ω , R N ) be such that det D u > as theset of x ∈ Ω for which the following are satisfied:i) the approximate differential of u at x exists and equals D u ( x ).ii) there exist w ∈ C ( R N , R N ) and a compact set K ⊂ Ω of density 1 at x such that u | K = w | K and D u | K = D w | K , iii) det D u ( x ) > is a set of full Lebesgue measure in Ω, i.e., | Ω \ Ω | = 0. Thisfollows from Theorem 3.1.8 in [20], Rademacher’s Theorem and Whitney’s Theorem. Twoimportant properties for a map are Lusin’s properties (N) and (N − ). Definition 2.3.
Let X ⊂ R n be a measurable set. We say that a measurable function u : X → R N satisfies Lusin’s condition (N) if for every A ⊂ X such that | A | = 0 we have | u ( A ) | = 0. We say that u satisfies condition (N − ) if for every A ⊂ R N such that | A | = 0we have | u − ( A ) | = 0.We will use the following result, which is a consequence of Proposition 2.1 (see, e.g., [6,Lemma 2.8] if necessary). Lemma 2.4.
Let u ∈ W , (Ω , R N ) . Then u | Ω d satisfies Lusin’s condition (N). Moreover,if det D u ( x ) = 0 for a.e. x ∈ Ω , then u satisfies Lusin’s (N − ) condition. Definition 2.5.
For any measurable set A of Ω, we define the geometric image of A under u as im G ( u , A ) = u ( A ∩ Ω )with Ω as in Definition 2.2.2.2. The surface energy.
In this subsection N ∈ N , N ≥ ⊂ R N is a boundedopen set. Let u ∈ W ,N − (Ω , R N ) be such that det D u ∈ L (Ω). We start by observingthat, when N = 3, cof D u is controlled in terms of | D u | . This follows, for example, fromthe following inequality that we will repeatedly use: Lemma 2.6. | F | ≥ √ | cof F | for all F ∈ R × , with optimal constant.Proof. The inequality is equivalent to | F | − | cof F | ≥
0. Both terms | F | and | cof F | areinvariant under multiplication by rotations. Therefore, by singular value decomposition,we can assume that F is diagonal with diagonal elements v , v , v . In this case, we have | F | − | cof F | = (cid:0) v + v + v (cid:1) − (cid:0) v v + v v + v v (cid:1) = 12 (cid:2) ( v − v ) + ( v − v ) + ( v − v ) (cid:3) ≥ . That the constant is optimal can be seen by considering F = I . (cid:3) The adjugate matrix adj A of A ∈ R N × N satisfies (det A ) I = A adj A , where I denotesthe identity matrix. The transpose of adj A is the cofactor cof A . Definition 2.7.
Let u ∈ H (Ω , R N ) be such that cof D u ∈ L (Ω , R N × N ) and det D u ∈ L (Ω).a) For every φ ∈ C c (Ω) and g ∈ C c ( R N , R N ) we define E u ( φ, g ) = Z Ω [ g ( u ( x )) · (cof D u ( x ) Dφ ( x )) + φ ( x ) div g ( u ( x )) det D u ( x )] d x . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 11 b) For all f ∈ C c (Ω × R N , R N ) we define E u ( f ) = Z Ω [ D x f ( x , u ( x )) · cof D u ( x ) + div y f ( x , u ( x )) det D u ( x )] d x and E ( u ) = sup {E u ( f ) : f ∈ C c (Ω × R N , R N ) , k f k L ∞ ≤ } . Clearly,
E ≤ E . The following result shows that if E vanishes, so does E . This impliesthat the divergence identities (1.5) are satisfied if and only if the surface energy E isidentically zero. The proof consists in using the continuity of f
7→ E u ( f ) and the densityof the linear span of products of functions of separated variables in C c ( R N , R N ) (see, e.g.,[40, Cor. 1.6.5]). Lemma 2.8.
Let u ∈ W ,N − (Ω , R N ) be such that det D u ∈ L (Ω) . Let V be an open setin Ω and suppose that E ( u | V ) = 0 . Then E ( u | V ) = 0 . In the rest of this section we take N = 3. The following result is a particular case of thecalculation of the energy E for the Cartesian product of two functions. Lemma 2.9.
Let ω ⊂ R be open and let I ⊂ R be open and bounded. Let v ∈ H ( ω, R ) .Then det D v ∈ L ( ω ) and E ( v ) = 0 . Define w : ω × I → R as w ( x , x , x ) =( v ( x , x ) , x ) . Then E ( w ) = 0 .Proof. Since det D v = ∂ x v ∂ x v − ∂ x v ∂ x v we find that | det D v | ≤ | D v | ∈ L ( ω ).Now for v ∈ H ( ω, R ) we can find a sequence of approximating maps v n ∈ C ∞ ( ω ) suchthat det D v n → det D v in L ( ω ) and cof D v n → cof D v in L ( ω ). Since E ( v n ) = 0 byintegration by parts, we find that E ( v ) = 0, by passing to the limit. For x ∈ R , write x = (ˆ x , x ) with ˆ x = ( x , x ), and analogously for y ∈ R . Using Lemma 2.8, it suffices toshow that ¯ E w ( φ, g ) = 0 for φ ∈ C c ( ω × I ) and g ∈ C c ( R , R ) of the form φ ( x ) = φ (ˆ x ) φ ( x ) and g ( y ) = (cid:18) g (ˆ y ) g ( y ) g (ˆ y ) g ( y ) (cid:19) , for some φ ∈ C c ( ω ) , φ ∈ C c ( I ) , g ∈ C c ( R , R ) , g ∈ C c ( R ) , g ∈ C c ( R ) , g ∈ C c ( R ) . For such φ and g we have Dφ ( x ) = (cid:18) φ ( x ) Dφ (ˆ x ) φ (ˆ x ) φ ′ ( x ) (cid:19) , div g ( y ) = g ( y ) div g (ˆ y ) + g (ˆ y ) g ′ ( y ) . On the other hand, for a.e. x ∈ ω × ID w ( x ) = (cid:18) D v (ˆ x ) (cid:19) , cof D w ( x ) = (cid:18) cof D v (ˆ x ) det D v (ˆ x ) (cid:19) , det D w ( x ) = det D v (ˆ x ) . Therefore, for a.e. x ∈ ω × I , g ( w ( x )) · (cof D w ( x ) Dφ ( x )) + φ ( x ) div g ( w ( x )) det D w ( x )= φ ( x ) g ( x ) h g ( v (ˆ x )) · (cid:16) cof D v (ˆ x ) Dφ (ˆ x ) (cid:17) + φ (ˆ x ) div g ( v (ˆ x )) det D v (ˆ x ) i + g ( v (ˆ x )) φ (ˆ x ) det D v (ˆ x ) [ g ( x ) φ ′ ( x ) + φ ( x ) g ′ ( x )] . (2.2)Now, since E ( v ) = 0 and E ( id I ) = 0 we have that Z ω h g ( v (ˆ x )) · (cid:16) cof D v (ˆ x ) Dφ (ˆ x ) (cid:17) + φ (ˆ x ) div g ( v (ˆ x )) det D v (ˆ x ) i dˆ x = 0and Z I [ g ( x ) φ ′ ( x ) + φ ( x ) g ′ ( x )] d x = 0so an integration of (2.2) in ω × I yields E w ( φ, g ) = 0. (cid:3) In the following, we show that the precomposition of a map of zero energy with a regularmap has also zero energy; a related result was shown in the proof of [31, Th. 7].
Lemma 2.10.
Let u ∈ H (Ω , R ) ∩ L ∞ (Ω , R ) be such that det D u ∈ L (Ω) and E ( u ) = 0 .Let L : R → R be locally Lipschitz. Then E ( L ◦ u ) = 0 .Proof. First assume that L is of class C . Let φ ∈ C c (Ω) and g ∈ C c ( R , R ). The key ofthe proof consists in showing that¯ E L ◦ u ( φ, g ) = ¯ E u ( φ, (adj D L )( g ◦ L )) . (2.3)The integrand corresponding to ¯ E L ◦ u ( φ, g ) is, for a.e. x ∈ Ω, g (( L ◦ u )( x )) · (cof D ( L ◦ u )( x ) Dφ ( x )) + φ ( x ) div g (( L ◦ u )( x )) det D ( L ◦ u )( x )= (adj D L ( u ( x )) g ( L ( u ( x )))) · (cof D u ( x )) Dφ ( x ))+ φ ( x ) div g ( L ( u ( x ))) det D L ( u ( x )) det D u ( x ) . (2.4)Now define ¯ g := (adj D L )( g ◦ L ). Then, for all y ∈ R ,div ¯ g ( y ) = X i,j =1 ∂∂y i [adj D L ( y ) ij g j ( L ( y ))] . Piola’s identity gives X i =1 ∂∂y i [adj D L ( y ) ij ] = 0 , j ∈ { , , } , ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 13 so, thanks to the matrix identity F adj F = (det F ) I valid for F ∈ R × ,div ¯ g ( y ) = X i,j =1 adj D L ( y ) ij ∂∂y i [ g j ( L ( y ))] = X i,j,k =1 adj D L ( y ) ij D g ( L ( y )) jk D L ( y ) ki = tr ( D L ( y ) adj D L ( y ) D g ( L ( y ))) = tr (det D L ( y ) D g ( L ( y )))= det D L ( y ) div g ( L ( y )) . With this, we find that the integrand of ¯ E u ( φ, ¯ g ) coincides with (2.4), so (2.3) is proved.As E ( u ) = 0 we obtain that ¯ E L ◦ u ( φ, g ) = 0.Now assume, as in the statement, that L is only locally Lipschitz, and take a sequence { L n } in C ( R , R ) such that L n → L a.e., D L n → D L a.e. andsup n ∈ N k L n k W , ∞ ( B ( , k u k L ∞ (Ω , R )) < ∞ . The existence of such approximating sequence follows from a classic result (see, e.g., [19,Th. 6.6.1]). By the first part of the proof,0 = Z Ω (cid:2) (adj D L n ( u ( x )) g ( L n ( u ( x )))) · (cof D ( u ( x ) Dφ ( x ))+ φ ( x ) div g ( L n ( u ( x ))) det D L n ( u ( x )) det D u ( x ) (cid:3) d x . Taking limits, we obtain that ¯ E L ◦ u ( φ, g ) = 0. By Lemma 2.8, E ( L ◦ u ) = 0. (cid:3) We recall now the definition of the distributional Jacobian determinant.
Definition 2.11.
Let u ∈ H (Ω , R ) ∩ L ∞ (Ω , R ). The distributional Jacobian Det D u of u is the distribution defined by h Det D u , ϕ i := − h adj D u u , Dϕ i = − Z Ω adj D u u · Dϕ for all ϕ ∈ C c (Ω).2.3. The axisymmetric setting.
In most of the paper we will work with axisymmetric(with respect to the x -axis) maps and domains, which are defined as follows. We say thatthe set Ω ⊂ R is axisymmetric if [ x ∈ Ω ( ∂B R ((0 , , | ( x , x ) | ) × { x } ) ⊂ Ω . When we define π : R → [0 , ∞ ) × R P : [0 , ∞ ) × R × R → R x ( | ( x , x ) | , x ) ( r, θ, x ) ( r cos θ, r sin θ, x ) , the axisymmetry of Ω is equivalent to the equalityΩ = { P ( r, θ, x ) : ( r, x ) ∈ π (Ω) , θ ∈ [0 , π ) } . (2.5) Given an axisymmetric set Ω, we say that u : Ω → R is axisymmetric if there exists v : π (Ω) → [0 , ∞ ) × R such that( u ◦ P )( r, θ, x ) = P ( v ( r, x ) , θ, v ( r, x )) , i.e. , u ( r cos θ, r sin θ, x ) = v ( r, x )(cos θ e + sin θ e ) + v ( r, x ) e (2.6)for all ( r, x , θ ) ∈ π (Ω) × [0 , π ). We will say that v is the function corresponding to u .This v is uniquely determined by u .If Ω and u are axisymmetric then so is u (Ω). Indeed, from (2.5) and (2.6) we obtainthat u (Ω) = { P ( v ( r, x ) , θ, v ( r, x )) : ( r, x ) ∈ π (Ω) , θ ∈ [0 , π ) } and π ( u (Ω)) = { ( π ◦ P )( v ( r, x ) , θ, v ( r, x )) : ( r, x ) ∈ π (Ω) , θ ∈ [0 , π ) } = v ( π (Ω)) , from which the equality u (Ω) = { P ( r, θ, x ) : ( r, x ) ∈ π ( u (Ω)) , θ ∈ [0 , π ) } follows.Given δ >
0, we define C δ as the (open, infinite, solid) cylinder of radius δ : C δ := { P ( r, θ, x ) : ( r, θ, x ) ∈ [0 , δ ) × [0 , π ) × R } . (2.7)We have π ( C δ ) = [0 , δ ) × R .2.4. Prescribing the boundary data.
As mentioned in the Introduction, we fix asmooth bounded open set e Ω of R such that Ω ⋐ e Ω and consider an orientation-preserving C diffeomorphism b from the closure of e Ω to R . We assume that Ω, e Ω, and b areaxisymmetric. The function u , originally defined in Ω, is extended to e Ω by settingΩ D := e Ω \ Ω , u = b in Ω D . We assume that the extension to e Ω, still called u , is in H ( e Ω , R ), as stated in thedefinition (1.6) of our function space A s . Regarding the class A rs of regular maps definedin (1.7), note that they are required to satisfy the divergence identities (1.5) not only in Ωbut in the larger domain e Ω where the maps in A s are defined. Expressed in terms of thesurface energy of Section 2.2, considering particularly the remark before Lemma 2.8, theclass A rs consists of the maps in A s having zero surface energy in e Ω: A rs = { u ∈ A s : E ( u ) = 0 in e Ω } . (2.8)Note that the property E ( u ) ≤ E ( b ) in the definition (1.6) of A s is natural for theDirichlet minimization problem studied in this paper. Also, observe that the conditiondet D u > u ∈ H ( e Ω , R ) such that E ( u ) ≤ E ( b ), thanks to the blow-up behaviour (1.3) of the neo-Hookean energy as theJacobian vanishes. ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 15
A family of good open sets.
Given a nonempty open set U ⋐ e Ω with a C bound-ary, we denote by d : e Ω → R the signed distance function to ∂U given by d ( x ) := dist( x, ∂U ) if x ∈ U x ∈ ∂U − dist( x, ∂U ) if x ∈ Ω \ U (2.9)and U t := { x ∈ e Ω; d ( x ) > t } , (2.10)for each t ∈ R . It is a classical result (see e.g. [18]) that there exists δ > t ∈ ( − δ, δ ), the set U t is open, compactly contained in e Ω and has a C boundary. Definition 2.12.
Let u ∈ H ( e Ω , R ) be such that det D u > U u as thefamily of nonempty open sets U ⋐ e Ω with a C boundary that satisfya) u | ∂U ∈ H ( ∂U, R ), and (cof ∇ u ) | ∂U ∈ L ( U, R × ),b) ∂U ⊂ Ω , H -a.e., where Ω is the set in Definition 2.2, and ( ∇ u | ∂U )( x ) = ∇ u ( x ) | T x ∂U for H -a.e. x ∈ ∂U ,c) lim ε → R ε (cid:12)(cid:12)(cid:12)R ∂U t | cof ∇ u | d H − R ∂U | cof ∇ u | d H (cid:12)(cid:12)(cid:12) d t = 0 , d) For every g ∈ C ( R , R ) with (adj D u )( g ◦ u ) ∈ L ( e Ω , R ),lim ε → Z ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∂U t g ( u ( x )) · (cof ∇ u ( x ) ν t ( x ))d H − Z ∂U g ( u ( x )) · (cof ∇ u ( x ) ν ( x ))d H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d t = 0where ν t denotes the unit outward normal to U t for each t ∈ (0 , ε ), and ν the unitoutward normal to U .Since we are imposing that u coincides with the C -diffeomorphism b in the exteriorDirichlet neighbourhood Ω D of ∂ Ω, without loss of generality we may assume that Ω ∈ U u .The following result is obtained using the coarea formula and Lebesgue’s differentiationtheorem for the first part, and Uryshon functions and Sard’s lemma for the second. Itguarantees that there are enough sets in U u . Lemma 2.13 (Lemma 2.11 in [33]) . Let u ∈ H ( e Ω , R ) be such that det D u > a.e. Let U ⋐ e Ω be a nonempty open set with a C boundary. Then U t ∈ U u for a.e. t ∈ ( − δ, δ ) .Moreover, for each compact K ⊂ e Ω there exists U ∈ U u such that K ⊂ U . Regularity, injectivity and weak convergence of the planar function.
Let C δ be as in Section 2.3. The link between the regularity of u and its associated 2 D map v isas follows. Lemma 2.14.
Let Ω be an axisymmetric domain, and u : Ω → R an axisymmetricmap with corresponding function v . Let δ > . Then, u ∈ H (Ω \ C δ , R ) if and only if v ∈ H ( π (Ω) \ ([0 , δ ] × R ) , R ) . Moreover, in this case, k u k H (Ω \ C δ , R ) ≤ π max {k x k L ∞ (Ω , R ) , δ − } k v k H ( π (Ω) \ ([0 ,δ ] × R ) , R ) , k v k H ( π (Ω) \ ([0 ,δ ] × R ) , R ) ≤ (2 πδ ) − k u k H (Ω \ ¯ C δ , R ) and for a.e. ( r, θ, x ) with ( r, x ) ∈ π (Ω) \ ([0 , δ ] × R ) and θ ∈ R , det D u ( P ( r, θ, x )) = v ( r, θ ) r det D v ( r, θ ) . (2.11) Proof.
First note that π (Ω \ C δ ) = π (Ω) \ ([0 , δ ] × R ) and, thanks to (2.6), for all ( r, x , θ ) ∈ π (Ω) × [0 , π ), | ( u ◦ P )( r, θ, x ) | = | v ( r, x ) | . (2.12)Suppose initially that u ∈ H (Ω \ C δ , R ). Let J be any open interval of length less than2 π . As P is a diffeomorphism in (0 , ∞ ) × J × R , we have that the map u ◦ P is locallySobolev in { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ([0 , δ ] × R ) , θ ∈ J } . (2.13)By (2.6) and applying P − , the map( r, θ, x ) ( v ( r, x ) , θ, v ( r, x )) (2.14)is locally Sobolev in the set (2.13), hence v is locally Sobolev in π (Ω) \ ([0 , δ ] × R ).Now assume v ∈ H ( π (Ω) \ ([0 , δ ] × R ) , R ). Then, for any bounded open interval J oflength less than 2 π , the map (2.14) is in H in the set (2.13). As P is locally Lipschitz,by (2.6), the map u ◦ P is locally Sobolev in the set (2.13). As P is a diffeomorphism in(0 , ∞ ) × J × R , we have that the map u is locally Sobolev in P ( { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ([0 , δ ] × R ) , θ ∈ J } ) . Taking now two open intervals J , J of length less than 2 π such that [0 , π ) ⊂ J ∪ J , weobtain that u is locally Sobolev in the set P ( { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ([0 , δ ] × R ) , θ ∈ J ∪ J } ) = Ω \ C δ . As for the inequalities of the norms, from (2.12) and the formula of integration in cylin-drical coordinates we obtain k u k L (Ω \ C δ , R ) ≤ π k x k L ∞ (Ω , R ) k v k L ( π (Ω) \ ([0 ,δ ] × R ) , R ) and k v k L ( π (Ω) \ ([0 ,δ ] × R ) , R ) ≤ (2 πδ ) − k u k L (Ω \ C δ , R ) . On the other hand, when we apply the chain rule to (2.6), it is easy to see (see, e.g., [36,App.] if necessary) that for a.e. ( r, x , θ ) ∈ π (Ω) × R , | D u ( P ( r, θ, x )) | = | D v ( r, x ) | + (cid:18) v ( r, x ) r (cid:19) , ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 17 whence k D v k L ( π (Ω) \ ([0 ,δ ] × R ) , R ) ≤ (2 πδ ) − k D u k L (Ω \ C δ , R ) . and k D u k L (Ω \ C δ , R ) ≤ π max {k x k L ∞ (Ω) , δ − } k v k H ( π (Ω) \ ([0 ,δ ] × R ) , R ) . Finally, formula (2.11) is immediate to check from the chain rule applied to (2.6) (see,(6.18) and if necessary, [36, App.]). (cid:3)
The orientation-preserving and injectivity conditions of u and v are related as follows. Lemma 2.15.
Let Ω be an axisymmetric domain. Let u ∈ H (Ω , R ) be axisymmetric,and let v be its corresponding function. The following hold:a) det D u > a.e. in Ω if and only if det D v > a.e. in π (Ω) .b) u is injective a.e. in Ω if and only if v is injective a.e in π (Ω) .Proof. For a) , assume first det D u > P is a local diffeomorphism in (0 , ∞ ) × R × R and the set { } × R × R has measure zero, we obtain that det D u ( P ( r, θ, x )) > r, θ, x ) in the set { ( r, θ, x ) : ( r, x ) ∈ π (Ω) , θ ∈ [0 , π ) } . (2.15)From Lemma 2.14 we obtain that formula (2.11) also holds for a.e. ( r, θ, x ). As v ≥ D v ( r, x ) > r, x ) ∈ π (Ω).Conversely, assume det D v > v > E be the set of points for which v = 0. Then Dv = 0 a.e. in E , so det D v = 0 a.e. in E . Therefore, E has measure zero. We conclude that det D u > b) , assume first u is injective a.e. The map P is injective in (0 , ∞ ) × [0 , π ) × R , itsinverse satisfies Lusin’s condition, and { } × R × R has measure zero. Therefore, u ◦ P isinjective a.e. in the set (2.15). By (2.6) and applying P − : P ((0 , ∞ ) × [0 , π ) × R ) → R ,the map (2.14) is injective a.e. in the set { ( r, θ, x ) : ( r, x ) ∈ π (Ω) , v ( r, x ) > , θ ∈ [0 , π ) } . Consequently, v is injective a.e. in the set { ( r, x ) : ( r, x ) ∈ π (Ω) , v ( r, x ) > } . Now let E be the set of ( r, θ, x ) such that ( r, x ) ∈ π (Ω), v ( r, x ) = 0 and θ ∈ [0 , π ).Then, by (2.6), in E \ ( { } × R × R ),( u ◦ P )( r, θ, x ) = (0 , , v ( r, x )) . Therefore, u is not injective in P ( E \ ( { } × R × R )), so P ( E \ ( { } × R × R )) has measurezero, and, consequently, E has measure zero. This shows that v is injective a.e.Conversely, assume that v is injective a.e. Then, the map (2.14) is injective a.e. in theset (2.15). By (2.6), the map u ◦ P is injective a.e. in the same set. As P is injective in(0 , ∞ ) × [0 , π ) × R and satisfies Lusin’s condition there, we obtain that u is injective a.e.in P ( { ( r, θ, x ) : ( r, x ) ∈ π (Ω) , r > , θ ∈ [0 , π ) } ) = Ω \ ( { (0 , } × R ) , and, hence, injective a.e. in Ω. (cid:3) The relationship between the weak convergences of a sequence of axisymmetric functionsand their associated functions is as follows. Weak convergence is denoted by ⇀ . Lemma 2.16.
Let Ω be an axisymmetric domain. For each j ∈ N , let u j ∈ H (Ω , R ) beaxisymmetric. Let u ∈ H (Ω , R ) , and assume that u j ⇀ u in H (Ω , R ) as j → ∞ . Thefollowing statements hold:i) u is axisymmetric.ii) Let v j and v be the corresponding functions of u j and u , respectively. Then v j ⇀ v in H ( π (Ω) \ ([0 , δ ] × R ) , R ) for each δ > .Proof. Up to a subsequence, u j → u a.e. in Ω as j → ∞ . As P : (0 , ∞ ) × R × R → ( R \ { (0 , } ) × R is a local diffeomorphism and { } × R × R has measure zero, we havethat u j ◦ P → u ◦ P a.e. in the set { ( r, θ, x ) : ( r, x ) ∈ π (Ω) , θ ∈ R } . as j → ∞ . On the other hand, by Lemma 2.14, for each δ >
0, the sequence { v j } j ∈ N isbounded in H ( π (Ω) \ ([0 , δ ] × R ) , R ). Therefore, there exists a function v δ ∈ H ( π (Ω) \ ([0 , δ ] × R ) , R ) such that v j ⇀ v δ in H ( π (Ω) \ ([0 , δ ] × R ) , R ) as j → ∞ . As the familyof sets π (Ω) \ ([0 , δ ] × R ) decreases with δ , we obtain a function v : π (Ω) \ ( { } × R ) → R such that v j ⇀ v in H ( π (Ω) \ ([0 , δ ] × R ) , R ) as j → ∞ , for each δ >
0. Therefore, up to asubsequence, v j → v a.e. in π (Ω) \ ( { } × R ). By defining v arbitrarily in π (Ω) ∩ ( { } × R ),we obtain that v j → v a.e. in π (Ω) and, consequently,( v j ( r, x ) , θ, v j ( r, x )) → ( v ( r, x ) , θ, v ( r, x ))and P ( v j ( r, x ) , θ, v j ( r, x )) → P ( v ( r, x ) , θ, v ( r, x ))for a.e. ( r, x , θ ) ∈ π (Ω) × R . As ( u j ◦ P )( r, θ, x ) = P ( v j ( r, x ) , θ, v j ( r, x )) for all( r, x , θ ) ∈ π (Ω) × R and j ∈ N , we obtain that (2.6) holds for a.e. ( r, x , θ ) ∈ π (Ω) × R . Byredefining v in a set of measure zero and redefining u through (2.6), we can assume that(2.6) holds for all ( r, x , θ ) ∈ π (Ω) × R . Therefore, u is axisymmetric with correspondingfunction v . Thus, i) is proved and ii) is also proved except for a subsequence. To showthat ii) holds for the whole sequence, by a standard argument it suffices to show that themap v constructed above is uniquely determined, and this is the case since the equality(2.6) defines v univocally. (cid:3) Arguing as in Lemma 2.16, one can show the following converse.
Lemma 2.17.
Let Ω be an axisymmetric domain. For each j ∈ N , let u j : Ω → R bean axisymmetric map with corresponding function v j . Let v : π (Ω) \ ( { } × R ) → R andassume that v ∈ H ( π (Ω) \ ([0 , δ ] × R ) , R ) and that v j ⇀ v in H ( π (Ω) \ ([0 , δ ] × R ) , R ) for every δ > . Define u : Ω \ R e → R by (2.6) . Then u j ⇀ u in H (Ω \ C δ ) for every δ > . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 19 Fine properties of maps in A s In this paper we consider the admissible classes A rs and A s defined in (2.8) and (1.6). Webegin by recalling that orientation-preserving H maps in 2 D are very regular, specificallythat they are continuous and satisfy both Lusin’s condition and the divergence identities.These properties are inherited by the 3 D axisymmetric maps u in A s away from thesymmetry axis.3.1. Regularity of maps in A s outside the axis of symmetry. We start with thefollowing regularity result.
Lemma 3.1.
Let Ω be an axisymmetric domain. Let u ∈ H (Ω , R ) be axisymmetric andsatisfy det D u > a.e., and let v be its corresponding function. Then:a) v has a representative that is continuous at each point of π (Ω) \ ( { } × R ) , differentiablea.e., and satisfies condition ( N ) in π (Ω) \ ( { }× R ) . Moreover, E ( v , π (Ω) \ ( { }× R )) = 0 .b) u has a representative that is continuous at each point of Ω \ R e , differentiable a.e.,and satisfies condition ( N ) in Ω \ R e . Moreover, E ( u , Ω \ R e ) = 0 .c) For each j ∈ N , let u j ∈ H (Ω , R ) be axisymmetric. Let u ∈ H (Ω , R ) , and assumethat u j ⇀ u in H (Ω , R ) as j → ∞ . If, in addition, det D u j > a.e. for all j ∈ N then v j → v uniformly in compact subsets of π (Ω) \ ( { } × R ) and u j → u uniformlyin compact subsets of Ω \ ( { (0 , } × R ) .Proof. Let δ >
0. By Lemma 2.14, v is in H ( π (Ω) \ ([0 , δ ] × R ) , R ) and, by Lemma 2.15,det D v > v has a representative v in π (Ω) \ ([0 , δ ] × R ) that is continuous, differentiablea.e. and satisfies the (N) property. That E ( v , π (Ω) \ ([0 , δ ] × R )) = 0 is also a classicalresult (see, e.g., [47, 54, 49, 51]) and can be proved by approximation as in Lemma 2.9. Asthis is true for every δ >
0, property a) is proved.We define the representative ¯ u of u through formula (2.6), but changing u , v by ¯ u , ¯ v ,respectively. As in Lemma 2.15, we readily obtain that ¯ u is continuous in Ω \ R e anddifferentiable a.e. We now show the (N) property for ¯ u . Let A be a null set in Ω \ R e ,and for each θ ∈ R define A θ := { ( r, x ) : P ( r, θ, x ) ∈ A } . Since L ( P − ( A )) = 0, we havethat L ( A θ ) = 0 for a.e. θ ∈ R . For any such θ we have that ¯ v ( A θ ) is L -null. By (2.6), P ( s, θ, y ) ∈ ¯ u ( A ) if and only if ( s, y ) ∈ ¯ v ( A θ ). Consequently, L ( ¯ u ( A )) = Z R Z π Z ∞ χ ¯ u ( A ) ( P ( s, θ, y )) s d s d θ d y = Z R Z π Z ∞ χ ¯ v ( A θ ) ( s, y ) s d s d θ d y . Therefore, for any
R > L ( ¯ u ( A ) ∩ B ( , R )) ≤ R Z R Z π Z ∞ χ ¯ v ( A θ ) ( s, y ) d s d θ d y = R Z π L (¯ v ( A θ )) d θ = 0 . As this is true for every
R >
0, we obtain that L ( ¯ u ( A )) = 0. Thus, ¯ u satisfies condition(N) in Ω \ R e . Let I ⊂ R be an open interval of length less than 2 π and define the function w in theset { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ( { } × R ) , θ ∈ I } as w ( r, θ, x ) := ( v ( r, x ) , θ, v ( r, x )).As shown above, E ( v , π (Ω) \ ( { } × R )) = 0, so by Lemma 2.9, E ( w ) = 0. By Lemma2.10, E ( P ◦ w ) = 0, so by (2.6), u ◦ P has zero surface energy in { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ( { } × R ) , θ ∈ I } . Let δ >
0. As P | ( δ, ∞ ) × I × R is a diffeomorphism that admits anextension to an open set containing the closure of ( δ, ∞ ) × I × R , by [31, Sect. 8] or [36,Sect. 6], u has zero surface energy in Ω ∩ P (( δ, ∞ ) × I × R ). Considering now two openintervals I and I of length less than 2 π , it is easy to check (see, if necessary, the proof of[46, Lemma 4.8]) that u has zero surface energy in Ω ∩ P (( δ, ∞ ) × ( I ∪ I ) × R ). Taking,additionally I and I such that [0 , π ] ⊂ I ∪ I , we obtain that u has zero surface energyin Ω \ C δ . Consequently, E ( u , Ω \ R e ) = 0.Now we show c) and assume that det D u j > j ∈ N . By Lemma 2.15,we have that det D v j > j ∈ N . Since the H norm of { v j } j ∈ N is boundedin π (Ω) \ ([0 , δ ] × R ) for each δ >
0, by a classic result on maps of finite distortion (see,e.g., [22, Lemma 2.1] for a precise reference), the family { v j } j ∈ N is equicontinuous in eachcompact set of π (Ω) \ ([0 , δ ] × R ), hence in each compact set of π (Ω) \ ( { } × R ). By theAscoli–Arzel`a theorem, and part ii) of Lemma 2.16, v j → v uniformly in each compactset of π (Ω) \ ( { } × R ), in principle up to a subsequence, but in fact the convergence holdsfor the whole sequence because v is uniquely determined. Having in mind (2.6), we obtainthat u j ◦ P → u ◦ P uniformly in each compact subset of { ( r, θ, x ) : ( r, x ) ∈ π (Ω) \ ( { } × R ) , θ ∈ R } . Then, as before, u j → u uniformly in each compact subset of Ω \ { (0 , } × R . (cid:3) The assumptions of Lemma 3.1 will hold in most of the paper, and, in this case, withoutfurther mention, u and v are taken to be the continuous representative of themselves inthe sets Ω \ R e and π (Ω) \ ( { } × R ), respectively.3.2. Topological degree.
We first recall how to define the classical Brouwer degree forcontinuous functions [17, 21]. Let N ≥
2. Let U ⊂ R N be a bounded open set. If u ∈ C ( U , R N ) then for every regular value y of u we setdeg( u , U, y ) = X x ∈ u − ( y ) ∩ U det D u ( x ) . (3.1)Note that the sum is finite since the pre-image of a regular value consists in isolated pointthanks to the inverse function theorem. We can show that the right-hand side of (3.1) isinvariant by homotopies. This allows to extend the definition (3.1) to every y / ∈ u ( ∂U ).This homotopy invariance can also be used to show that the definition depends only onthe boundary values of u . If u is only in C ( ∂U, R N ), it is again the homotopy invariancewhich allows to define the degree of u , since in this case we may extend u to a continuousmap in U by Tietze’s theorem and setdeg( u , U, · ) = deg( v , U, · ) , where v is any map in C ( U , R N ) which is homotopic to the extension of u . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 21 If U is of class C and u ∈ C ( ∂U, R N ), by using (3.1), Sard’s theorem and the divergenceidentities Div (cid:0) (adj D u ) g ◦ u (cid:1) = (div g ) ◦ u det D u ∀ g ∈ C c ( R , R )we can make a change of variables and integrate by parts to obtain Z R N deg( u , U, y ) div g ( y )d y = Z ∂U ( g ◦ u ) · (cof D u ν ) d H N − . (3.2)This formula can be used as the definition of the degree for maps in W ,N − ∩ L ∞ ( ∂U, R N )as noticed by Brezis & Nirenberg [10]. For any open set U having a positive distance awayfrom the symmetry axis R it is possible to use the classical degree since there every mapin A s has a continuous representative (Lemma 3.1). However, for open sets U crossing theaxis (where maps in A s may have singularities) we use the Brezis–Nirenberg degree. Definition 3.2.
Let U ⊂ R N be a bounded open set. For any u ∈ C ( ∂U, R N ) and any y ∈ R N \ u ( ∂U ) we denote by deg( u , U, y ) the classical topological degree of u with respectto y . Suppose now that U ⊂ R N is a C bounded open set and u ∈ W ,N − ( ∂U, R N ) ∩ L ∞ ( ∂U, R N ). Then the degree of u , denoted by deg( u , U, · ), is defined as the only L function which satisfies Z R N deg( u , U, y ) div g ( y )d y = Z ∂U ( g ◦ u ) · (cof D u ν ) d H N − , for all g ∈ C ∞ ( R N , R N ).To see that this definition makes sense we refer to [10] or [13, Remark 3.3]. Also,using (3.2) for a sequence of smooth maps approximating u we can see that for any u ∈ C ( ∂U, R N ) ∩ W ,N − ( ∂U, R N − ) such that L N (cid:0) u ( ∂U ) (cid:1) = 0 the two definitions are consistent(as stated in [50, Prop. 2.1.2]). Thanks to Lemma 3.1, our maps u satisfiy the (N) property,so the condition on u ( ∂U ) is satisfied for all regular open sets U that are a distance apartfrom the axis. On the one hand, by the continuity property of the degree (see e.g. [21]) wecan see that the topological image of a bounded open set is open. This will give us adequateambient spaces to work with in the deformed configuration, see Equation (3.10). On theother hand, it is by working with the Brezis–Nirenberg degree that the Sobolev regularity ofthe inverses (crucial for the lower semicontinuity result presented in this paper) is obtainedin the presence of singularities (Proposition 3.14).We will invoke many previous results about the degree and related concepts (such asthe topological image or the condition INV; see below). Most of the references that weuse in this work use the degree with slightly different assumptions on u . In particular,M¨uller and Spector [50] use the degree for W ,p maps with p >
2. In their context, if u ∈ W ,p (Ω , R ) then for “almost all” C open sets U ⋐ Ω one has that u ∈ W ,p ( ∂U, R ),hence (the precise representative of) u | ∂U is continuous and the degree is defined. Theworks [33, 6] are based on [50] and also follow this approach. On the other hand, Contiand De Lellis [13] use the degree for H ∩ L ∞ maps, for the which the above continuityproperty does not hold. The works [32, 33] are based on [13] and also follow this approach.In full truth, when we invoke results of the papers above involving the degree (or its related concepts), we should prove them again, because the assumptions or even the definitionsare not exactly the same. Nevertheless, since their proofs apply to our case with only smallmodifications, we will not provide a proof.3.3. Topological image for the classical degree.
An important part of our analysisrefers only to open sets U that either are a distance apart from the symmetry axis orenclose entirely the closed segment L := Ω ∩ R e (3.3)where the singularities can occur. To be precise, we use the setting of Section 2.4 andfrequently deal with open sets U ⊂ e Ω such that ∂U ∩ L = ∅ . Since u = b in the Dirichletregion Ω D = e Ω \ Ω, the map u is continuous in ∂U and, hence, the classical degreedeg( u , U, · ) is well defined. For those sets U the topological image is defined as follows. Definition 3.3.
Let N ≥
2. Let U be a bounded open set of R N and let u ∈ C ( ∂U, R N ).We define im T ( u , U ), the topological image of U under u , as the set of y ∈ R N \ u ( ∂U )such that deg( u , U, y ) = 0.The following elementary property will be useful. Lemma 3.4.
Let U be a bounded open set of R N . If u ∈ C ( ∂U, R N ) then ∂ im T ( u , U ) ⊂ u ( ∂U ) .Proof. Let y ∈ ∂ im T ( u , U ). Then there exist sequences { y j } j ∈ N ⊂ im T ( u , U ) and { z j } j ∈ N ⊂ R N \ im T ( u , U ) both converging to y . Assume, for a contradiction, that y / ∈ u ( ∂U ). Asdeg( u , U, y j ) = 0 for all j ∈ N , by the continuity of the degree, deg( u , U, y ) = 0. Since u ( ∂U ) is compact and since y / ∈ u ( ∂U ) we have dist( y, u ( ∂U )) >
0. Thus, since z j → y necessarily all but a finite number of z j are in R N \ u ( ∂U ). Then, for those j , by defini-tion, we have deg( u , U, z j ) = 0 and, by the continuity of the degree deg( u , U, y ) = 0, acontradiction. (cid:3) In the axisymmetric setting, the following relation between the topological images of u and v holds. Lemma 3.5.
Let Ω ⊂ R be an axisymmetric domain. Let u ∈ H (Ω , R ) be axisymmetricand satisfy det D u > a.e., and let v be its corresponding function. Let U ⊂ Ω be anaxisymmetric open set such that U ∩ R e = ∅ . Then im T ( v , π ( U )) = π (im T ( u , U )) .Proof. Let z ∈ R \ v ( ∂π ( U )) and θ ∈ R . Let I be an open interval of length less than 2 π containing θ . Set U I := { ( r, θ ′ , x ) : ( r, x ) ∈ π ( U ) , θ ′ ∈ I } . By the product property forthe degree (see, e.g., [11, Th. 8.7]),deg (( v , id R , v ) , U I , ( z , θ, z )) = deg ( v , π ( U ) , z ) deg( id R , I, θ ) = deg ( v , π ( U ) , z ) , where id R is the identity map in R . As P : U I → R can be extended to an orientation-preserving diffeomorphism in an open set containing U I we obtain by the composition ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 23 formula for the degree (see, e.g., [17, Th. 5.1]) thatdeg (( v , id R , v ) , U I , ( z , θ, z )) = deg ( P ◦ ( v , id R , v ) , U I , P ( z , θ, z ))= deg ( u ◦ P , U I , P ( z , θ, z )) , where in the last formula we have used (2.6). Applying again the composition formula weobtain deg ( u ◦ P , U I , P ( z , θ, z )) = deg ( u , P ( U I ) , P ( z , θ, z )) . Altogether, we have shown thatdeg ( v , π ( U ) , z ) = deg ( u , P ( U I ) , P ( z , θ, z )) . Now we show that P ( z , θ, z ) / ∈ u (cid:0) U \ P ( U I ) (cid:1) . (3.4)Indeed, let x ∈ U \ P ( U I ). Then x = P ( r cos θ ′ , r sin θ ′ , x ) with ( r, x ) ∈ π ( U ) and θ ′ ∈ R \ ( I + 2 π Z ). By (2.6), u ( x ) = P ( v ( r, x ) , θ ′ , v ( r, x )), which implies (3.4). In turn,(3.4) and the excision property of the degree (see, e.g., [17, Th. 3.1] or [11, Th. 8.4]) yielddeg ( u , P ( U I ) , P ( z , θ, z )) = deg ( u , U, P ( z , θ, z )) , which, together with (3.4) shows thatdeg ( v , π ( U ) , z ) = deg ( u , U, P ( z , θ, z )) . (3.5)Recapitulating, we have shown that if z ∈ R \ v ( ∂π ( U )) and θ ∈ R then P ( z , θ, z ) / ∈ u ( ∂U ) and formula (3.5) holds.Now let z ∈ im T ( v , π ( U )). Then z / ∈ v ( ∂π ( U )) and deg ( v , π ( U ) , z ) = 0. By (3.5),deg ( u , U, P ( z , θ, z )) = 0 for any θ ∈ R , so P ( z , θ, z ) ∈ im T ( u , U ). Consequently, z = ( π ◦ P )( z , θ, z ) ∈ π (im T ( u , U )).To prove the converse inclusion, we start with the following simple facts:i) If ( r, x ) ∈ π ( U ) and θ ∈ R then P ( r, θ, x ) ∈ U .ii) If ( r, x ) / ∈ π ( U ) and θ ∈ R then P ( r, θ, x ) / ∈ U .iii) If ( r, x ) ∈ ∂π ( U ) and θ ∈ R then P ( r, θ, x ) ∈ ∂U .Property ii) is obvious, while i) is a consequence of the axisymmetry of U . Let us showiii). Let ( r, x ) ∈ ∂π ( U ) and θ ∈ R . Note that π ( U ) is an open set in R , so ∂π ( U ) = π ( U ) \ π ( U ) . As ( r, x ) ∈ π ( U ), an elementary argument based on i) and the continuity of P showsthat P ( r, θ, x ) ∈ U . On the other hand, property ii) shows that P ( r, θ, x ) / ∈ U , so P ( r, θ, x ) ∈ ∂U and iii) is proved.We are in a position to show the inclusion π (im T ( u , U )) ⊂ im T ( v , π ( U )). Let z ∈ π (im T ( u , U )). Then there exists θ ∈ R such that P ( z , θ, z ) ∈ im T ( u , U ). Therefore, P ( z , θ, z ) / ∈ u ( ∂U ) and deg ( u , U, P ( z , θ, z )) = 0 . (3.6) We shall show that z / ∈ v ( ∂π ( U )) by assuming that z = v ( r, x ) for some ( r, x ) ∈ π ( U ).Due to (2.6), P ( z , θ, z ) = P ( v ( r, x ) , θ, v ( r, x )) = u ◦ P ( r, θ, x ) , so, by (3.6), P ( r, θ, x ) / ∈ ∂U , and, by iii), z / ∈ v ( ∂π ( U )). Thanks to (3.5) and (3.6),deg ( v , π ( U ) , z ) = 0, so z ∈ im T ( v , π ( U )). (cid:3) Topological image of the singular segment.
Throughout this section assumethat b , Ω, e Ω, and Ω D are as in Section 2.4. Note that all u ∈ A s satisfy the propertiesstated in Lemmas 3.7–3.8.Away from the segment L = Ω ∩ R e condition INV is defined as follows. Definition 3.6.
Let U be a bounded open set in R . If u ∈ C ( U, R ), we say that u satisfiesproperty (INV) in U provided that for every point x ∈ U and a.e. r ∈ (0 , dist( x , ∂U )):(a) u ( x ) ∈ im T ( u , B ( x , r )) for a.e. x ∈ B ( x , r )(b) u ( x ) / ∈ im T ( u , B ( x , r )) for a.e. x ∈ e Ω \ B ( x , r ).The degree of any map u in A s with respect to any open set U separated from thesymmetry axis coincides a.e. with the number of preimages by u (with approximate dif-ferentiability) in that open set. This is proved in Lemma 3.7 below and relies on the fineregularity properties satisfied away from the axis and on the preservation of orientation. Asa consequence, Lemma 3.8 obtains the nonnegativity of the degrees and the monotonicityof the topological images of open sets not containing the singular segment L . Lemma 3.7.
Suppose that u ∈ H ( e Ω , R ) is axisymmetric and satisfies det D u > a.e.and u = b in Ω D . Then:(a) u is continuous in e Ω \ L and E ( u ) = 0 in e Ω \ L .(b) For any U ∈ U u (see Definition 2.12) such that U ∩ L = ∅ deg( u , U, · ) = N ( u , Ω d ∩ U, · ) a.e. and im G ( u , U ) = im T ( u , U ) a.e. , (3.7) where Ω d is the set of approximate differentiability.(c) u satisfies condition (INV) in e Ω \ L if and only if u is injective a.e. In particular, allmaps in A s satisfy (INV) in e Ω \ L .Proof. Part (a) follows from Lemma 3.1 and [46, Lemma 4.8]. Parts (b) and (c) can beobtained with the same proof of [6, Th. 4.1 and Lemma 5.1.(a)]. (cid:3)
We will need the following monotonicity property of the topological image (cf. [50,Lemma 7.3], [13, Lemma 3.12]). Recall that if U ⊂ e Ω is an open set with a C boundarywe denote by U t = { x ∈ U ; d ( x ) ≥ t } for t ∈ R where d is the signed distance functiondefined by (2.9). Lemma 3.8.
Let u ∈ H ( e Ω , R ) be an injective axisymmetric map which satisfies det D u > a.e. and u = b in Ω D . Let U and V be open sets in e Ω . Suppose that U δ ⋐ V − δ ⋐ e Ω \ L for some δ > . Then, for a.e. s, t ∈ ( − δ, δ ) we have im T ( u , U s ) ⊂ im T ( u , V t ) . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 25
Proof.
From Lemma 3.7, with the same proof of [6, Prop. 4.3], we obtain that im T ( u , U s ) ⊂ im T ( u , V t ). Use is made of the excision property of the classical degree, combined with theobservation that deg( u , V t \ U − s , · ) ≥ y ∈ im T ( u , U s ) but y is not in im T ( u , V t ).Then, by Lemma 3.4, y necessarily lies on u ( ∂V t ), say y = u ( x ) with x ∈ ∂V t . Sinceim T ( u , U s ) is open and u | ∂V t is continuous, there exists η > u ( x ) ∈ im T ( u , U s ) for all x ∈ ∂V t ∩ B ( x , η ) . On the other hand, without loss of generality, U s and V t may be assumed to belong to U u .By [50, Lemma 2.5], for every x ∈ ∂V t ∩ Ω D (cid:0) im G ( u , e Ω \ V t ) , u ( x ) (cid:1) ≥ . Since im T ( u , U s ) is open, it follows that for every x ∈ ∂V t ∩ Ω ∩ B ( x , η ) D (cid:0) im G ( u , e Ω \ V t ) ∩ im T ( u , U s ) , u ( x ) (cid:1) ≥ . By (3.7) this would imply that L (cid:0) im G ( u , e Ω \ V t ) ∩ im G ( u , U s ) (cid:1) >
0, contradicting that u is injective a.e., unless the set ∂V t ∩ Ω ∩ B ( x , η ) were empty. However, that is impossibleby property b) of Definition 2.12. (cid:3) Recall that Ω b := b (Ω) and e Ω b := b ( e Ω) . Definition 3.9.
Let u ∈ A s .a) We define the topological image of e Ω \ L by u asim T ( u , e Ω \ L ) := [ U ∈U u U ⋐ e Ω \ L im T ( u , U ) . b) We define the topological image of L by u asim T ( u , L ) := e Ω b \ im T ( u , e Ω \ L ) . Lemma 3.10. If u ∈ A s , then:(a) u ( x ) ∈ Ω b for every x ∈ Ω \ L , and(b) im T ( u , e Ω \ L ) ⊂ e Ω b .Proof. Suppose that u ( x ′ , z ) lies outside Ω b for some ( x ′ , z ) ∈ Ω \ L . Let η > (cid:0) u ( x ′ , z ) , Ω b (cid:1) > η and (cid:8) y ∈ R \ Ω b : dist (cid:0) y , Ω b (cid:1) < η (cid:9) ⊂ b (Ω D )(recall the notation Ω D = e Ω \ Ω). It is possible to find such η because b (Ω D ) lies outside Ω b and b is a diffeomorphism. Since u is continuous, we can find a small open neighbourhood E ⊂ R containing x ′ such thatdist (cid:0) u ( x ′ , z ) , Ω b (cid:1) > η x ′ ∈ E. Maps in A s are injective a.e. so there exists a Lebesgue-null set N such that u | e Ω \ ( L ∪ N ) isinjective. Without loss of generality, we may assume that N is disjoint with Ω D , because b is injective. By Fubini’s theorem, L (cid:0) { z ∈ R : ( x ′ , z ) ∈ N } (cid:1) = 0 for a.e. x ′ ∈ E . Fix anysuch x ′ .Let z be the first z > z such that dist (cid:0) u ( x ′ , z ) , Ω b (cid:1) = η , which exists because theexpression on the left is continuous with respect to z . Note that dist (cid:0) u ( x ′ , z ) , Ω b ) = 0when ( x ′ , z ) ∈ ∂ Ω, hence z is smaller than that value (or those values of z ) and the wholevertical segment { x ′ } × ( z , z ) lies inside Ω. By the choice made of η , u ( x ′ , z ) belongsto b (Ω D ) = u (Ω D ). But u restricted to the complement of N is injective, and we areassuming that Ω D is part of that complement. Therefore, ( x ′ , z ) necessarily belongs to N ′ := { z ∈ R : ( x ′ , z ) ∈ N } . In fact, the set S := { z ∈ [ z , z ] : dist (cid:0) u ( x ′ , z ) , Ω b (cid:1) < η } , which contains z = z , is entirely contained in N ′ (recall that dist (cid:0) u ( x ′ , z ) , Ω b (cid:1) ≥ η for any z ∈ S thanks to the definition of z and z ). On the other hand, using that z dist (cid:0) u ( x ′ , z ) , Ω b (cid:1) is continuous, it is possible to find δ > z ∈ ( z − δ, z ] ⇒ dist (cid:0) u ( x ′ , z ) , Ω b (cid:1) < η . But this implies that S , and, hence, N ′ , contain the interval ( z − δ, z ]. This contradictsthat N ′ has measure zero, finishing the proof of (a).For (b) we are to prove that im T ( u , U ) ⊂ e Ω b for every U ∈ U u with U ⋐ e Ω \ L . Fix anysuch U . Using (3.7) we obtain thatim T ( u , U ) = im G ( u , U ∩ Ω \ L ) ∪ im G ( u , U \ Ω) a.e.The first image is contained in Ω b by Part (a); that set, in turn, is contained in e Ω b . Thesecond is contained in b ( U \ Ω) because u = b in Ω D . Therefore,im T ( u , U ) ⊂ Ω b ∪ b ( U \ Ω) a.e. (3.8)Taking into account that im T ( u , U ) is open, that relation yieldsim T ( u , U ) ⊂ Ω b ∪ b ( U \ Ω) ⊂ b ( U ∪ Ω) ⊂ e Ω b . Indeed, take y ∈ im T ( u , U ). Let B be a ball centred at y and contained in im T ( u , U ). By(3.8) it is possible to find a sequence ( y j ) of points in B ∩ (cid:0) Ω b ∪ b ( U \ Ω) (cid:1) that convergesto y . Hence y belongs to the closure of Ω b ∪ b ( U \ Ω), finishing the proof. (cid:3)
Note that im T ( u , e Ω \ L ) is open as a union of open sets and hence im T ( u , L ) is closed.Also, by Lemma 3.10.(b), im T ( u , e Ω \ L ) = e Ω b \ im T ( u , L ) . (3.9) ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 27
So as to have an idea of the role played by im T ( u , L ), note that in the construction of Conti& De Lellis [13] this set consists (apart from the corresponding segment in the symmetryaxis) of the sphere ∂B (cid:0) (0 , , ) , (cid:1) , which may be regarded as new surface inside the elasticbody created by the singular map u .The following exhaustion result can be obtained, e.g., by applying Lemma 2.13 to theset K of points in e Ω at a distance at least δ > L ∪ ∂ e Ω. That im T ( u , e Ω \ L ) can bewritten as a union of a countable family of local topological images is then a corollary ofLemma 3.8. Lemma 3.11.
Let u ∈ H ( e Ω , R ) be such that det D u > a.e. Then there exists asequence { U k } k ∈ N in U u such that U k ⋐ U k +1 ⋐ e Ω \ L for all k ∈ N , and e Ω \ L = ∞ [ k =1 U k , im T ( u , e Ω \ L ) = ∞ [ k =1 im T ( u , U k ) . Sobolev regularity of inverses of functions in A s . As b : ˜Ω → R is a diffeo-morphism we have that Ω b = im T ( b , Ω). Moreover, for u ∈ A s , as u = b in e Ω \ Ω = Ω D ,we have that the trace of u coincides with the trace of b on ∂ Ω. As b | ∂ Ω is continu-ous, it is a representative of u | ∂ Ω , hence the degree deg( u , Ω , · ) and the topological imageim T ( u , Ω) are defined. Moreover, they are equal to deg( b , Ω , · ) and im T ( b , Ω), respectively.In particular, the following equalities will used several times:Ω b = b (Ω) = im T ( b , Ω) = im T ( u , Ω) , e Ω b = im T ( u , e Ω) . (3.10)Let Ω be the set of Definition 2.2. Note that since maps in A s are defined in e Ω, thenΩ also contains points outside Ω, in fact it is of full measure in e Ω. It was proved in [31,Lemma 3] that if u is one-to-one a.e. then Ω is one of the sets of full measure to whichthe restriction of u results in a map that is (fully) injective. Definition 3.12.
Let u ∈ A s . We define its inverse as the map u − : im G ( u , Ω) → R that sends every y ∈ im G ( u , Ω) to the only x ∈ Ω such that u ( x ) = y .We now prove that when u has zero surface energy in e Ω then also the geometric imageof e Ω (not only its topological image) coincides with e Ω b (up to a Lebesgue-null set). Mapsin A s are thus forced to fill the holes they may create at one part of the body with thematerial coming from a different part of it. The first step for the proof is to establish (3.7)also for open sets U enclosing the singular segment L . (In passing, we are also showingthat the a.e. injectivity of maps in A rs is automatic, since they inherit it from the boundarydata b .) Proposition 3.13.
Suppose that u ∈ H ( e Ω , R ) is axisymmetric and satisfies det D u > a.e., u = b in Ω D , and E ( u ) = 0 in e Ω . Then(a) u ∈ L ∞ (Ω , R ) , Det D u = det D u , and u is injective a.e.(b) im G ( u , e Ω) = e Ω b a.e. (c) For any U ∈ U u (see Definition 2.12) deg( u , U, · ) = χ im G ( u ,U ) a.e. (3.11) In particular, when ∂U ∩ L = ∅ (so that the classical degree and the topological imageis well defined), im G ( u , U ) = im T ( u , U ) a.e. (3.12) Proof.
The results were proved in [6, Th. 4.1] for maps u ∈ W ,p with p >
2. Here we onlyexplain the very minor modifications needed for the generalization to our H setting. Nowthat we have E ( u ) = 0 in e Ω (as opposed to Lemma 3.7 where essentially any axisymmetricmap was considered, so that E ( u ) = 0 only in e Ω \ L ), arguing exactly as in [6, Th. 4.1] weobtain both (3.12) and deg( u , U, · ) = N ( u , Ω d ∩ U, · ) a.e. , (3.13)where Ω d is the set of approximate differentiability, for any U ∈ U u such that ∂U ∩ L = ∅ (as opposed to Lemma 3.7 where the stronger restriction U ∩ L = ∅ was imposed). Inparticular, applying (3.13) to any U ∈ U u such that Ω ⋐ U ⋐ e Ω, we find that for a.e. x ∈ Ω u ( x ) ∈ im G ( u , U ) a.e. = im T ( u , U ) = im T ( b , U ) ⊂ e Ω b , thus proving the L ∞ bound. In addition, N ( u , Ω d ∩ U, · ) a.e. = deg( u , U, · ) = deg( b , U, · ) . As b is an orientation-preserving diffeomorphism,deg( b , U, · ) = b ( U ) , R \ b ( U ) , undefined in b ( ∂U )and b ( ∂U ) has measure zero. We conclude that N ( u , Ω d ∩ U, · ) = deg( b , U, · ) = χ b ( U ) a.e.,which implies that u is injective a.e. in U . As this is true for all U ∈ U u with Ω ⋐ U , and e Ωcan be written as the union of countably many such U , we conclude that u is injective a.e.and im G ( u , e Ω) = b ( e Ω) a.e. Since the L ∞ bound has already been established, the identityDet D u = det D u can be proved exactly as in [6, Thm. 4.1].Take now an arbitrary U ∈ U u (on whose boundary u is not necessarily continuous).Proceeding as in [6, Thm. 4.1] one still obtains that there exists c ∈ Z such that N ( u , Ω d ∩ U, · ) − deg( u , U, · ) = c a.e. , where deg( u , U, · ) is now the Brezis–Nirenberg degree (see Definition 3.2 and the remarkafter the definition). Since u is injective a.e., the first term coincides a.e. (see Lemma 2.4)with χ im G ( u ,U ) . As u ∈ L ∞ (Ω , R ), there exists a set N ⊂ Ω of measure zero such that u (Ω \ N ) ⊂ B ( , k u k L ∞ ). Therefore, im G ( u , Ω) ⊂ B ( , k u k L ∞ ) ∪ u (Ω ∩ N ). The set u (Ω ∩ N ) has measure zero thanks to Lemma 2.4. Thus, χ im G ( u ,U ) = 0 a.e. outside R \ B ( , k u k L ∞ ). By Definition 3.2, deg( u , ∂U, · ) = 0 a.e. outside B ( , k u k L ∞ ). Consequently, c = 0 and (3.11) holds. (cid:3) ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 29
Next, we show that the inverse of a map u in A rs has W , regularity. This followsessentially from [33, Th. 3.4]; nevertheless, some clarifying statements are in order. Thearguments in [33] apply for H maps satisfying properties (a) and (b) of condition INV(Definition 3.6) for every x in the domain where u is defined (in this case e Ω). In particular,they must be satisfied also for points x in the segment L = Ω ∩ R e where u maypotentially have singularities. Since u | ∂B ( x ,r ) cannot be ensured to be continuous when x ∈ L , then the topological images—and, hence, condition INV—are defined in [33] usingthe Brezis–Nirenberg degree (see Definition 3.2). Hence, we need to show that u satisfiesthat generalized invertibility condition and thus fulfills the hypothesis of [33, Thm. 3.4]. Proposition 3.14.
Let u ∈ A s .(a) If U ∈ U u and U ∩ L = ∅ , then u − ∈ W , (im T ( u , U ) , R ) .(b) If, furthermore, u ∈ A rs , then u − ∈ W , ( e Ω b , R ) .In both cases, D u − ( u ( x )) = D u ( x ) − = adj D u ( x )det D u ( x ) for a.e. x ∈ U ( resp., x ∈ e Ω) . (3.14) Proof.
Following [13], in the H setting the sets A u ,U := { y ∈ R : deg( u , U, y ) = 0 } are given an auxiliary and less prominent role, the actual topological images being nowdefined as im TBN ( u , U ) := { y ∈ R : D (cid:0) A u ,U , y (cid:1) = 1 } . The superscript ‘BN’ has been added here to indicate that use is made of the Brezis–Nirenberg degree. That notation, and that definition of topological image, will only appearin this proof of Proposition 3.14.Part I: if u ∈ A rs then for any open set U ∈ U u (see Definition 2.12)im G ( u , U ) = im TBN ( u , U ) a.e. (3.15)The proof consists in recalling (3.11), which implies thatim G ( u , U ) = A u ,U a.e. (3.16)Now, by Lebesgue’s differentiation theorem,im G ( u , U ) = { y ∈ R : D (im G ( u , U ) , y ) = 1 } a.e.Using (3.16) as well, we conclude (3.15).Part II: maps in A rs satsify INV in the whole of e Ω. Let U ∈ U u and assume that u | U \ N is injective for some set N ⊂ U of measure zero. Take a ∈ U and define r a := dist( a , ∂U ).Then B ( a , r ) ∈ U u | U for a.e. r ∈ (0 , r a ). Fix any such r . For all x ∈ B ( a , r ) ∩ Ω we have that u ( x ) ∈ im G ( u , B ( a , r )). By (3.15) and Lemma 2.4, we infer that u ( x ) ∈ im TBN ( u , B ( a , r )) for a.e. x ∈ B ( a , r ). Now, for all x ∈ U \ ( B ( a , r ) ∪ N ), by the injectivity, u ( x ) / ∈ im G ( u , B ( a , r )). As before, u ( x ) / ∈ im TBN ( u , B ( a , r )) for a.e. x ∈ U \ B ( a , r ). We have then shown that u | U satisfies the condition INV for H maps (with the topologicalimages defined using the Brezis–Nirenberg degree). As e Ω can be written as the union ofcountably many U ∈ U u , we conclude that u satisfies condition INV.Part III: regularity of the inverse when u ∈ A rs . Once condition INV for H maps hasbeen established, we obtain by [33, Th. 3.4] that the extension of u − by zero to all R isin SBV and the restriction of D u − to im TBN ( u , U ) is absolutely continuous with respectto the Lebesgue measure for any U ∈ U u . Apply this to any U ∈ U u such that Ω ⋐ U ⋐ e Ω. Since b is a C orientation-preserving diffeomorphism, applying the measure-theoreticinverse function theorem in [50, Lemma 2.5] we find that im TBN ( b , U ) = im T ( b , U ). Asim T ( b , U ) is open, we obtain that u − ∈ W , (im T ( b , U ) , R ). Since u − = b − in e Ω b \ im T ( b , U ), it follows that u − ∈ W , ( e Ω b , R ) as desired. Finally, (3.14) is proved in [33,Thm. 3.4].Part IV: regularity of the inverse when u is only in A s . Fix U ∈ U u with U ∩ L = ∅ .First recall that im G ( u , U ) = im T ( u , U ) a.e. (Lemma 3.7), hence almost every point inim T ( u , U ) lies in the set where u − is defined. Using (3.7) and arguing as in Parts I andII we obtain that im G ( u , U ) = im TBN ( u , U ) a.e. and that u satisfies the condition INVfor H maps (with the topological images defined using the Brezis–Nirenberg degree) in e Ω \ L . Since, by Lemma 3.7.(a), also E ( u ) = 0 in e Ω \ L , then the map u | e Ω \ L satisfies all thehypothesis of [33, Thm. 3.4]. It follows that the extension by zero of u − is in SBV and therestriction of D u − to im TBN ( u , U ) is absolutely continuous with respect to the Lebesguemeasure, and that (3.14) holds. It is easy to see that im T ( u , U ) ⊂ im TBN ( u , U ): indeed,any point in the topological image of the left is an interior point (because it is defined usingthe classical degree), hence the set A u ,U naturally has density one at that point (see theremark after Definition 3.2). Therefore, D u − im T ( u , U ) is absolutely continuous and u − | im T ( u ,U ) is in W , , as claimed. We have denoted by the restriction of a measure. (cid:3) Lemma 3.15.
Let u ∈ A s . Then, im T ( u , e Ω \ L ) = im G ( u , e Ω \ L ) = im G ( u , e Ω) a.e.Furthermore u − ∈ W , ( e Ω b \ im T ( u , L ) , R ) and formula (3.14) holds for a.e. x ∈ Ω \ L .Proof. Let { U k } k ∈ N be the countable family of Lemma 3.11. Since from Proposition 3.14, u − ∈ W , (im T ( u , U k ) , R ), this implies that u − ∈ W , (im T ( u , e Ω \ L ) , R ). Using (3.9)we find that u − ∈ W , ( e Ω b \ im T ( u , L ) , R ). Moreover, formula (3.14) holds for a.e. x ∈ U k , from which we get immediately that it also holds for a.e. x ∈ Ω \ L . Using thatformula, and a change of variables, we find that Z im T ( u , e Ω \ L ) | D u − ( y ) | d y = Z e Ω | cof D u ( x ) | d x < ∞ To see this we use that in order for a function f to be locally in a Sobolev space it is enough to verifythat for every point y an open set V can be found such that y belongs to V , V is contained in the domain,and f is Sobolev in V . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 31 since u ∈ H ( e Ω , R ). Therefore, u − ∈ W , (im T ( u , e Ω \ L ) , R ).By Lemma 3.7, im T ( u , U k ) = im G ( u , U k ) a.e. for all k ∈ N . Hence,im G ( u , e Ω) a.e. = im G ( u , e Ω \ L ) = [ k ∈ N im G ( u , U k ) a.e. = [ k ∈ N im T ( u , U k ) = im T ( u , e Ω \ L ) , where Lemma 2.1 was used in the first equality (since L has zero Lebesgue measure). (cid:3) Weak and pointwise convergence of inverses.
The following shows that theinverse of an axisymmetric map is axisymmetric.
Lemma 3.16.
Let u ∈ H ( e Ω , R ) be axisymmetric and injective a.e. Let v be its corre-sponding D function. Then there exist representatives of u − and v − such that u − isaxisymmetric with corresponding function v − .Proof. It was shown in the proof of Lemma 2.15 that u ◦ P is injective a.e., the map w : ( r, θ, x ) ( v ( r, x ) , θ, v ( r, x )) defined by (2.14) in the set (2.13) is injective a.e.,and v is injective a.e.Taking inverses in (2.6), we obtain that for a.e. y ∈ e Ω b = im T ( u , e Ω), P − (cid:0) u − ( y ) (cid:1) = w − (cid:0) P − ( y ) (cid:1) . (3.17)Now we use that, e Ω b is axisymmetric and thus e Ω b = { P ( r ′ , θ, x ′ ); ( r ′ , x ′ ) ∈ π ( e Ω b ) , θ ∈ [0 , π ) } . We also use that w − can be expressed in terms of the inverse of v : w − : ( r ′ , θ, x ′ ) ( v − ( r ′ , x ′ ) , θ, v − ( r ′ , x ′ )) . Thus writing y = P ( r ′ , θ, x ′ ) and applying P to (3.17) we find u − ◦ P ( r ′ , θ, x ′ ) = P ( v − ( r ′ , x ′ ) , θ, v − ( r ′ , x ′ )) , for a.e. ( r ′ , θ, x ′ ) ∈ π ( e Ω b ). This means that, since we can redefine u − and v − on sets ofnull measure, u − is axisymmetric and the associated map is v − . (cid:3) The following result on the convergence of the inverses will be used many times.
Lemma 3.17.
For each j ∈ N , let u j , u ∈ H ( e Ω , R ) be axisymmetric. Assume that u j ⇀ u in H ( e Ω , R ) as j → ∞ . Suppose that det D u j > a.e. for all j ∈ N and det D u > a.e., and that u j and u are invertible a.e. Then u − j → u − a.e.Proof. By Lemma 2.16, u is axisymmetric. Let v j and v be the corresponding 2 D functionsto u j and u , respectively. By Lemmas 2.14, 2.15 and 3.1, det D v j > v j is injectivea.e., v j ∈ H ( π ( e Ω) \ ([0 , δ ] × R ))) for any δ >
0, and analogously for v . Moreover, v j ⇀ v in H ( π ( e Ω) \ ([0 , δ ] × R ))) for each δ >
0, and E ( v j , π ( e Ω) \ ( { } × R )) = 0, and analogouslyfor v .By [6, Th. 6.3], v − j → v − a.e. By Lemma 2.15, we see that u is injective a.e. and, byLemma 3.16 that u − is axisymmetric with corresponding function v − . Now, the sameproof of Lemma 2.16 or Lemma 2.17 (applied to the inverses) shows that u − j → u − a.e. (cid:3) The horizontal components of the inverse have no singular parts on im T ( u , L ) . For general maps in A s the equality e Ω b = im G ( u , e Ω) does not hold in general, as can be seenby the example of a classical radially symmetric cavitation ( e Ω b contains the cavity openedby u whereas im G ( u , e Ω) does not). Furthermore, even when im G ( u , e Ω) does coincide a.e.with e Ω b , and even though Lemma 3.15 shows that the inverse is Sobolev when restricted toim T ( u , e Ω \ L ), that inverse is not necessarily in W , ( e Ω b , R ). An example is offered by thelimit map in the construction by Conti & De Lellis [13], where im T ( u , L ) consists (apartfrom the symmetry axis) of the sphere ∂B (cid:0) (0 , , ) , (cid:1) in the deformed configuration and u − has a jump across this sphere. Nevertheless, in the following lemma we show that anysuch singularities in u − are exclusively due to the vertical component u − of the inverse,whereas its horizontal components u − and u − do enjoy (or admit an extension to thewhole of e Ω b that enjoys) of a Sobolev regularity. This gives a first explanation of the originof the admissible space B that we propose in this paper (see its definition (1.8) in theIntroduction) for the study of the axisymetric neo-Hookean problem.From now on, for α ∈ { , , } , we denote by u − α the α -th component of u − . Recallthat equalities (3.10) hold and that e Ω b = im T ( u , e Ω \ L ) ∪ im T ( u , L ) (3.18)by Definition 3.9 and Lemma 3.10.(b).We shall need the following gluing theorem for BV functions [2, Thm. 3.84]. Proposition 3.18.
Let
N, m ≥ . Let Ω ⊂ R N be an open set. Let u , v ∈ BV (Ω , R m ) and let E be a set of finite perimeter in Ω , with ∂ ∗ E ∩ Ω oriented by ν E . Let u + ∂ ∗ E , v − ∂ ∗ E be the traces of u and v on ∂ ∗ E , which are defined for H -a.e. point of ∂ ∗ E . Set w = u χ E + v χ Ω \ E . Then w ∈ BV (Ω , R m ) ⇔ Z ∂ ∗ E ∩ Ω | u + ∂ ∗ E − v − ∂ ∗ E | d H N − < ∞ . If w ∈ BV (Ω , R m ) the measure D w is representable by D w = D u E + ( u + ∂ ∗ E − v − ∂ ∗ E ) ⊗ ν E H N − ( ∂ ∗ E ∩ Ω) + D v E , where E and E , respectively, denote the set of points at which E has density zero andone. Proposition 3.19.
Let u ∈ A s and α = 1 , . Denote by d u − α : e Ω b → R the map d u − α = ( u − α in im T ( u , e Ω \ L ) , in im T ( u , L ) . Then, d u − α ∈ W , ( e Ω b ) and d u − α has a precise representative whose restriction to the com-plement of a certain H -null set is continuous.Proof. Note that im T ( u , e Ω \ L ) a.e. = im G ( u , e Ω). Let v be the planar function correspondingto u . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 33
Part I: covering the 2 D domain π (Ω) \ ( { } × R ) with an increasing sequence of goodopen sets, ever closer to the singular segment.Thanks to Lemma 2.14, v ∈ H ( π ( e Ω) \ ([0 , δ ] × R ) , R ) for each δ >
0. Then, by Lemma2.13, for a.e. small δ > an open set E ( δ ) ⋐ π ( e Ω) \ ( { } × R ), with a C boundary, such that ∂E ( δ ) ∩ π (Ω) = { ( r, x ) ∈ π (Ω) : r = δ } and E ( δ ) ∈ U v (that is, all the analogous properties of Definition 2.12 are satisfied for theplanar map v ). Indeed, it can be seen that for every small c > C open set E ⋐ π ( e Ω \ ( { } × R ) such that ∂E ∩ π (Ω) = { ( r, x ) ∈ π (Ω) : r = c } (begin with the set ( { c } × R ) ∩ π (Ω), which consists of a finite number of segments and isnonempty because c is small; stretch this set vertically so as to obtain a new finite unionof segments containing the former ones but having their endpoints on π (Ω D ) = π ( e Ω \ Ω);then close the loop—or loops—with a C curve entirely contained in π (Ω D )). ApplyingLemma 2.13 to E we find that E t ∈ U v for a.e. t >
0, with E t defined in (2.10). It can beseen that ∂E t ∩ π (Ω) = { ( r, x ) ∈ π (Ω) : r = c − t } . Recall that E t ∈ U v for every small c > t >
0. Since a.e. small δ > δ = c − t for an appropriate choice of c and t , the claim follows.Part II: for orientation-preserving continuous maps satisfying the divergence identities,the topological image of a point is the classical image of that point.Define (cf. [6, Def. 5.6]) the topological image of a point ( r, x ) asim T (cid:0) v , ( r, x ) (cid:1) := \ <ρ
Part VI: H -continuity of d u − α . For each k ∈ N define w k : π ( e Ω b ) → R as w k = ( v − in im T (cid:0) v , π ( e Ω) \ (cid:0) [0 , δ k ] × R (cid:1) ,δ k otherwise,where v − is defined in terms of b − in π (cid:16) b (cid:0) e Ω \ (Ω ∪ C δ k ) (cid:1)(cid:17) . By Parts IV and V, thefunction w k is continuous at every point in π ( e Ω b ) \ T . Sincesup π ( e Ω b ) \ T | w − w k | = δ k , we have that w k → w uniformly in π ( e Ω b ) \ T as k → ∞ .For the bound | w − w k | ≤ δ k in the image of π ( e Ω) ∩ (cid:0) (0 , δ k ] × R (cid:1) we use that if ( r, x ) ∈ π (Ω) with r > v ( r, x ) ∈ π (Ω b ) \ T then v ( r, x ) ∈ π (cid:0) im T ( u , e Ω \ L ) (cid:1) , w is continuousat v ( r, x ), and w (cid:0) v ( r, x ) (cid:1) = r . That can be proved similarly as Part IV, finding k suchthat δ k < r and assuming that ( s, y ) = v ( r, x ) is both on v ( ∂E k ) and on v ( ∂E k +1 ).Therefore, w | π ( e Ω b ) \ T is continuous. Since d u − e + d u − e = w (cos θ e + sin θ e ) we havethen that d u − α has a precise representative whose restriction to the complement of a certainset of zero H -measure (the preimage by π of T ) is continuous.Part VII: Sobolev regularity of d u − α . Let V k ⋐ e Ω be the good (three-dimensional) openset such that π ( V k ) = E k . By [32, Prop. 2.17.(vi)], im T ( u , V k ) has finite perimeter and ∂ ∗ im T ( u , V k ) = im G ( u , ∂V k ) up to an H -null set. The set b (cid:0)e Ω \ (Ω ∪ C δ k ) (cid:1) also has finiteperimeter since b is diffeomorphism up to the boundary of e Ω. By Part V and Lemma 3.5,im T ( u , V k ) ∪ b (cid:0)e Ω \ (Ω ∪ C δ k ) (cid:1) = im T ( u , e Ω \ C δ k ) (3.25)for every k ∈ N . Hence, im T ( u , e Ω \ C δ k ) is a set of finite perimeter. By Part IV of theproof of Proposition 3.14, the map u − V k : R → R given by u − V k ( y ) = ( u − ( y ) , y ∈ im T ( u , V k ) , , y ∈ R \ im T ( u , V k )is in SBV ( R , R ) and D u − V k im T ( u , V k ) is absolutely continuous. Applying Proposition3.18 to u − V k and to b − , with im T ( u , V k ) as the set of finite perimeter in the hypotheses ofthat gluing theorem, we obtain that the map y ∈ e Ω b ( u − ( y ) , y ∈ im T ( u , V k ) b − ( y ) , y ∈ e Ω b \ im T ( u , V k )is in SBV ( e Ω b , R ), with derivative given by D u − V k im T ( u , V k ) + D b − (cid:0)e Ω b \ im T ( u , V k ) (cid:1) + (cid:0) ( u − ) + − ( b − ) − (cid:1) ν im T ( u ,V k ) H im G ( u , ∂V k ) . (The set im T ( u , V k ) has neither density zero nor one at H -a.e. point in ∂ im T ( u , V k ) thanksto [50, Lemma 2.5]) Since u = b in Ω D , taking (3.25) into account, the map can be rewrittenas y ∈ e Ω b ( u − ( y ) , y ∈ im T ( u , e Ω \ C δ k ) b − ( y ) , otherwise , with a corresponding rewriting for the derivative. At this point, taking into account (3.25),we apply Proposition 3.18 again, now to the first two components of the above map andto the function y = ( s cos θ, s sin θ, y ) ∈ e Ω b δ k (cos θ, sin θ )(which belongs to W , ( e Ω b , R )), with im T ( u , e Ω \ C δ k ) as the set of finite perimeter in thehypothesis of that gluing theorem, to find that the map W k : e Ω b → R given by W k : y = ( s cos θ, s sin θ, y ) ∈ e Ω b ((cid:0) u − ( y ) , u − ( y ) (cid:1) y ∈ im T ( u , e Ω \ C δ k ) δ k (cos θ, sin θ ) otherwise, (3.26)is in SBV ( e Ω b , R ), with derivative given by D ( u − , u − ) im T ( u , e Ω \ C δ k ) + δ k ( − sin θ, cos θ ) ⊗ Dθ (cid:0)e Ω b \ im T ( u , e Ω \ C δ k ) (cid:1) + (cid:0) ( u − , u − ) + − δ k (cos θ, sin θ ) (cid:1) ν im T ( u , e Ω \ C δk ) H (cid:0)e Ω b ∩ ∂ ∗ im T ( u , e Ω \ C δ k ) (cid:1) . However, by Lemma 3.5, the radial component of what would be the planar map cor-responding to W k in (3.26) is precisely w k , so by Part VI we know that the jump( u − , u − ) + − δ k (cos θ, sin θ ) is zero for H -a.e. point on e Ω b ∩ ∂ ∗ im T ( u , e Ω \ C δ k ). Therefore,the maps W k under consideration belong to W , ( e Ω b ).The uniform convergence w k → w of Part VI translates, in particular, into the a.e. point-wise convergence of the maps W k in (3.26) to the map ( d u − , d u − ) in the statement of theproposition. On the other hand, Lemma 2.14 shows that the gradients of the maps W k are equiintegrable because Z A | D W k | d y ≤ Z A ∩ im T ( u , e Ω \ L ) | D u − | d y + Z A ∩ e Ω b δ k | Dθ | d y for any measurable subset A ⊂ π ( e Ω b ). Therefore, the limit ( d u − , d u − ) also belongs to W , ( e Ω b , R ), finishing the proof. (cid:3) Existence of minimizers of the neo-Hookean energy in theaxisymmetric class A s As mentioned in the introduction, Ball and Murat’s result [5] that W ,p -quasiconvexityis necessary for sequential weak lower semicontinuity in W ,p reveals a serious obstructionto prove the existence of minimizers for the neo-Hookean energy E (see (1.2)), since theexample of cavitation shows that this energy functional is not W ,p -quasiconvex in 3 D forany p <
3. Nevertheless, morally, when attention is restricted to the axisymmetric class
ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 39 A s cavitation no longer implies that the neo-Hookean energy fails to be W , -quasiconvex.Indeed, there the energy functional is in truth defined on π (Ω), which is a 2 D subdo-main of the right half-plane. Therefore, the opening of cavities by a given map is partof its behaviour on the free boundary π ( L ) = π (Ω) ∩ ( { } × R ), and does not contradict W , -quasiconvexity. All in all, one may wonder if by restricting the attention to the ax-isymmetric setting the neo-Hookean energy becomes lower semicontinuous. Lemma 4.1below, which plays an important role in the proof of the main theorem of this paper (seeSection 6), shows that this is indeed the case. Lemma 4.1.
Let { u n } n be a sequence in A s . Then there exists u ∈ A s such that, up to asubsequence, u n ⇀ u in H ( e Ω , R ) , det D u n ⇀ det D u in L ( e Ω) ,χ im G ( u n , e Ω) → χ im G ( u , e Ω) a.e. as n → ∞ , and E ( u ) ≤ lim inf n →∞ E ( u n ) . (4.1) Proof.
Since E ( u n ) ≤ E ( b ) for all n ∈ N , we have, thanks to (1.2) and Poincar´e’s inequalitytogether with the boundary condition u n = b in Ω D , that { u n } n ∈ N is bounded in H ( e Ω , R ).Thus there exists u ∈ H ( e Ω , R ) such that, up to a subsequence u n ⇀ u in H . Since, upto a subsequence, u n → u a.e. we have that u = b a.e. on e Ω \ Ω. Moreover, by Lemma2.16, u is axisymmetric. Besides, we have sup n R ˜Ω H (det D u n ) < + ∞ . By using the DeLa Vall´ee Poussin criterion, we find that there exists d ∈ L ( e Ω) such thatdet D u n ⇀ d in L ( e Ω) . We claim that d >
0. Indeed, from the fact that det D u n > d ≥ d were zero in a set A of positive measure, then it would imply (for a subsequence)that det D u n → L ( A ) and a.e. in A . Thanks to the assumptions on H , we wouldhave H (det D u n ) → ∞ a.e. in A and, by using Fatou’s lemma, E ( u n ) → ∞ , which isimpossible. Therefore, d > e Ω. By Lemma 3.1, E ( u n , e Ω \ R e ) = 0 for all n ∈ N .Then, by [30, Th. 2], d = det D u a.e. and u is injective a.e. Thus u ∈ A s . Also, againby [30, Th. 2], for a.e. δ >
0, and χ im G ( u n , e Ω \ C δ ) → χ im G ( u , e Ω \ C δ ) a.e. as n → ∞ . From here,using the equiintegrability of the Jacobians, it is easy to prove that χ im G ( u n , e Ω) → χ im G ( u , e Ω) in L ( e Ω). Passing to a subsequence we obtain the stated a.e. convergence.Thanks to the weak continuity of the Jacobian, the lower semicontinuity of the term R e Ω H (det D u ) follows from the convexity of H , by standard lower semicontinuity results.Therefore, E is sequentially lower semicontinuous for the weak convergence in H , i.e.,(4.1) holds, and, in particular, E ( u ) ≤ E ( b ). (cid:3) Even though the regularity problem of showing that minimizers cannot have dipoles ispresumably easier to attack using Theorem 1.1 (because the augmented energy functionalmakes manifest more sharply the cost of a singularity), it is interesting to see that inthe axisymmetric setting the neo-Hookean energy E does have a minimizer. This followsfrom Lemma 4.1, which not only established the lower semicontinuity of E but proved the sequential weak compactness of A s as well. The proof is a standard application of thedirect method and shall be omitted. Theorem 4.2.
The energy E attains its minimum in A s . Weak limits of regular maps
In this section we investigate the properties of maps in the weak H closure A rs of theclass of regular maps. We start by proving that A rs is contained in the space B defined in(1.8). Proposition 5.1.
Let u ∈ A rs . Theni) u belongs to A s .ii) im G ( u , Ω) = Ω b a.e. and L (im T ( u , L )) = 0 .iii) u − ∈ BV ( e Ω b , R ) and supp D s u − ⊂ im T ( u , L ) . Moreover, k u − k BV ( e Ω b , R ) ≤ M forsome M > not depending on u .iv) u − α ∈ W , ( e Ω b ) for α = 1 , .Proof. Let { u n } n ∈ N ⊂ A rs satisfy u n ⇀ u in H ( e Ω , R ). By Lemma 4.1, u ∈ A s ,det D u n ⇀ det D u in L ( ˜Ω), and χ im G ( u n , e Ω) → χ im G ( u , e Ω) a.e. Now, by Proposition 3.13.(b),im G ( u n , e Ω) = e Ω b a.e. for every n ∈ N , hence u inherits this property. By (3.18) and Lemma3.15, it then follows that L (im T ( u , L )) = 0, completing the proof of ii) .From Proposition 3.14 we have that u − n ∈ W , ( e Ω b , R ) for all n ∈ N and k D u − n k L ( e Ω b , R × ) = k cof D u n k L ( e Ω , R × ) ≤ k D u n k L (Ω , R × ) + k D b k L (Ω D ; R × ) ≤ E ( u n ) + C ≤ E ( b ) + C where we have used Lemma 2.6.On the other hand the image of each u − n is contained in Ω, so k u − n k L ∞ ( e Ω b , R ) and,hence k u − n k L ( e Ω b , R ) are bounded by a constant only depending on Ω and e Ω b . Thus,by the theorem of compactness in BV we find that, up to a subsequence, there exists w ∈ BV ( e Ω b , R ) such that u − n → w in L ( e Ω b , R ) and a.e. in e Ω b . By Lemma 3.17, w = u − a.e. Finally, by Lemma 3.15 we have supp D s u − ⊂ im T ( u , L ). This shows iii) .Since L (im T ( u , L )) = 0, the functions u − α and d u − α (see Proposition 3.19) coincide a.e.Thus, by Proposition 3.19, u − α ∈ W , ( e Ω b ), which shows iv) . (cid:3) For u ∈ A rs we have, by Proposition 5.1, that u − ∈ BV ( e Ω b , R ) and we introduce thefollowing decomposition of the distributional derivative of u − : D u − = ∇ u − + D s u − = ∇ u − + D j u − + D c u − . In this decomposition ∇ u − denotes the absolutely continuous part of D u − with respect tothe Lebesgue measure, D s u − is the singular part which can be furthermore decomposed ina jump part denoted by D j u − and a Cantor part D c u − . This decomposition is standard(see, e.g., [2, Sect. 3.9]). Moreover, we denote by J u − the set of jump points of u − .We fix a Borel orientation ν of J u − , and, with respect to this orientation, the lateral ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 41 traces of u − are denoted by ( u − ) + and ( u − ) − . Analogously, the jump is defined as[ u − ] := ( u − ) + − ( u − ) − .The following lemma, which uses many ideas of [31, Th. 2], relates the surface energy E u with the singular part of D u − . With a small abuse of notation, given φ ∈ C c (Ω), wedefine [ φ ◦ u − ] in J u − as φ ◦ ( u − ) + − φ ◦ ( u − ) − . Recall that u − initially is definedonly on im G ( u , e Ω) but if im G ( u , e Ω) = e Ω b a.e. then u − is defined a.e. in the open set e Ω b .Assume that u − is the precise representative of itself. Lemma 5.2.
Let u ∈ H ( e Ω , R ) ∩ L ∞ (Ω , R ) be such that det D u ∈ L ( e Ω) and det D u > a.e. Let φ ∈ C c ( e Ω) and g ∈ C c ( R ) . Suppose that im G ( u , e Ω) = e Ω b a.e., u is injective a.e.and u − ∈ BV ( e Ω b , R ) . Then E u ( φ, g ) = −h D s ( φ ◦ u − ) , g i = − Z e Ω b ∇ φ ( u − ( y )) ⊗ g ( y ) · d D c u − ( y ) − Z J u − [ φ ◦ u − ] g · ν d H . Proof.
By the change of variables formula and using that im G ( u , e Ω) = e Ω b a.e., we find E u ( φ, g ) = Z e Ω b (cid:2) g ( y ) · D u ( u − ( y )) − T Dφ ( u − ( y )) + φ ( u − ( y )) div g ( y ) (cid:3) d y . (5.1)By the chain rule for BV functions (see, e.g., [2, Th. 3.96]), φ ◦ u − ∈ BV ( e Ω b , R ) and ∇ ( φ ◦ u − ) = ∇ φ ( u − ) ∇ u − ,D s ( φ ◦ u − ) = ∇ φ ( u − ) D c u − + [ φ ◦ u − ] ⊗ ν u H n − J u − , (5.2)By Lemma 3.15, ∇ u − ( u ( x )) = ∇ u ( x ) − for a.e. x ∈ e Ω. This and (5.2) imply that, fora.e. y ∈ ˜Ω b , ∇ ( φ ◦ u − )( y ) = ∇ φ ( u − ( y )) ∇ u ( u − ( y )) − . Therefore, Z e Ω b g ( y ) · D u ( u − ( y )) − T Dφ ( u − ( y ))d y = Z e Ω b ∇ ( φ ◦ u − )( y ) · g ( y )d y = h∇ ( φ ◦ u − ) , g i . (5.3)On the other hand, by definition of distributional derivative, Z e Ω b φ ( u − ( y )) div g ( y )d y = −h D ( φ ◦ u − ) , g i . (5.4)Putting together (5.1), (5.3) and (5.4) we obtain E u ( φ, g ) = h∇ ( φ ◦ u − ) − D ( φ ◦ u − ) , g i = −h D s ( φ ◦ u − ) , g i . This, together with (5.2), concludes the proof. (cid:3)
Recall the definition of the singular segment L = Ω ∩ R e in (3.3). Proposition 5.3.
Let u ∈ A rs . Then u − ( y ) ∈ L for | D c u − | -a.e. y ∈ e Ω b and ( u − ) ± ( y ) ∈ L for H -a.e. y ∈ J u − .Proof. Without loss of generality, u − is the precise representative of itself.Let g ∈ C c ( R , R ) and φ ∈ C c ( e Ω \ R e ). Then, there exists δ > φ ∈ C c ( e Ω \ C δ ). By Lemma 3.1, we have that E u ( φ, g ) = 0, so due to Lemma 5.2 we obtainthat Z e Ω b ∇ φ ( u − ( y )) ⊗ g ( y ) · d D c u − ( y ) + Z J u − [ φ ◦ u − ] g · ν d H = 0 . (5.5)By approximation, the previous equality, which does not involve derivatives of g , is alsovalid for every bounded Borel g : R → R .Let D c u − = A | D c u − | be the polar decomposition of D c u − , so A : e Ω b → R × isBorel, | D c u − | -integrable and | A | = 1 in | D c u − | -a.e. e Ω b . Let y ∈ e Ω b be a | D c u − | -Lebesgue point of u − , i.e.,lim r → + R B ( y ,r ) | u − ( y ) − u − ( y ) | d | D c u − | ( y ) | D c u − | ( B ( y , r )) = 0 , (5.6)and note that | D c u − | -a.e. point of e Ω b satisfies that. Let us suppose that u − ( y ) / ∈ L andlet us take a closed cube Q ⊂ e Ω centered at u − ( y ) with Q ∩ L = ∅ . Consider the Borelset U := { y ∈ e Ω b : u − ( y ) ∈ Q and u − is approximately continuous at y } . Given any ψ ∈ C ( R ), take φ ∈ C c ( e Ω \ R e ) such that φ ( x ) = ψ ( x α ) for all x ∈ Q . For1 ≤ α, i ≤ r > g = sgn ψ ′ sgn A αi χ B ( y ,r ) ∩ U e i and deducethat Z { y ∈ B ( y ,r ); u − ( y ) ∈ Q } sgn ψ ′ sgn A αi (cid:0) ∇ φ ( u − ( y )) ⊗ e i (cid:1) · A d | D c u − | = 0 . This can also be written as Z { y ∈ B ( y ,r ); u − ( y ) ∈ Q } | ψ ′ ( u − α ( y )) || A αi | d | D c u − | = 0 . We use the previous equality first with ψ ( t ) = cos t and then with ψ ( t ) = sin t . We sumthe two equalities and use that | cos t | + | sin t | ≥ Z { y ∈ B ( y ,r ); u − ( y ) ∈ Q } | A αi | d | D c u − | = 0 . We then sum this equality for 1 ≤ α, i ≤ | D c u − | (cid:0) { y ∈ B ( y , r ); u − ( y ) ∈ Q } (cid:1) = 0 . This equality implies that | u − ( y ) − u − ( y ) | > diam Q/ | D c u − | -a.e. y ∈ B ( y , r ).As this is true for all r >
0, this is a contradiction with (5.6). Therefore, | D c u − | -a.e. y ∈ e Ω b satisfies u − ( y ) ∈ L . ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 43
Now we show that ( u − ) ± ( y ) ∈ L for H -a.e. y ∈ J u − . Let y ∈ J u − be a H J u − -Lebesgue point for both ( u − ) + and ( u − ) − , i.e.,lim r → + R J u − ∩ B ( y ,r ) | ( u − ) ± ( y ) − ( u − ) ± ( y ) | d H ( y ) H ( J u − ∩ B ( y , r )) = 0 , (5.7)and note that H -a.e. point in J u − satisfies that. For each r >
0, we apply (5.5) to g = χ J u − ∩ B ( y ,r ) ν and deduce that Z J u − ∩ B ( y ,r ) [ φ ◦ u − ]d H = 0 . This and (5.7) imply that [ φ ◦ u − ]( y ) = 0. If ( u − ) + ( y ) / ∈ L and ( u − ) − ( y ) ∈ L ,we choose φ ∈ C c ( e Ω \ R e ) such that φ ( u − ) + ( y ) = 0 and reach a contradiction with[ φ ◦ u − ]( y ) = 0. Analogously if ( u − ) + ( y ) ∈ L and ( u − ) − ( y ) / ∈ L . If ( u − ) + ( y ) / ∈ L and ( u − ) − ( y ) / ∈ L , we choose φ ∈ C c ( e Ω \ R e ) such that φ (( u − ) + ( y )) = φ (( u − ) − ( y )),which contradicts [ φ ◦ u − ]( y ) = 0. Hence, the only possibility is that ( u − ) + ( y ) ∈ L and ( u − ) − ( y ) ∈ L . (cid:3) Lower bound for the relaxed energy and a more explicit alternativevariational problem
In this section we study the energetic cost for a weak limit of functions in A rs to leave A rs . Note that condition INV is not satisfied, in general, by functions in A rs : this is in factone of the main features of the counterexample of [13, Sect. 6]. This is due to the lackof equiintegrability of the cofactors and, as a consequence, the theory of [32] cannot beapplied.A standard diagonal argument shows that A rs is closed under the weak convergence of H (Ω , R ). From Proposition 5.1 we see that the energy F ( u ) = E ( u ) + 2 k D s u − k M (Ω b , R × ) is well defined on A rs and that F ( u ) = E ( u ) + 2 k D s u − k M (Ω b , R × ) . (6.1)We start with the following lemma, which plays a role of an energy-area inequality andshould be compared with Lemma 2.6. In many problems with lack of compactness, theloss of compactness is caused by sequences of maps realizing asymptotically the equalityin such inequality. Lemma 6.1.
Let u ∈ A s . Then | adj D u e | ≤ | D u | . This inequality is optimal and cannot be attained by a map in A s . Proof.
With the expressions of D u and cof D u in terms of the associated 2 D map v , cf.(6.19), we find | adj D u e | = | v | r (cid:0) | ∂ r v | + | ∂ x v | (cid:1) / ≤ (cid:18) | v | r + | ∂ r v | + | ∂ x v | (cid:19) ≤ | D u | . The equality implies v r = ( | ∂ r v | + | ∂ x v | ) / and ∇ v = 0. This cannot be attained bya map in A s , since ∇ v = 0 implies det D v = 0, so det D u = 0. Nonetheless, the constantis optimal in A s , as can be checked by considering v ( r, x ) = r and v ( r, x ) = εx for ε ց
0, which corresponds to u ( x ) = ( x , x , εx ). (cid:3) The following lower semicontinuity result is the cornerstone of the strategy that we areproposing in this paper for the study of the regularity of the minimizers of the neo-Hookeanenergy. In fact, the study of the fine properties obtained in the previous sections was carriedout in order to provide a rigorous expression of the ideas contained in this proposition.
Proposition 6.2.
The energy F defined in (6.1) is sequentially lower semicontinuous in A rs for the weak convergence in H ( e Ω , R ) .Proof. Recall from Proposition 5.1 that A rs ⊂ A s . Let { u k } k ∈ N be a sequence in A rs tending weakly in H ( e Ω , R ) to u ∈ A rs . Thanks to Proposition 5.1 iii), the BV norm of u − k is bounded, so, due to Lemma 3.17, we have that, up to a subsequence, u − k ∗ ⇀ u − in BV ( e Ω b , R ) and a.e. By Lemma 4.1, we have that det D u k ⇀ det D u in L ( e Ω) and,because of the convexity of H , Z e Ω H (det D u ) ≤ lim inf k →∞ Z e Ω H (det D u k ) . (6.2)We first prove that the sequence (det D u − k ) is equiintegrable. This can be proved asin [6, Prop. 7.8]. Indeed, define H : (0 , ∞ ) → R as H ( t ) := tH (1 /t ). Then H growssuperlinearly at infinity and Z e Ω b H (det D u − k )d y = Z e Ω H (det D u k ) ≤ E ( b ) . Thus the equiintegrability result follows from the De La Vall´ee Poussin criterion.Now, let ε be any positive number. Recall that, for δ > C δ is given by (2.7). Choose δ >
0, such that Z C δ ∩ e Ω | D u | d x < ε. (6.3)Because of the axial symmetry, it can be seen that the sequence { χ e Ω \ C δ cof D u k } k ∈ N is equiintegrable, cf. [36, Th. 1.3]. This is due to the the fact that the corresponding2D- maps v k are bounded in H (cid:16) π (cid:16)e Ω \ C δ (cid:17) , R (cid:17) and then by a result of M¨uller [48],since we also have det D v k > D v k are equiintegrable. Now we obtain theequiintegrability result for { χ e Ω \ C δ cof D u k } k ∈ N by expressing the cofactor matrix in terms ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 45 of the 2D map v k and observing that one entry is det D v k and the others are products ofa sequence converging strongly in L by a sequence converging weakly in L ; cf. (6.19) inthe Appendix.Hence, there exists η >
0, independent of k , such that if A ⊂ e Ω is measurable, | A | < η ⇒ Z A \ C δ | cof D u k | d x < ε, ∀ k ∈ N . (6.4)Given any open subset V of e Ω b (which we shall later choose to be a thin neighbourhoodof im T ( u , L )), and any good δ < δ , Z V |∇ ( u − k ) | d y = Z u − k ( V ) ∩ C δ | adj ∇ u k e | + Z u − k ( V ) \ C δ | adj ∇ u k e | d x , (6.5)where we have used that ∇ u − = adj D u det D u and det D u > | adj D u e | ≤ | D u | . Hence, the first integral in the right-hand side of (6.5) is boundedby the integral of | D u k | in C δ ∩ e Ω. As for the second integral, note that | u − k ( V ) \ C δ | ≤ Z V det D u − k d y . (6.6)Since L (cid:0) im T ( u , L ) (cid:1) = 0 combining (6.6), (6.4), and the equiintegrability of { det D u − k } k ∈ N it is possible to find δ >
0, with δ < δ and an open set V ⊂ e Ω b such thatim T ( u , L ) ⊂ V and Z u − k ( V ) \ C δ | cof D u k | d x < ε, ∀ k ∈ N . (6.7)By virtue of (6.7), for this V we have that, for all k ∈ N , Z V |∇ ( u − k ) | d y ≤ Z C δ ∩ e Ω | D u k | d x + ε. (6.8)Therefore, by using Proposition 5.1.iv), we have that, for all k ∈ N , k D ( u − k ) k M ( V, R × ) = Z V |∇ ( u − k ) | d y + k D s u − k k M ( V, R × ) ≤ Z C δ ∩ e Ω | D u k | d x + ε + k D s u − k k M ( e Ω b , R × ) . Hencelim inf k →∞ k D ( u − k ) k M ( V, R × ) ≤ ε + lim inf k →∞ " Z C δ ∩ e Ω | D u k | d x + k D s u − k k M ( e Ω b , R × ) . (6.9) As u − k ∗ ⇀ u − in BV ( e Ω b , R ), by using Proposition 5.1.iii) and iv) we have that k D s u − k M ( e Ω b , R × ) = k D s u − k M ( V, R × ) = k D s ( u − ) k M ( V, R × ) (6.10) ≤ k D ( u − ) k M ( V, R × ) ≤ lim inf k →∞ k D ( u − k ) k M ( V, R × ) . Putting together (6.9) and (6.10) we find k D s ( u − ) k M ( e Ω b , R × ) ≤ ε + lim inf k →∞ " Z C δ ∩ e Ω | D u k | d x + k D s u − k k M ( e Ω b , R × ) . (6.11)As u k ⇀ u in H ( e Ω , R × ),12 Z e Ω \ C δ | D u | d x ≤ lim inf k →∞ Z e Ω \ C δ | D u k | d x . (6.12)Gathering (6.3), (6.12), we obtain12 Z C δ ∩ e Ω | D u | d x + 12 Z e Ω \ C δ | D u | d x ≤ ε + lim inf k →∞ Z e Ω \ C δ | D u k | d x . (6.13)Now we use (6.11) and (6.13) along with the property that lim inf k → + ∞ a k +lim inf k → + ∞ b k ≤ lim inf k → + ∞ ( a k + b k ) to obtain k D s u − k M ( e Ω b , R × ) + 12 Z e Ω | D u | d x ≤ ε + lim inf k →∞ (cid:20) Z e Ω | D u k | d x + k D s u − k k M ( e Ω b , R × ) (cid:21) . The last property being true for every ε > Z e Ω | D u | + k D s u − k M ( e Ω b , R × ) ≤ lim inf k →∞ (cid:20) Z e Ω | D u k | d x + k D s u − k k M ( e Ω b , R × ) (cid:21) . (6.14)The proof of the proposition is concluded by using (6.2) and (6.14). (cid:3) The following observations about the proof of Proposition 6.2 are in order.Inequality (6.8) is indicative of the antagonistic situation we are facing: while the Dirich-let energy of the limit map in the small cylinder C δ is less than ε , see (6.3), the Dirichletenergy of the approximating sequence u k , even in the smaller cylinder C δ , is very largeand invested in creating large gradients for the inverse. These large gradients will giverise to the singular part of the inverse of the limit map, a pathological behaviour of whichthe gradient of the limit map has absolutely no memory. In contrast, by considering therelaxation of the neo-Hookean energy, given the limit map u one is forced to consider theprocess in which it was obtained from regular maps u k , bringing back into the calculationthe singular Dirichlet energy that concentrated around the symmetry axis.The second equality in (6.10) aids in appreciating the necessity of the Sobolev regularityresult for the horizontal components of the inverse—Proposition 3.19. Without it, theestimate in (6.5) would have been made for the whole cofactor matrix, yielding in (6.1) thesuboptimal prefactor √ × × ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 47 that D s u − is supported in im T ( u , L ), itself a consequence of the Sobolev regularity of theinverses in im T ( u , L )—Proposition 3.14 and Lemma 3.15.Note that what has been proved in Proposition 6.2 is that if u k is a sequence in A rs with weak H (Ω , R )-limit u then both inequalities (6.14) and (6.2) hold. If, in addition, { u k } k ⊂ A rs then we have in particular Z Ω | D u | + 2 k D s u − k M (Ω b , R × ) ≤ lim inf k →∞ Z Ω | D u k | . Finally, note also that, from Lemma 3.19 or Proposition 5.1 iv), as well as Proposition5.1 ii) k D s u − k M ( e Ω b , R × ) = k D s u − k M ( e Ω b , R ) = k D s u − k M ( V, R ) , for any open set V ⊂ e Ω with im T ( u , L ) ⊂ V . This fact has been used crucially in the proofof Proposition 6.2.The proof of Proposition 6.2 follows an important idea already used in several variationalproblems with concentration: in order to estimate the energetic cost of such a concentrationphenomenon one has to estimate the energy of the sequence considered in a neighbourhoodof the concentration set. However, here we face an additional difficulty: the energetic costis described by k D s u − k M ( e Ω b , R × ) and hence we have to consider the image by u k of aneighbourhood of the concentration set (the e -axis) and this image depends on k .Since we are in the presence of a problem of lack of compactness it is natural to try anddescribe the space A rs and the relaxed energy defined on this space by E rel ( u ) := inf { lim inf n →∞ E ( u n ) : { u n } n ∈ N ∈ A rs and u n ⇀ u in H (Ω , R ) } . (6.15)It is well known that definition (6.15) of E rel is equivalent to saying that E rel is thelargest lower semicontinuous functional in A rs (for the weak topology of H ( e Ω , R )) that isbelow E in A rs . Since F is lower semicontinuous in A rs for the weak topology of H ( e Ω , R )and F = E in A rs , we have that E rel ≥ F in A rs . (6.16)In particular, E rel = E in A rs . (6.17)It is tempting to conjecture that the equality E rel = F holds at least for some specialchoices of function H . In view of Proposition 6.2 (and its consequence, the inequality E rel ≥ F in A rs ), it remains to characterize A rs and to show that for any u ∈ A rs thereexists a sequence { u n } n ∈ N ⊂ A rs converging weakly to u in H (Ω , R ) such thatlim n →∞ E ( u n ) = F ( u ) . There are serious difficulties in constructing this sequence { u n } n ∈ N (if it exists at all). Oneof them relies on the restrictions of being orientation-preserving and injective a.e., eventhough there are some partial results in this direction (see [39, 38, 14, 46, 16] and thereferences therein). Nevertheless, in our forthcoming paper we discuss the example u of Conti and De Lellis [13], which in fact was one of the motivations for the current work;improving their construction we will be able to prove the equality E rel ( u ) = F ( u ).At any rate, the interest of defining the relaxed energy in an abstract way is to be ableto prove that it attains its infimum in A rs and that the initial energy attains its minimumin A rs if and only if there exists a minimizer of E rel in A rs which is in A rs . These two factsare classical in the theory of relaxation and follow from abstract arguments. We recall theproof of these facts in Proposition 6.3 for the comfort of the reader. Then we show thatthe energy F satisfies analogous properties and, hence, can be a substitute E rel : this is theobject of Theorem 6.4. Proposition 6.3.
The following statements hold:i) The relaxed energy E rel attains its minimum on A rs .ii) Every minimizing sequence for inf u ∈A rs E ( u ) converges, up to a subsequence, towardsa minimizer of E rel in A rs .iii) The neo-Hookean energy E attains its minimum on A rs if and only if a minimizer of E rel is in A rs .Proof. From a diagonal argument, A rs is closed for the weak topology in H (Ω , R ) and,from the abstract definition of E rel , we can see that it is lower semicontinuous for the weakconvergence in H (Ω , R ). Moreover, E rel is coercive since Z Ω | D u | ≤ E ( u )and the functional u R Ω | D u | is lower semicontinuous, so Z Ω | D u | ≤ E rel ( u ) . (The coercivity can also be proved by invoking the much more difficult Proposition 6.2:see (6.16)) The conclusion of the first assertion follows.For the second assertion, we take any minimizing sequence { u n } n ⊂ A rs of E . Since thesequence is bounded in H ( e Ω , R ), up to a subsequence we find that u n ⇀ u in H ( e Ω , R ).By definition, u ∈ A rs . To show that u is a minimizer of E rel we take any w ∈ A rs . Bydefinition of E rel , there exists a sequence { w k } k in A rs such that E rel ( w ) = lim k →∞ E ( w k ).Then, E rel ( u ) ≤ lim inf n →∞ E rel ( u n ) = lim n →∞ E ( u n ) = inf A rs E ≤ E ( w k )for every k ∈ N , so E rel ( u ) ≤ lim k →∞ E ( w k ) = E rel ( w ) . For the third assertion, we first assume that there exists a minimizer u of E rel in A rs such that u ∈ A rs . We then have E rel ( u ) ≤ E rel ( w ) for any w ∈ A rs . But, thanks to(6.17), we find that E ( u ) ≤ E ( w ) for all w ∈ A rs . That is, u is a minimizer of E in A rs .Now we assume that E attains its minimum on A rs : we denote by u one such minimizer.Let us show that u is also a minimizer of E rel in A rs , so consider any w ∈ A rs . As before, ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 49 there exists a sequence { w n } n in A rs such that E rel ( w ) = lim n →∞ E ( w n ). Then, E ( u ) = inf A rs E ≤ E ( w n )for every n ∈ N , so by taking limits we obtain E ( u ) ≤ E rel ( w ). (cid:3) Theorem 6.4.
There exists a minimizer of F in A rs . Moreover, if there exists a minimizerof F in A rs which is also in A rs then the infimum of E in A rs is attained.Proof. Recall that A rs is closed for the weak convergence in H ( e Ω , R ). It is also boundedin H ( e Ω , R ). Now from Proposition 6.2 we also have that F is lower semicontinuous in A rs . Clearly, F is coercive in A rs . This readily implies the existence of minimizers.As for the second part of the statement, we assume that there exists a minimizer u of F in A rs such that u ∈ A rs . We then have F ( u ) ≤ F ( w ) for any w ∈ A rs . But since F = E in A rs , we find that E ( u ) ≤ E ( w ) for all w ∈ A rs . That is, u is a minimizer of E in A rs . (cid:3) In the same vein, we have the following result.
Theorem 6.5.
The energy E attains its infimum on A rs .Proof. Indeed from Lemma 4.1 we have that A rs ⊂ A s and that E is lower semicontinuouson A s . Moreover, A rs is closed for the weak convergence in H . As noted before, E iscoercive in A s . These are the three main ingredients to obtain the conclusion. (cid:3) Regarding the difference between Theorems 6.5 and 4.2, the reader is reminded of theLavrentiev phenomenon associated to cavitation [4]. Compared to Theorem 1.1, as men-tioned in the comment before Theorem 4.2, showing that minimizers of the neo-Hookeanenergy cannot have dipoles is likely to be easier with the functional having the additionalterm, since it makes manifest more sharply the cost of a singularity.It would be nice to have an explicit characterization of A rs . Even though this charac-terization is still missing, we are able to prove the existence of minimizers of the energy F in the explicit space B defined in (1.8) which is a priori larger than A rs . Indeed, fromProposition 5.1 we have that A rs ⊂ B ⊂ A s . Besides, the energy F is well defined on B ,it controls the BV norm of the inverses, and a slight adaptation of Proposition 6.2 yieldsthe lower semicontinuity of F in B . Lemma 6.6.
Let u ∈ B . Then k D u − k M ( e Ω b , R × ) ≤ F ( u ) . Proof.
By Lemma 2.6, F ( u ) ≥ Z e Ω | cof D u | d x + 2 k D s u − k M ( e Ω b , R × ) ≥ Z im G ( u , e Ω) |∇ u − | d y + k D s u − k M ( e Ω b , R × ) = k D u − k M ( e Ω b , R × ) . (cid:3) Proposition 6.7.
The energy F is sequentially lower semicontinuous in B for the weakconvergence in H ( e Ω , R ) .Proof. Let { u k } k ∈ N be a sequence in B tending weakly in H ( e Ω , R ) to u ∈ B . We canassume that lim inf k →∞ F ( u k ) < ∞ . In particular, sup k ∈ N k D u − k k M ( e Ω b , R × ) < ∞ . As k u k k L ∞ ( e Ω , R ) and, hence, k u k k L ( e Ω , R ) are bounded, the BV norm of u − k is bounded, so,due to Lemma 3.17, we have that, up to a subsequence, u − k ∗ ⇀ u − in BV ( e Ω b , R ) anda.e. From here, the proof is the same as in Proposition 6.2. (cid:3) Proof of Theorem 1.1.
Let { u n } n ⊂ B be a minimizing sequence for F in B . Clearlysup n F ( u n ) < ∞ , and we can assume that u n ⇀ u in H ( e Ω , R ). Since, by Proposition 6.7, F is lower semicontinuous for the weak convergence in H of maps in B , it suffices to showthat the weak limit u is in B . We know from Lemma 4.1 that u ∈ A s , det D u n ⇀ det D u in L ( e Ω) and im G ( u n , e Ω) → im G ( u , e Ω) a.e. In particular, e Ω b = im G ( u , e Ω) a.e. and from(3.18), im T ( u , L ) is a null Lebesgue set. Now we use Lemma 6.6 to say that F ( u n ) ≥ k D u − n k M ( e Ω b , R × ) . As { u − n } n ∈ N is bounded in L ∞ ( e Ω b , R ), we find that u − n is bounded in BV ( e Ω b , R ). Upto a subsequence, thanks to Lemma 3.17, we have that u − n → u − in L ( e Ω b , R ) and a.e.,with u − ∈ BV ( e Ω b , R ). From Proposition 3.19 we also infer that u − , u − are in W , ( e Ω b ).This proves that u minimizes F in B .The other statement of Theorem 1.1 can be shown as in the proofs of Proposition 6.3 orTheorem 6.4. (cid:3) We point out that Theorem 1.1 allows us to transform the problem of lack of compactnessinto a problem of regularity without appealing to a relaxation result. With Theorem 1.1we have at hand an explicit minimization problem: the energy and space of minimizationare explicit. Furthermore, if this problem admits a minimizer which is in A rs then it is alsoa minimizer of F (and, hence E ) in A rs . Studying the regularity of minimizers of F is leftfor a future work.The following is a summary of existence results we obtained in this article.Spaces A rs A rs A rs A rs B A s Energies
E E E rel
F F E
Existence of min. ? yes Th. 6.5 yes Prop. 6.3 yes Th. 6.4 yes Th. 1.1 yes Th. 4.2
ELAXATION OF THE NEO-HOOKEAN ENERGY IN 3D 51
Appendix: Working with axially symmetric maps
We recall from the Appendix in [36] that if u : Ω → R is axisymmetric and is given incylindrical coordinates by u ( r cos θ, r sin θ, x ) = v ( r, x ) e r + v ( r, x ) e then D u = ∂ r v ∂ x v v r ∂ r v ∂ x v , (6.18)cof D u = v r ∂ x v − v r ∂ r v Dv − v r ∂ x v v r ∂ r v , (6.19)det D u = 1 r v det D v , and the Dirichlet energy is given by Z Ω | D u | = 2 π Z π (Ω) (cid:0) | ∂ r v | + | ∂ x v | (cid:1) r d r d x + 2 π Z π (Ω) v r d r d x . Acknowledgements
We gratefully acknowledge J. Ball for his observation (see Section 4) that in the axi-symmetric setting cavitation does not truly show that the neo-Hookean energy fails to be W , -quasiconvex. R.R is partially supported by the ANR project BLADE Jr. ANR-18-CE40-0023. References [1]
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Dipartimento di Matematica e Geoscienze, Universit`a degli Studi diTrieste, Via Weiss 2 - 34128 Trieste, Italy.
Email address : [email protected] (Duvan Henao) Faculty of Mathematics, Pontificia Universidad Cat´olica de Chile, Vicu˜naMackenna 4860, Macul, Santiago, Chile
Email address : [email protected] (Carlos Mora-Corral) Faculty of Sciences, Universidad Aut´onoma de Madrid, C/ Tom´as yValiente 7, E-28049 Madrid, Spain.
Email address : [email protected] (R´emy Rodiac) Universit´e Paris-Saclay, CNRS, Laboratoire de Math´ematiques d’Orsay,91405, Orsay, France,
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