Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems of Fracture Mechanics
aa r X i v : . [ m a t h . A P ] D ec Some sufficient conditions for lower semicontinuity in SBD andapplications to minimum problems of Fracture Mechanics.
Giuliano Gargiulo ∗ Elvira Zappale † Abstract
We provide some lower semicontinuity results in the space of special functions of bounded deformationfor energies of the type Z J u Θ( u + , u − , ν u ) d H N − , [ u ] · ν u ≥ H N − − a. e. on J u , and give some examples and applications to minimum problems.Keywords: Lower semicontinuity, fracture, special functions of bounded deformation, joint convexity, BV -ellipticity. This study is motivated by the results contained in [7, 8, 9] where it has been studied, both from themechanical and computational viewpoint with several techniques, in the regime of linearized elasticity, thepropagation of the fracture in a cracked body with a dissipative energy a la Barenblatt, i.e. of the type R Γ φ ([ u ] · ν u , [ u ] · τ u ) d H N − , where Γ denotes the unknown crack site, [ u ] · ν u , [ u ] · τ u represent the detachmentand the sliding components respectively, of the opening of the fracture [ u ] and, the energy density φ has theform φ ([ u ] · ν u , [ u ] · τ u ) = u ] · ν u = [ u ] · τ u = 0 ,K if [ u ] · ν u ≥ , + ∞ if [ u ] · ν u < K is a suitable positive constant. It has to be emphasized that the energy density φ in (1.1) alsotakes into account an infinitesimal noninterpenetration constraint, i.e. all the deformations u pertaining tothe effective description of the energy must satisfy [ u ] · ν u ≥ H N − a.e. on Γ.More precisely, the subsequent analysis aims to extend some of the results contained in [24, 25, 26]. Infact the target of those results was providing a mathematical justification to the minimization procedureadopted in [7, 8, 9] which appears at each time step, when studying the propagation of the fracture usingthe quasistatic evolution method, as introduced in [23] and developed in many other papers (see for instance[17, 18] for the first formulation in terms of free discontinuity problem in the nonlinear elasticity setting,[15] for the linear case, see also the more recent papers [16, 22, 14] among a wide literature). Indeed inorder to derive, from the mathematical viewpoint, the properties of the energy φ above which guaranteelower semicontinuity with respect to the natural convergences (2.13) ÷ (2.15) below, in order to generalizethe energetic model contained in [7, 8, 9] and finally to extend the lower semicontinuity results for surfaceintegrals contained in [12], the following result has been proved in [24]: ∗ Dipartimento di Scienze Biologiche ed Ambientali, Universit´a degli Studi del Sannio, via Port’Arsa, 82100, Benevento, Italy.Email: [email protected] † DIIMA, Universit`a degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy.Email: [email protected] heorem 1.1. Let Ω be a bounded open subset of R N , let Φ := { ϕ : [0 , + ∞ [ → [0 , + ∞ [ , ϕ convex, subadditive and nondecreasing } (1.2) and let ϕ ∈ Φ . Let { u h } be a sequence in SBD (Ω) , such that [ u h ] · ν u h ≥ H N − -a.e. on J u h for every h ,converging to u in L (Ω; R N ) satisfying (2.12) below, with a function γ : [0 , + ∞ [ → [0 , ∞ [ nondecreasing andverifying the superlinearity condition (2.11) below. Then [ u ] · ν u ≥ H N − − a.e. on J u , (1.3) and Z J u ϕ ([ u ] · ν u ) d H N − ≤ lim inf h → + ∞ Z J uh ϕ ([ u h ] · ν u h ) d H N − . (1.4)It can be easily seen that the class Φ in (1.2) includes functions of the type φ above, but it has also tobe remarked that, in general, the functions in Φ can be truly convex. Indeed, typical examples of functionsin Φ are given by ϕ : s ∈ R + (1 + s p ) p , p ≥
1, but in practice this class of functions does not perfectly fitthe mechanical framework, where actually a ‘concave-type’ behavior is expected.In fact, the present paper originates from the desire of finding a wider class of functions, containingthe function φ in [7, 8, 9], including energy densities with a more general dependence on the opening ofthe fracture [ u ] and on the normal of the crack site ν u rather than just on their scalar product [ u ] · ν u asfor ϕ in (1.2) or possibly exhibiting a dependence from the ‘traces’ on the two sides of the crack site, andwhich still ensures lower semicontinuity. A first result in this direction, i.e. the lower semicontinuity of R J u Ψ([ u ] , ν u ) d H N − with respect to convergences (2.13) ÷ (2.15), together with a characterization of suchintegrands (see [25, Theorem 4.5]), has been achieved in [25, Theorem 1.2], where the classΨ : ( a, p ) ∈ R N × S N − sup ξ ∈ S N − | p · ξ | ψ ( | a · ξ | ) , (1.5)has been introduced, with ψ : [0 , + ∞ [ → [0 , + ∞ [ lower semicontinuous, nondecreasing, subadditive (moregenerally a lower semicontinuos function such that ψ ( | · | ) is subadditive). For a more detailed discussion seeRemark 3.2 below.With the aim of considering surface energies whose densities have explicit dependence on the two different one-sided Lebesgue limits (see Section 2 below) and on the normal to the jump site, we introduce here theclass Θ and prove Theorem 1.3 stated below. Definition 1.2.
Let Θ , with a notational abuse, be the class of functions of the form Θ : ( i, j, p ) ∈ R N × R N × S N − → sup ξ ∈ S N − f ( i · ξ, j · ξ, p · ξ ) (1.6) where f : R → ]0 , + ∞ [ is a continuous BV -elliptic function. (See Definition 2.7 for BV -ellipticity.) Theorem 1.3.
Let Ω be a bounded open subset of R N , let γ : [0 , + ∞ [ → [0 , + ∞ [ be a non-decreasing functionverifying condition (2.11) and let Θ be as in (1.6) where f is a continuous BV -elliptic function in the senseof Definition 2.7. Let { u h } be a sequence in SBD (Ω) satisfying the bound (2.12), such that [ u h ] · ν u h ≥ , H N − -a.e. on J u h for every h and converging to u in L (Ω; R N ) . Then (1.3) holds and Z J u Θ( u + , u − , ν u ) d H N − ≤ lim inf h → + ∞ Z J uh Θ( u + h , u − h , ν u h ) d H N − . (1.7)The structure of the paper is the following. In Section 2 the main results from Geometric Measure Theoryconcerning spaces of functions with bounded deformation and special functions of bounded variation, arerecalled. Section 3 is devoted to Theorem 1.3 and to related minimun problems.2 Notations and preliminary results
In this paper Ω will be a bounded open subset of R N . We shall usually suppose, when not explicitlymentioned, (essentially to avoid trivial cases) that N >
1. Let u ∈ L (Ω; R m ), the set of Lebesgue points of u is denoted by Ω u . Equivalently x ∈ Ω u if and only if there exists a (necessarily unique) ˜ u ( x ) ∈ R m suchthat lim ̺ → + ̺ N Z B ̺ ( x ) | u ( y ) − ˜ u ( x ) | dy = 0 . A function u ∈ L (Ω; R m ) is said to be of bounded variation, and we write u ∈ BV (Ω; R m ) if itsdistributional gradient Du is an m × N matrix of finite Radon measures in Ω, Du ∈ M b (Ω; M m × N ).Furthermore the following Lebesgue ( Radon − Nykodim ) decomposition holds Du = ∇ u L N + D s u , where D s u , the singular part with respect the Lebesgue measure L N , can be split as D j u + D c u , with D j u therestriction of Du to Ω \ Ω u , and D c u the restriction of D s u to Ω u . For the above, and further details on BVfunctions, see e.g. [6]. BD (Ω) is the space of vector fields with bounded deformation and it is defined as the set of vector fields u = ( u , . . . , u N ) ∈ L (Ω; R N ) whose distributional gradient Du = { D i u j } has the symmetric part Eu = { E ij u } , E ij u = ( D i u j + D j u i ) / M b (Ω; M N × Nsym ), the space of bounded Radon measures in Ω with values in M N × Nsym (thespace of symmetric N × N matrices). For u ∈ BD (Ω), the jump set J u is defined as the set of points x ∈ Ωwhere u has two different one-sided Lebesgue limits u + ( x ) and u − ( x ), with respect to a suitable direction ν u ( x ) ∈ S N − = { ξ ∈ R N : | ξ | = 1 } , i.e.lim ̺ → + ̺ N Z B ± ̺ ( x,ν u ( x )) | u ( y ) − u ± ( x ) | dy = 0 , (2.1)where B ± ̺ ( x, ν u ( x )) = { y ∈ R N : | y − x | < ̺, ( y − x ) · ( ± ν u ( x )) > } , (( u + , u − , ν u ) are determined withinpermutation to ( u − , u + , − ν u )), accordingly we shall assume that all the subsequent integrands f ( i, j, p ) willbe compatible with this permutation, i.e. f ( i, j, p ) = f ( j, i, − p ). Ambrosio, Coscia and Dal Maso [5] provedthat for every u ∈ BD (Ω) the jump set J u is Borel measurable and countably ( H N − , N −
1) rectifiableand ν u ( x ) is normal to the approximate tangent space to J u at x for H N − -a.e. x ∈ J u , where H N − is the( N − u ∈ BD (Ω), the Lebesgue decomposition of Eu is Eu = E a u + E s u with E a u the absolutely continuous part and E s u the singular part with respect to the Lebesgue measure L N . E u denotes the density of E a u with respect to L N , i.e. E a u = E u L N . We recall that E s u can befurther decomposed as E s u = E j u + E c u with E j u , the jump part of Eu , i.e. the restriction of E s u to J u and E c u the Cantor part of Eu , i.e. therestriction of E s u to Ω \ J u . In [5] it has been shown that E j u = ( u + − u − ) ⊙ ν u H N − ⌊ J u (2.2)where ⊙ denotes the symmetric tensor product, defined by a ⊙ b := ( a ⊗ b + b ⊗ a ) / a, b ∈ R N ,and H N − ⌊ J u denotes the restriction of H N − to J u , i.e. ( H N − ⌊ J u )( B ) = H N − ( B ∩ J u ) for every Borelset B ⊆ Ω, (and we then write B ∈ B (Ω)). Moreover it has been also proved that | E c u | ( B ) = 0 for every B ∈ B (Ω) such that H N − ( B ) < + ∞ , where | · | stands for the total variation. In the sequel, for every u ∈ L loc (Ω; R N ) we denote by [ u ] the vector u + − u − . For any y, ξ ∈ R N , ξ = 0, and any B ∈ B (Ω) let π ξ := { y ∈ R N : y · ξ = 0 } ,B ξy := { t ∈ R : y + tξ ∈ B } ,B ξ := { y ∈ π ξ : B ξy = ∅} , (2.3)3.e. π ξ is the hyperplane orthogonal to ξ , passing through the origin and B ξ = p ξ ( B ), where p ξ , denotes theorthogonal projection onto π ξ . B ξy is the one-dimensional section of B on the straight line passing through y in the direction of ξ .Given a function u : B → R N , defined on a subset B of R N , for every y, ξ ∈ R N , ξ = 0, the function u ξy : B ξy → R is defined by u ξy ( t ) := u ξ ( y + tξ ) = u ( y + tξ ) · ξ for all t ∈ B ξy . (2.4)Following [5] we can say that a vector field u belongs to BD (Ω) if and only if its ’ projected sections ’ u ξy belong to BV (Ω ξy ). More precisely the following Structure Theorem (cf. [5, Theorem 4.5]) holds. Theorem 2.1.
Let u ∈ BD (Ω) and let ξ ∈ R N with ξ = 0 . Then(i) E a uξ · ξ = R Ω ξ D a u ξy d H N − ( y ) , | E a uξ · ξ | = R Ω ξ | D a u ξy | d H N − ( y ) .(ii) For H N − -almost every y ∈ Ω ξ , the functions u ξy and ˜ u ξy (the Lebesgue representative of u , cf. formula(2.5) in [5]) belong to BV (Ω ξy ) and coincide L -almost everywhere on Ω ξy , the measures | Du ξy | and V ˜ u ξy (the pointwise variation of ˜ u ξt cf. formula (2.8) in [5]) coincide on Ω ξy ,and E u ( y + tξ ) ξ · ξ = ∇ u ξy ( t ) =(˜ u ξy ) ′ ( t ) for L -almost every t ∈ Ω ξy .(iii) E j uξ · ξ = R Ω ξ D j u ξy d H N − ( y ) , | E j uξ · ξ | = R Ω ξ | D j u ξy | d H N − ( y ) .(iv) ( J ξu ) ξy = J u ξy for H N − -almost every y ∈ Ω ξ and for every t ∈ ( J ξu ) ξy u + ( y + tξ ) · ξ = ( u ξy ) + ( t ) = lim s → t + ˜ u ξy ( s ) u − ( y + tξ ) · ξ = ( u ξy ) − ( t ) = lim s → t − ˜ u ξy ( s ) , where the normals to J u and J u ξy are oriented so that ν u · ξ ≥ and ν u ξy = 1 .(v) E c uξ · ξ = R Ω ξ D c u ξy d H N − ( y ) , | E c uξ · ξ | = R Ω ξ | D c u ξy | d H N − ( y ) . The space
SBD (Ω) of special vector fields with bounded deformation is defined as the set of all u ∈ BD (Ω)such that E c u = 0, or, in other words Eu = E u L N + [ u ] ⊙ ν u H N − ⌊ J u We also recall that if Ω ⊂ R , then the space SBD (Ω) coincides with the space of real valued specialfunctions of bounded variations
SBV (Ω), consisting of the functions whose distributional gradient is a Radonmeasure with no Cantor part (see [6] for a comprehensive treatment of the subject).Furthermore we restate [5, Proposition 4.7] to be exploited in the sequel.
Proposition 2.2.
Let u ∈ BD (Ω) and let ξ , . . . , ξ N be a basis of R N . Then the following three conditionsare equivalent:(i) u ∈ SBD (Ω) .(ii) For every ξ = ξ i + ξ j with ≤ i, j ≤ n , we have u ξy ∈ SBV (Ω ξy ) for H N − -almost every y ∈ Ω ξ .(iii) The measure | E s u | is concentrated on a Borel set B ⊂ Ω which is σ -finite with respect to H N − . Moreover, following [5] we give:
Definition 2.3.
For any u ∈ BD (Ω) we define the non-negative Borel measure λ u on Ω as λ u ( B ) := 12 ω N − Z S N − λ ξu ( B ) d H N − ( ξ ) ∀ B ∈ B (Ω) , (2.5) where, for every ξ ∈ S N − λ ξu ( B ) := Z Ω ξ H ( J u ξy ∩ B ξy ) d H N − ( y ) ∀ B ∈ B (Ω) . (2.6)4et J ξu := { x ∈ J u : [ u ] · ξ = 0 } , (2.7)we recall that H N − ( J u \ J ξu ) = 0 for H N − − a.e. ξ ∈ S N − . (2.8)The following result is a consequence of the Structure Theorem Theorem 2.4.
For every u ∈ BD (Ω) and any ξ ∈ S N − , λ ξu ( B ) = Z J ξu ∩ B | ν u · ξ | d H N − ∀ B ∈ B (Ω) , (2.9) where ν u is the approximate unit normal to J u . Moreover λ u = H N − ⌊ J u . A standard approximation argument by simple functions, proves, more generally, that for every Borel function g : Ω → [0 , + ∞ ], it results Z J ξu ∩ B g ( y ) | ν u · ξ | d H N − ( y ) = Z Ω ξ Z p ξ ( J ξu ∩ B ) g ( y + tξ ) d H ( t ) d H N − ( y ) (2.10)for any ξ ∈ S N − .We recall the following compactness result for sequences in SBD proved in [12, Theorem 1.1 and Remark2.3].
Theorem 2.5.
Let γ : [0 , + ∞ [ → [0 , + ∞ [ be a non-decreasing function such that lim t → + ∞ γ ( t ) t = + ∞ . (2.11) Let { u h } be a sequence in SBD (Ω) such that k u h k L ∞ (Ω; R N ) + Z Ω γ ( |E u h | ) dx + H N − ( J u h ) ≤ K (2.12) for some constant K independent of h . Then there exists a subsequence, still denoted by { u h } , and a function u ∈ SBD (Ω) such that u h → u strongly in L loc (Ω; R N ) , (2.13) E u h ⇀ E u weakly in L (Ω; M N × Nsym ) , (2.14) E j u h ⇀ E j u weakly* in M b (Ω; M N × Nsym ) , (2.15) H N − ( J u ) ≤ lim inf h → + ∞ H N − ( J u h ) . (2.16)We will also make use of the following result from Measure Theory [6, Lemma 2.35] Lemma 2.6.
Let λ be a positive σ -finite Borel measure in Ω and let ϕ i : Ω → [0 , ∞ ] , i ∈ N , be Borelfunctions. Then Z Ω sup i ϕ i dλ = sup (X i ∈ I Z A i ϕ i dλ ) where the supremum ranges over all finite sets I ⊂ N and all families { A i } i ∈ I of pairwise disjoint open setswith compact closure in Ω . BV -ellipticityand joint convexity, (the first notion was already introduced in [3, 4] in order to describe sufficient conditionsfor lower semicontinuity in SBV for surface integrals).We stress that the definitions below we are referring to (i.e. BV -ellipticity and joint convexity)), appearslightly different from those stated in [6], but we emphasize that for the applications to lower semicontinuityproblems with respect to convergence (2.13) ÷ (2.16) we have in mind, they can be considered as ‘equivalent’.Indeed, what really matters to that aim, is to have the sequences { u h } ⊂ SBD (Ω) with range in a suitablecompact set of R N , (related to the considered energy density). This fact is evident in the arguments used inthe proofs of lower semicontinuity results in the original articles (see [2] and also [6]).Let Q ν be an open cube of R N , centred at 0, with side lenght 1 and faces either parallel or orthogonalto ν ∈ S N − and let u i,j,ν be the function defined as u i,j,ν = (cid:26) i if y · ν > ,j if y · ν < Definition 2.7.
Let T ⊂ R m be a finite set, and f : T × T × S N − → [0 , + ∞ [ . A function f is said to be BV -elliptic if Z J ν f ( v + , v − , ν v ) d H N − ≥ f ( i, j, ν ) (2.17) for any bounded piecewise constant function v : Q ν → T such that { v = u i,j,ν } ⊂⊂ Q ν and any triplet ( i, j, ν ) in the domain of f . A function f : R m × R m × S N − → [0 , + ∞ [ is said BV -elliptic if it verifies (2.17) for any finite set T ⊂ R m .In the sequel, with an abuse of notations we will use the same symbol for any BV -elliptic function andits positive 1-homogeneous extension in the last variable. Definition 2.8.
Let f : R m × R m × R N → [0 , + ∞ ] . We say that f is jointly convex if f ( i, j, p ) = sup h ∈ N { ( g h ( i ) − g h ( j )) · p } ∀ ( i, j, p ) ∈ R m × R m × R N , for some sequence { g h } ⊂ [ C ( R m )] N . The above notion was introduced in [2] with the name of regular ‘bi-convexity’, see Lemma 3.4 therein.We also recall, as proven in [2] (see also [6]), that joint convexity implies BV -ellipticity, and the equiva-lence between the two notions is still an open problem, even if there are some classes of function for which thetwo notions are proven to be equivalent (see [2, Example 5.1] and Example 3.5 herein). On the other hand BV -ellipticity is very difficult to verify in practice, whereas this is not the case for joint convexity. More-over, necessarily any jointly convex function is lower semicontinuous and f ( i, j, p ) = f ( j, i, − p ) , f ( i, i, p ) =0 ∀ i, j ∈ R m , p ∈ R N , f ( i, j, · ) is positively 1-homogeneous and convex, ∀ i, j ∈ R m .In [2] (see Theorem 3.3 therein) it has been proven the following theorem that will be invoked in theproof of Theorem 1.3. Theorem 2.9.
Let f : R m × R m × S N − → [0 , + ∞ [ be a continuous BV - elliptic function. Let { u h } ⊂ SBV (Ω; R m ) be a sequence converging in L (Ω; R m ) to u such that k u h k L ∞ and H N − ( J u h ) are boundedand {|∇ u h |} is equiintegrable. Then u ∈ SBV (Ω; R d ) and Z J u f ( u + , u − , ν u ) d H N − ≤ lim inf h Z J uh f ( u + h , u − h , ν h ) d H N − . Assuming f jointly convex, one can allow f to take the value + ∞ and not necessarily be continous, as ithas been proven in [2, Theorem 3.6], (see also [6, Theorem 5.22].) Theorem 2.10.
Let K ′ ⊂ R m be a compact set and let f : R m × R m × S N − → [0 , + ∞ ] be a jointly convexfunction. Let { u h } ⊂ SBV (Ω; R m ) be a sequence converging in L (Ω; R m ) to u such that u h ∈ K ′ a.e. in Ω , {|∇ u h |} is equiintegrable and H N − ( J u h ) bounded. Then u ∈ SBV (Ω; R d ) , u ∈ K ′ a.e. in Ω and, Z J u f ( u + , u − , ν u ) d H N − ≤ lim inf h Z J uh f ( u + h , u − h , ν h ) d H N − .
6e observe that the assumption inf f > f defined on R m × R m × S N − by a function defined just on the compact set K ′ × K ′ × S N − . On the other hand it wouldbe enough to require such a density jointly convex just on K ′ × K ′ × S N − obtaining it through functions { g h } ⊂ ( C ( K ′ )) N in place of { g h } ⊂ ( C ( R m )) N when giving Definition 2.8, since the sequence { u h } inTheorem 2.10 has range in K ′ . In fact this latter approach has been followed in [6], but the present choiceallows a more transparent comparison of the lower semicontinuity results Theorem 1.3 and Proposition 3.3with the results contained in [24] and [25], see Theorems 1.1 herein and [25, Theorem 1.2]. We start this section by providing a lower semicontinuity lemma along directions that will be to a greatdegree exploited in the proof of Theorem 1.3. The proof develops in analogy with a similar result in [25],essentially exploiting the slicing method for
SBD fields introduced in [5, 12], and we write it here for reader’sconvenience.
Lemma 3.1.
Let f be a continuous BV -elliptic function as in Definition 2.7. Let Ω be a bounded open subsetof R N . Let { u h } be a sequence in SBD (Ω) satisfying the bound (2.12), such that [ u h ] · ν u h ≥ H N − -a.e.on J u h for every h and converging to u in L (Ω; R N ) . Then Z J u f (cid:0) u + ( y ) · ξ, u − ( y ) · ξ, ν u · ξ (cid:1) d H N − ( y ) ≤ lim inf h Z J uh f (cid:0) u + h ( y ) · ξ, u − h ( y ) · ξ, ν u h · ξ (cid:1) d H N − ( y ) (3.1) for H N − -a.e. ξ ∈ S N − .Proof. Let { u h } ⊂ SBD (Ω) satisfying the bound (2.12) and converging to u in L (Ω; R N ). Theorem 2.5ensures that u ∈ SBD (Ω).Let ξ ∈ S N − , and let p ξ : J u → π ξ be the orthogonal projection onto π ξ . First we observe that ( iv )in Theorem 2.1 guarantees that one can choose the normals to J u , J u h , J u ξy and J u hξy oriented so that ν u · ξ, ν u h · ξ ≥ ν u ξy = ν u hξy = 1.This fact and Proposition 2.2 ensure that for H N − - a.e. y ∈ Ω ξ it results( u ξy ) + ( t ) = ( u · ξ ) + ( y + tξ ) and ( u ξy ) − ( t ) = ( u · ξ ) − ( y + tξ ) for every t ∈ J u ξy and,( u hξy ) + ( t ) = ( u h · ξ ) + ( y + tξ ) and( u hξy ) − ( t ) = ( u h · ξ ) − ( y + tξ ) for every t ∈ J u hξy , (3.2)with u ξy , u hξy ∈ SBV (Ω ξy ) for H N − -a.e. y ∈ Ω ξ .Thus we can restate (3.1) as Z J u | ν u · ξ | f (cid:0) u + ( y ) · ξ, u − ( y ) · ξ, (cid:1) d H N − ( y ) ≤ lim inf h Z J uh | ν u h · ξ | f (cid:0) u + h ( y ) · ξ, u − h ( y ) · ξ, (cid:1) d H N − ( y ) (3.3)On the other hand, by (2.7) and (2.8), we have Z J u | ξ · ν u | f ( u + ( y ) · ξ, u − ( y · ξ ) , d H N − ( y ) = Z J ξu | ξ · ν u | f ( u + ( y ) · ξ, u − ( y ) · ξ, d H N − ( y ) , Z J uh | ξ · ν u h | f ( u + h ( y ) · ξ, u − h ( y ) · ξ, d H N − ( y ) = Z J ξuh | ξ · ν u h | f ( u + h ( y ) · ξ, u − h ( y ) · ξ, d H N − ( y ) (3.4)7or every h ∈ N and for H N − -a.e. ξ ∈ S N − . (3.2), (3.4), (2.10) guarantee the existence of N ⊂ S N − suchthat H N − ( N ) = 0 and Z J u | ξ · ν u | f ( u + ( y ) · ξ, u − ( y ) · ξ, d H N − ( y ) = Z Ω ξ h Z J uξy f (( u ξy ) + ( t ) , ( u ξt ) − ( t ) , d H ( t ) i d H N − ( y ) , Z J uh | ξ · ν u h | f ( u + h ( y ) · ξ, u − h ( y ) · ξ, d H N − ( y ) = Z Ω ξ h Z J uhξy f (( u hξy ) + ( t ) , ( u hξy ) − ( t ) , d H ( t ) i d H N − ( y ) , for every h ∈ N and for every ξ ∈ S N − \ N .Consequently the proof will be completed once we show that Z Ω ξ h Z J uξy f (( u ξy ) + ( t ) , ( u ξt ) − ( t ) , d H ( t ) i d H N − ( y ) ≤ lim inf h → + ∞ Z Ω ξ h Z J uhξy f (( u hξy ) + ( t ) , ( u hξy ) − ( t ) , d H ( t ) i d H N − ( y ) (3.5)for every ξ ∈ S N − \ N .To this end, for each ξ ∈ S N − \ N consider a subsequence { u k } ≡ { u h k } such thatlim inf h → + ∞ Z J uhξy f (( u hξy ) + ( t ) , ( u hξy ) − ( t ) , d H ( t ) = lim k → + ∞ Z J ukξy f (( u kξy ) + ( t ) , ( u kξy ) − ( t ) , d H ( t ) . (3.6)Next consider a further subsequence (denoted by { u j } ≡ { u k j } ) such thatlim j → + ∞ H N − ( J u j ) = lim inf k → + ∞ H N − ( J u k ) . (3.7)We want to show that the assumptions of Theorem 2.9 in dimension one are satisfied.By (ii) in Theorem 2.1 (i.e. E u j ( y + tξ ) · ξ = ( u jξ ) ′ y ( t ) for H N − -a.e. y ∈ Ω ξ and for L -a.e. t ∈ Ω ξy )and by Fubini-Tonelli’s theorem, for any ξ ∈ S N − \ N we can define I y,ξ ( u j ) = R Ω ξy γ ( | u ′ jξy ( t ) | ) dt , where u jξy ( t ) = u j ( y + tξ ) · ξ and we have Z π ξ I y,ξ ( u j ) d H N − ( y ) = Z Ω γ ( |E u j ( x ) ξ · ξ | ) dx. Since { u j } satisfies the bound (2.12) and γ is non-decreasing, it follows that Z π ξ I y,ξ ( u j ) d H N − ( y ) ≤ Z Ω γ ( |E u j ( x ) | ) dx ≤ K, (3.8)for every ξ ∈ S N − \ N and for H N − -a.e. y ∈ Ω ξ . It is also easily seen that, from the bound on k u j k L ∞ ,deriving from the global bound (2.12), k u jξy k L ∞ (Ω ξy ) ≤ K. (3.9)From (3.8), (2.10) and (2.12) for every ξ ∈ S N − \ N it results that there exists a constant C ≡ C ( K ) suchthat lim inf j → + ∞ Z π ξ [ I y,ξ ( u j ) + H ( J u jξy )] d H N − ( y ) ≤ C < + ∞ . Let us fix ξ ∈ S N − \ N (such that the previous inequality holds). Using Fubini-Tonelli’s theorem and con-vergence in measure for L - converging sequences, we can extract a subsequence { u m } = { u j m } (dependingon ξ ) such that lim m → + ∞ Z π ξ [ I y,ξ ( u m ) + H ( J u mξy )] d H N − ( y ) =lim inf j → + ∞ Z π ξ [ I y,ξ ( u j ) + H ( J u jξy )] d H N − ( y ) ≤ C < + ∞ , (3.10)8nd for a.e. y ∈ Ω ξ , u ξm,y ∈ SBV (Ω ξy ) and u mξy → u ξy in L loc (Ω ξy ), with u ξy ∈ SBV (Ω ξy ).Let ξ ∈ S N − \ N : by (3.10) and Fatou’s lemma, for H N − -a.e. y ∈ Ω ξ , it resultslim inf m → + ∞ [ I y,ξ ( u m ) + H ( J u mξy )] < + ∞ . (3.11)Let us fix N Ω ξ ⊂ Ω ξ and a point y ∈ Ω ξ \ N Ω ξ , such that H N − ( N Ω ξ ) = 0, (3.11), (3.9) hold and suchthat u mξy ∈ SBV (Ω ξy ) for any m . Passing to a further subsequence { u l } ≡ { u m l } we can assume that thereexists a constant C ′ such thatlim inf m → + ∞ [ I y,ξ ( u m ) + H ( J u mξy )] = lim l → + ∞ [ I y,ξ ( u l ) + H ( J u lξy )] ≤ C ′ . This means that { u lξy } ∈ SBV (Ω ξy ) and satisfies all the assumptions of Theorem 2.9 for each interval(connected component) I ⊂ Ω ξy . Consequently (3.6), ( iv ) of Theorem 2.1 and, Theorem 2.9 guarantee that Z J uξy f (( u ξy ) + ( t ) , ( u ξy ) − ( t ) , d H ( t ) ≤ lim l → + ∞ Z J ulξy f (( u lξy ) + ( t ) , ( u lξy ) − ( t ) , d H ( t ) =lim inf h → + ∞ Z J uhξy f (( u hξy ) + ( t ) , ( u hξy ) − ( t ) , d H ( t ) (3.12)for H N − -a.e. ξ ∈ S N − and for H N − -a.e. y ∈ Ω ξ .The lower semicontinuity stated in (3.5) now follows from Fatou’s lemma, which completes the proof.Now we are in position to prove Theorem 1.3. Proof of Theorem 1.3.
We preliminarly observe that (1.3) follows by Theorem 1.1, thus it only remains toprove (1.7) and this will be achieved essentially through the applications of Lemma 3.1 and Lemma 2.6.The continuity of f allows us to assume ξ in (1.2) varying in any countable subset of S N − . It will bechosen in S N − \ N , N being the H N − exceptional set introduced in Lemma 3.1, and it will be denoted by A , with elements ξ α .By superadditivity of liminf:lim inf h → + ∞ Z J uh Θ( u + h , u − h , ν u h ) d H N − ≥ X α lim inf h → + ∞ Z J uh ∩ A α f ( ξ α · u + h , ξ α · u − h , ξ · ν u h ) d H N − for any finite family of pairwise disjoint open sets A α ⊂ Ω.By Lemma 3.1 we havelim inf h → + ∞ Z J uh f ( ξ α · u + h , ξ α · u − h , ν u h ) d H N − ≥ Z J u f ( ξ α · u + , ξ α · u − h , ν u h d H N − for every ξ α ∈ A . Thereforelim inf h → + ∞ Z J uh Θ( u + h , u − h , ν u h ) d H N − ≥ X α Z J u ∩ A α f ( ξ α · u + , ξ α · u − , ξ α · ν u ) d H N − for every ξ α ∈ A and for any finite family of pairwise disjoint open sets A α ⊂ Ω.By Lemma 2.6 we can interchange integration and supremum over all such families, thus gettinglim inf h → + ∞ Z J uh Θ( u + h , u − h , ν u h ) d H N − ≥ Z J u Θ( u + , u − , ν u ) d H N − , whence (1.7) follows and this concludes the proof. 9 emark 3.2. It is worthwhile to observe that φ in (1.1) of [7, 8, 9] can be recast in terms of a suitable Θ in (1.6) requiring in the model that the noninterpenetration constraint (1.3) is verified. In fact it suffices toconsider (as already observed in [25]) Θ( i, j, p ) = sup ξ ∈ S N − K | p · ξ | , i.e. f ( a , a , b ) = ψ ( | a − a | ) θ ( b ) for suitable ψ and θ (see 2 of Examples 3.5 below), with ψ = ψ const : t ∈ [0 , + ∞ [ → K, K > , and θ = | · | , from which one deduces that Θ = Θ const : ( i, j, p ) ∈ R N × R N × S N − → K .Moreover we recall as emphasized in [25, Remark 4.8] that the constant functions K represent the onlyintersections between the classes Ψ in (1.5) and Φ in (1.2). On the other hand, the fact that the classes (1.2)and (1.6) do differ is not very surprising and, indeed, also the techniques adopted to prove the related lowersemicontinuity results Theorem 1.1 and Theorem 1.3 (and its simplified version given in [25, Theorem 1.2])are very different, the first relying essentially on Geometric Measure Theory and the second on the structureof the Special fields with Bounded Deformation together with the characterization of lower semicontinuity in SBV , enlightened in [5, 12] and in [2].
We observe that, while joint convexity entails BV -ellipticity, on the other hand, one can replace the BV -elliptic function f in (1.6) by a jointly convex one, which may take also the value + ∞ , i.e.Θ : ( i, j, p ) ∈ R N × R N × S N − → sup ξ ∈ S N − f ( i · ξ, j · ξ, p · ξ ) (3.13)where f : R × R × R → ]0 , + ∞ ] is a jointly convex function as in Definition 2.8.Thus the following result holds, which is indipendently obtained and not stated as a Corollary of Theorem1.3, since we may avoid to require f continuous and finite. Proposition 3.3.
Let Ω be a bounded open subset of R N , let γ : [0 , + ∞ [ → [0 , + ∞ [ be a non-decreasingfunction verifying condition (2.11) and let Θ be as in (3.13) where f is a jointly convex function. Let { u h } be a sequence in SBD (Ω) satisfying the bound (2.12), such that [ u h ] · ν u h ≥ , H N − -a.e. on J u h for every h and converging to u in L (Ω; R N ) . Then (1.3) and (1.7) hold.Proof. First we assume f continuous. Under this extra assumption, the proof develops as in Theorem 1.3making use of Theorem 2.9 in place of Theorem 2.10, when stating and proving the analogue of Lemma 3.1.Then for general jointly convex f , it is enough to observe that by Definition 2.8, f can be approximatedby a non decreasing sequence of continuous jointly convex functions, namely f k ( a , a , b ) = sup h ≤ k { ( g h ( a ) − g h ( a )) · b } .Furthermore, for every k ∈ N , let Θ k : R N × R N × S N − → [0; + ∞ ] be the functional defined byΘ k ( i, j, p ) := sup ξ ∈ S N − f k ( i · ξ, j · ξ, p · ξ ) (3.14)Clearly, Θ( i, j, p ) = sup k ∈ N Θ k ( i, j, p ) . (3.15)Since this supremum is actually a monotone limit, monotone convergence theorem gives Z J u Θ( u + , u − , ν u ) d H N − = lim k → + ∞ Z J u Θ k ( u + , u − , ν u ) d H N − On the other hand, the first part of the proof ensures that each functional R J u Θ k ( u + , u − ; ν u ) d H N − issequentially lower semicontinuous with respect to the L - strong convergence along all the sequences { u n } ∈ SBD (Ω) satisfying the bound (2.12), so that Z J u Θ( u + , u − , ν u ) d H N − ≤ lim inf k → + ∞ Z J uh Θ( u + h .u − h , ν u h ) d H N − which concludes the proof. 10 emark 3.4. We emphasize that Theorem 1.3 and Proposition 3.3 still hold with obvious adaptations if onereplaces the integrand Θ in (1.6) (or (3.13) respectively) by Θ( i, j, p ) := sup ξ ∈ S N − f ξ ( i · ξ, j · ξ, p · ξ ) with f ξ as in (1.6) (or (3.13) respectively) continuously depending on ξ ∈ S N − .It is worthwhile to observe that, looking at the proof of Lemma 3.1, Proposition 3.3 provides lower semi-continuity along sequences { u h } satisfying (2.12) also for energy densities Θ obtained via (3.13) by functions f jointly convex just on sets of the type K ′ × K ′ × R , insofar as K ′ is such that u h ( x ) ∈ K ′ for a.e. x andall h . In the sequel, taking also into account the models proposed in [6, Example 5.23], we first state theproperties inherited by the function Θ in Theorem 1.3 and then we provide some examples, essentially inthe case where f is jointly convex.We observe that Definition 1.2 easily entails that Θ has the following properties, see also [25, Proposition4.2]:(i) Θ( Oi, Oj, Op ) = Θ( i, j, p )for every orthogonal matrix O ∈ R N × N , i, j ∈ R N and p ∈ S N − .(ii) Θ( i, j, · ) is an even function, if f is even in the last variable. Moreover Θ( i, j, p ) = Θ( j, i, − p ) . (iii) If f is continuous, then Θ is continuous on R N × R N × S N − .(iv) Θ( i, j, p ) is subbaditive, in the sense that Θ( i, j, p ) ≤ Θ( i, k, p ) + Θ( k, j, p ), for every p ∈ S N − .(v) If f is bounded, Θ is also bounded on R N × R N × S N − . Examples 3.5.
Let a , a , b ∈ R , possible choices of a jointly convex function f in Definition 1.2 are thefollowing, see [6, Example 5.23]:1 f ( a , a , b ) = (cid:26) ( g ( a ) + g ( a )) | b | if a = a , if a = a and g : R → [0 , + ∞ [ is continuous2 f ( a , a , b ) = ψ ( | a − a | ) θ ( b ) with ψ lower semicontinuous, increasing and subadditive, and θ even,positively 1-homogeneous and convex. It is worthwhile also to mention that for this class of functionsjoint convexity and BV -ellipticity are equivalent as proven in [2] (see Proposition 5.1 and subsequentobservations therein).Observe that θ ( · ) = | · | recovers the result, obtained with different techniques in [25, Theorem 1.2]. Seealso Proposition 3.3 herein.3 f ( a , a , b ) = δ ( a , a ) θ ( b ) , with δ : R × R → [0 , + ∞ [ , continuous, positive, symmetric and satisfyingthe triangle inequality, (for instance δ ( a , a ) = | g ( a ) − g ( a ) | , with g : R → [0 , + ∞ [ continuous, and θ ( · ) = | · | . As an application of Theorem 1.3 and Proposition 3.3 some existence results may be proven.We emphasize that they strongly rely on some recent lower semicontinuity results for bulk energies in
SBD due to Ebobisse [19] and to Lu and Yang (see [28, Theorem 2.8 and Theorem 4.1]).
Theorem 3.6.
Let s > and let V : Ω × M N × N sym → [0 , + ∞ ) be a Carath´eodory function satisfying • for a.e. x ∈ Ω , for every ξ ∈ M N × N sym , C | ξ | s ≤ V ( x, ξ ) ≤ ρ ( x ) + C (1 + | ξ | s ) (3.16) for some constant C > and a function ρ ∈ L (Ω) , for a.e. x ∈ Ω , V ( x , · ) is symmetric quasiconvex, i.e., V ( x , ξ ) ≤ | A | Z A V ( x , ξ + E ϕ ( x )) dx for every bounded open subset A of R N , for every ϕ ∈ W , ∞ ( A ; R N ) and ξ ∈ M N × N sym .Let h ∈ L (Ω; R N ) and let { H ( x ) } x ∈ Ω be a uniformly bounded family of closed subsets of R N . Let Θ : R N × R N × S N − → [0 , + ∞ [ , be a continuous function as in (1.6). Then the constrained minimum problem min u ∈ SBD (Ω) , [ u ] · ν u ≥ a.e. in J u ,u ( x ) ∈ H ( x ) a.e. in Ω (cid:26)Z Ω V ( x, E u ) dx + H N − ( J u ) + Z J u Θ( u + , u − , ν u ) d H N − + Z Ω h · udx (cid:27) (3.17) admits a solution.Proof. The hypotheses on { H ( x ) } x ∈ Ω guarantee that every minimizing sequence { u h } is bounded in L ∞ .Therefore, by using the Direct Methods of the Calculus of Variations and, by virtue of Theorem 2.5 aboveand Theorem 1.3, [28, Theorem 2.8] we get a solution.By the same token invoking [28, Theorem 4.1] the following result can be proven Theorem 3.7.
Let V : Ω × R N × M N × N sym → [0 , + ∞ ] . Assume that for every x ∈ Ω and for every u ∈ R N and for every A ∈ M N × N sym : • V ( x, u, · ) is convex and lower semicontinuous on M N × Nsym ; • V ( · , u, A ) is measurable in Ω ; • for a.e. x ∈ Ω and for all u ∈ R N and η > there exists δ > such that V ( x, u, ξ ) − V ( x, v, ξ ) ≤ η (1 + V ( x, v, ξ )) for all v ∈ R N with | u − v | ≤ δ and for all ξ ∈ M N × N sym ; • there exist γ > , s > such that V ( x, u, ξ ) ≥ γ | ξ | s , for every x ∈ Ω and for every ( u, ξ ) ∈ R N × M N × N sym . Let h ∈ L (Ω; R N ) and let { H ( x ) } x ∈ Ω be a uniformly bounded family of closed subsets of R N . Let Θ : R N × R N × S N − → [0 , + ∞ [ , be a continuous function as in (1.6). Then the constrained minimum problem min u ∈ SBD (Ω) , [ u ] · ν u ≥ a.e. in J u ,u ( x ) ∈ H ( x ) a.e. in Ω (cid:26)Z Ω V ( x, u, E u ) dx + H N − ( J u ) + Z J u Θ( u + , u − , ν u ) d H N − + Z Ω h · udx (cid:27) (3.18) admits a solution. Other choices of the forces h , appearing in the minimum problems above, are also possible: we refer to[28]. Analogously the function Θ can be chosen as in (3.13) and it is enough to invoke Proposition 3.3. References [1]
L. Ambrosio , A Compactness Theorem for a Special Class of Functions of Bounded Variation , Boll.Un. Mat. Ital. , , (1989), 857-881.[2]
L. Ambrosio , Existence theory for a new class of variational problems , Arch. Rational Mech. Anal., , (1990), 291-322. 123]
L. Ambrosio, A. Braides , Functionals defined on partitions in sets of finite perimeter I: Integralrepresentation and Γ convergence , J. Math. Pures Appl., (1990), 285-306.[4] L. Ambrosio, A. Braides , Functionals defined on partitions in sets of finite perimeter II: semiconti-nuity, relaxation, homogenization , J. Math. Pures Appl., (1990), 307-333.[5] L. Ambrosio, A. Coscia, G. Dal Maso , Fine Properties of Functions in BD , Arch. Rational Mech.Anal., (1997), 201-238.[6]
L. Ambrosio, N. Fusco, D. Pallara , Functions of Bounded Variations and Free Discontinuity Prob-lems, Oxford Science Publication, Clarendon Press, Oxford, 2000.[7]
M. Angelillo, E. Babilio, A. Fortunato , A computational approach to quasi-static propagation ofbrittle fracture , Proceedings of the Colloquium Lagrangianum, 2002, Ravello.[8]
M. Angelillo, E. Babilio, A. Fortunato , A numerical approach to irreversible fracture as a freediscontinuity problem , Proceedings of the Colloquium Lagrangianum, 2003, Montepellier.[9]
M. Angelillo, E. Babilio , Comparing numerical solutions for the propagation of brittle fractures basedon local energy minimization with classical fracture mechanics results , Atti del XIX Convegno AIMETA,Ancona 14-17 settembre 2009[10]
G. Anzellotti , A class of convex noncoercive functionals and masonry-like materials , Annales del’Institut Henri Poincar´e. Analyse non lin´eaire, , n.4, (1985), 261-307.[11] G. I. Barenblatt , The mathematical theory of equilibrium cracks in brittle fracture , Advances inApplied Mechanics, , (1962), 55-129.[12] G. Bellettini, A. Coscia, G. Dal Maso , Compactness and Lower semicontinuity in SBD , Math.Z., ,(1998), 337-351.[13]
G. Bouchitt´e, G. Buttazzo , New Lower Semicontinuity Results for Nonconvex Functionals definedon Measures , Nonlinear Analysis, Theory, Methods and Applications, , No7, (1990), 679-692.[14] B. Bourdin, G. A. Francfort, J.-J. Marigo , The variational approach to fracture , J. Elasticity, (2008), 5148.[15] A. Chambolle , A density result in two-dimensional linearized elasticity, and applications ,Arch. Ra-tional Mech. Anal., (2003), 211233.[16]
G. Dal Maso, G. A. Francfort, R. Toader , Quasistatic crack growth in nonlinear elasticity , Arch.Ration. Mech. Anal., (2005), 165225.[17]
G. Dal Maso, R. Toader , A model for the quasi-static growth of brittle fractures: existence andapproximation results , Arch. Rational Mech. Anal., (2002), 101135.[18]
G. Dal Maso, R. Toader , A model for the quasi-static growth of brittle fractures based on localminimization , Math. Models Methods Appl. Sci., , No. 12, (2002), 1773-1799.[19] F. Ebobisse , A lower semicontinuity result for some integral functionals in the space SBD , NonlinearAnal., Theory Methods Appl. , No.7 (A), (2005), 1333-1351.[20] F. Ebobisse , On lower semicontinuity of integral functionals in LD (Ω) Ric. Mat., , No.1, (2000),65- 76.[21] H. Federer , Geometric Measure Theory, Springer-Verlag Berlin, 1969.[22]
G. A. Francfort, C. J. Larsen , Existence and convergence for quasi-static evolution in brittlefracture , Comm. Pure Appl. Math., , (2003), 1465-1500.[23] G. A. Francfort, J.-J. Marigo , Revisiting brittle fracture as an energy minimization problem , J.Mech. Phys. Solids, 46 (1998), 1319-1342. 1324]
G. Gargiulo, E. Zappale , A Lower Semicontinuity result in
SBD , J. Conv. Anal., , (2008), n.1,191-200.[25] G. Gargiulo, E. Zappale , A lower semicontinuity result in SBD for surface integral functionals ofFracture Mechanics , submitted[26]
G. Gargiulo, E. Zappale , Some sufficient conditions for lower semicontinuity in
SBD , Atti del XIXConvegno AIMETA, Ancona 14-17 settembre 2009[27]
A. Griffith , The phenomena of rupture and flows in solids , Phil. Trans. Roy. Soc. London, -A,(1920), 163-198.[28]
Zhong-Xue Lu, Xiao-Ping Yang , Existence of free discontinuity problems in
SBD (Ω), NonlinearAnalysis, , (2009), 332-340.[29] R. Temam , Probl´emes math´ematiques en plasticit´e, Paris Gauthiers-Villars, 1983.[30]
R. Temam, G. Strang , Functions of bounded deformation , Arch. Rational Mech. Anal.,75