Asymptotic Analysis of Second Order Nonlocal Cahn-Hilliard-Type Functionals
aa r X i v : . [ m a t h . A P ] S e p Asymptotic Analysis of Second Order NonlocalCahn-Hilliard-Type Functionals
Gianni Dal MasoSISSA,Via Bonomea 265, 34136 Trieste, ItalyIrene FonsecaDepartment of Mathematical Sciences,Carnegie Mellon University,Pittsburgh PA 15213-3890, USAGiovanni LeoniDepartment of Mathematical Sciences,Carnegie Mellon University,Pittsburgh PA 15213-3890, USAOctober 17, 2018
Abstract
In this paper the study of a nonlocal second order Cahn–Hilliard-typesingularly perturbed family of functions is undertaken. The kernels con-sidered include those leading to Gagliardo fractional seminorms for gradi-ents. Using Γ convergence the integral representation of the limit energyis characterized leading to an anisotropic surface energy on interfaces sep-arating different phases.
In the van der Waals–Cahn–Hilliard theory of phase transitions [15], [38], [47],[28], the total energy is given by1 ε Z Ω W ( u ( x )) dx + ε Z Ω |∇ u ( x ) | dx, (1.1)where the open bounded set Ω ⊂ R n represents a container, u : Ω → R is thefluid density, and W : R → [0 , + ∞ ) is a double-well potential vanishing only at1he phases − ε R Ω |∇ u ( x ) | dx penalizes rapid changesof the density u , and it plays the role of an interfacial energy. This problemhas been extensively studied in the last four decades (see, e.g., [8], [9], [10], [24],[34], [35], [37], [36], [44], [45]).Higher order perturbations were considered in the study of shape deforma-tion of unilamellar membranes undergoing inplane phase separation (see, e.g.,[30], [46], [31, 40]). A simplified local version of that model (see [40]) leads tothe study of a Ginzburg-Landau-type energy1 ε Z Ω W ( u ( x )) dx + qε Z Ω |∇ u ( x ) | dx + ε Z Ω (cid:12)(cid:12) ∇ u ( x ) (cid:12)(cid:12) dx , (1.2)where q ∈ R . This functional is also related to the Swift–Hohenberg equation(see [43]). When q = 0, the functional reduces to the second order version of(1.1), to be precise, 1 ε Z Ω W ( u ( x )) dx + ε Z Ω (cid:12)(cid:12) ∇ u ( x ) (cid:12)(cid:12) dx , (1.3)which was studied in [23]. The case q > |∇ u | replaced by | ∆ u | . The case q < ε R Ω |∇ u ( x ) | dx replaced by a nonlocal term, leading to the energy1 ε Z Ω W ( u ( x )) dx + ε Z Ω Z Ω J ε ( x − y ) | u ( x ) − u ( y ) | dxdy , (1.4)where J ε ( x ) := 1 ε n J (cid:16) xε (cid:17) (1.5)and the kernel J : R n → [0 , + ∞ ) is an even measurable function such that Z R n J ( x )( | x | ∧ | x | ) dx =: M J < + ∞ , (1.6)with a ∧ b := min { a, b } . Functionals of the form (1.4) arise in equilibriumstatistical mechanics as free energies of continuum limits of Ising spin systemson lattices. In that setting, u is a macroscopic magnetization density and J stands for a ferromagnetic Kac potential (see [3]). Note that (1.6) is satisfied if J is integrable and has compact support. Another important case is when J ( x ) = | x | − n − s with 12 < s < , (1.7)so that J ε ( x ) = ε s | x | − n − s , which leads to Gagliardo’s seminorm for the frac-tional Sobolev space H s ( R n ) (see [20], [25] [32]). A functional related to (1.4)2ith kernel (1.7) has been studied in [4], [5], and [39] for 0 < s < L p version in dimension n = 1).The motivation in [39] was the renewed interest in the fractional Laplacian(see, e.g., [14] and the references therein), and nonlocal characterizations offractional Sobolev spaces ([6], [11], [12], [33] and the references therein).Another important application of this type of nonlocal singular perturba-tion functionals is in the study of dislocations in elastic materials exhibitingmicrostructure (see, e.g., [13], [18], [26]).In this paper we consider a nonlocal version of (1.3), to be precise, we studythe functional F ε ( u ) := 1 ε Z Ω W ( u ( x )) dx + ε Z Ω Z Ω J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dxdy (1.8)for u ∈ W , (Ω), where Ω ⊂ R n , n ≥
2, is a bounded open set with Lipschitzboundary, the double-well potential W : R → [0 , + ∞ ) is a continuous functionwith W − ( { } ) = {− , +1 } satisfying appropriate coercivity and growth con-ditions, and J ε is given by (1.5). We assume a non-degeneracy hypothesis (see(2.2)) on the even measurable kernel J : R n → [0 , + ∞ ), and that (1.6) holds.We establish compactness in L (Ω) for energy bounded sequences, and inorder to study the asymptotic behavior of (1.8) as ε → + , we use the notion ofΓ-convergence (see [19]) with respect to the metric in L (Ω) and we identify theΓ-limit of F ε . As it is usual, we extend F ε ( u ) to be + ∞ for u ∈ L (Ω) \ W , (Ω).Our first main result is the following theorem. Theorem 1.1 (Compactness)
Assume that W and J satisfy (2.3)–(2.6) and(1.6), (2.2), respectively. Let { u ε } ⊂ W , (Ω) ∩ L (Ω) be such that M := sup ε F ε ( u ε ) < + ∞ . (1.9) Then there exists a sequence ε j → + such that { u ε j } converges in L (Ω) tosome function u ∈ BV (Ω; {− , } ) . The proof of this theorem is more involved than the corresponding one in[2] due to the presence of gradients in the nonlocal term. This prevents us fromusing standard arguments in which discontinuities in u may be allowed. Wefirst prove compactness in n = 1, and then use a slicing technique to treat thehigher dimensional case.To state the Γ convergence result, we need to introduce some notation. Given n ≥ ν ∈ S n − := ∂B (0), let ν , . . . , ν n be an orthonormal basis in R n with ν n = ν . Here, and in what follows, we denote by B r ( x ) the open ball in R n centered at x and with radius r . Let V ν := { x ∈ R n : | x · ν i | < / i = 1 , . . . , n − } , (1.10) Q ν := { x ∈ R n : | x · ν i | < / i = 1 , . . . , n } , (1.11)3et W , ν ,...,ν n − be the set of all functions v ∈ W , ( R n ) such that v ( x + ν i ) = v ( x )for a.e. x ∈ R n and for every i = 1 , . . . , n −
1, and let X ν := { v ∈ W , ν ,...,ν n − : v ( x ) = ± x ∈ R n with ± x · ν ≥ / } (1.12)When n = 1 take ν = ± V ν := R , Q ν := ( − / , / X ν be the spaceof all functions v ∈ W , ( R ) such that v ( x ) = ± x ∈ R with ± x ≥ / ψ ( ν ) := inf <ε< inf v ∈ X ν F νε ( v ) , (1.13)where F νε ( u ) := 1 ε Z Q ν W ( u ( x )) dx + ε Z V ν Z R n J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dxdy . Finally, we define F : L (Ω) → [0 , + ∞ ] by F ( u ) := Z S u ψ ( ν u ) d H n − if u ∈ BV (Ω; {− , } ) , + ∞ otherwise in L (Ω) , (1.14)where S u is the jump set of u , ν u is the approximate normal to S u , and H n − is the ( n − Theorem 1.2 ( Γ -Limit) Assume that W and J satisfy (2.2)–(2.6) and (1.6),respectively. Then for every ε j → + the sequence {F ε j } Γ -converges to F in L (Ω) . Although the general structure of the proof is standard, there are remarkabletechnical difficulties due to the nonlocality of the perturbation and the presenceof gradients.This paper is organized as follows. After a brief section on preliminaries, onSection 3 in order to establish compactness in dimension n = 1, we prove aninterpolation result, which allows us to control the L norm of u ′ in terms ofthe full energy (see Lemma 3.5). Section 4 is devoted to compactness in higherdimensions, and here again we obtain the equivalent to the interpolation Lemma3.5 (see Lemma 4.3). As it is classical in this type of problems, it is importantto be able to modify admissible sequences near the boundary of their domainwithout increasing the limit energy. We address this in Theorem 5.1 in Section5. Section 6 concerns the Γ-liminf inequality, and in Section 7 we construct therecovery sequence for the Γ-limsup inequality. In what follows, in addition to (1.6) we also assume that the kernel J : R n → [0 , + ∞ ) has the following property: there exist γ J > δ J ∈ (0 , c J >
0, such4hat for all ξ ∈ S n − there are α ( ξ ) < β ( ξ ) satisfying − γ J ≤ α ( ξ ) ≤ α ( ξ ) + δ J ≤ β ( ξ ) ≤ γ J (2.1)and Z β ( ξ ) α ( ξ ) J ( tξ ) | t | n − dt ≤ c J . (2.2) Remark 2.1
For example, condition (2.2) holds if there exist < r < R and a > such that J ( x ) ≥ a for every x ∈ R n with r < | x | < R . Indeed, it is enoughto set γ J = R , δ J = R − r , α ( ξ ) = r , β ( ξ ) = R , and c J = ( na ) − ( r − n − R − n ) . We assume that the double-well potential is a continuous function W : R → [0 , + ∞ ) such that W − ( { } ) = {− , } , (2.3)( | s | − ≤ c W W ( s ) for all s ∈ R , (2.4) W is increasing on [1 , + ∞ ) and on [ − , − a W ] , (2.5) W is decreasing on ( −∞ , −
1] and on [1 − a W , , (2.6)for some constants c W > a W ∈ (0 , s ≤ | s + 1 | ≥ , then | s − | = | s | − s − ≤ | s | − + 4 ≤ c W W ( s ) + m W W ( s ), where m W := min {|| s |− |≥ } W ( s ) > . (2.7)Together with (2.4) this leads to the estimate( s − ≤ ˆ c W W ( s ) for all s ∈ R with | s + 1 | ≥ , (2.8)where ˆ c W := 2 c W + m W . Similarly, it can be shown that( s + 1) ≤ ˆ c W W ( s ) for all s ∈ R with | s − | ≥ . (2.9)We recall that Ω ⊂ R n is a bounded open set with Lipschitz boundary. Forevery ε > u ∈ L (Ω) consider the functional F ε ( u ) := (cid:26) W ε ( u ) + J ε ( u ) if u ∈ W , (Ω) ∩ L (Ω) , + ∞ otherwise, (2.10)where W ε ( u ) := 1 ε Z Ω W ( u ( x )) dx for u ∈ L (Ω) , (2.11)and J ε ( u ) := ε Z Ω Z Ω J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dxdy for u ∈ W , (Ω) . (2.12)5n the sequel, we will use a localized version of (2.10). To be precise, giventwo open sets A , B ⊂ R n we define W ε ( u, A ) := 1 ε Z A W ( u ( x )) dx (2.13)for u ∈ L ( A ), and J ε ( u, A, B ) := ε Z A Z B J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dxdy (2.14)for u ∈ W , ( A ∪ B ). When A = B we set F ε ( u, A ) := W ε ( u, A ) + J ε ( u, A, A ) and J ε ( u, A ) := J ε ( u, A, A ) (2.15)for u ∈ W , ( A ) ∩ L ( A ).Since J is even, by Fubini’s theorem for all u ∈ W , ( A ∪ B ) we have that J ε ( u, A, B ) = J ε ( u, B, A ) . (2.16)Moreover, if A ∩ B = Ø we have J ε ( u, A ∪ B ) = J ε ( u, A ) + 2 J ε ( u, A, B ) + J ε ( u, B ) . (2.17)In the compactness theorem we use a slicing argument based on the followingpreliminary result. Given a vector ξ ∈ S n − , the hyperplane through the originorthogonal to ξ is denoted by Π ξ , that is,Π ξ := { x ∈ R n : x · ξ = 0 } . (2.18)If E ⊂ R n and y ∈ Π ξ , then we define E ξy := { t ∈ R : y + tξ ∈ E } . (2.19)The next result is a particular case of the affine Blaschke–Petkantschin for-mula, for which we refer to [41, Theorem 7.2.7]. Proposition 2.2
Let E ⊂ R n be a Borel set and let g : E × E → [0 , + ∞ ] be aBorel function. Then Z E Z E g ( x, y ) dxdy = 12 Z S n − Z Π ξ Z E ξz Z E ξz g ( z + sξ, z + tξ ) | t − s | n − dsdtd H n − ( z ) d H n − ( ξ ) . Proof.
For the convenience of the reader we present a proof. We extend g tobe zero outside E × E . Using the change of variables τ = t − s , we obtain Z R g ( z + sξ, z + tξ ) | t − s | n − ds = Z R g ( z + tξ − τ ξ, z + tξ ) | τ | n − dτ , Z Π ξ Z R Z R g ( z + sξ, z + tξ ) | t − s | n − dsdtd H n − ( z )= Z R n Z R g ( y − τ ξ, y ) | τ | n − dτ dy . Exchanging the order of integration and using integration in spherical coordi-nates we have12 Z S n − Z Π ξ Z R Z R g ( z + sξ, z + tξ ) | t − s | n − dsdtd H n − ( z ) d H n − ( ξ )= 12 Z R n Z S n − Z R g ( y − τ ξ, y ) | τ | n − dτ d H n − ( ξ ) dy = Z R n Z R n g ( x, y ) dxdy , which concludes the proof.For ξ ∈ S n − and ε > J ξ : R → [0 , + ∞ ) by J ξ ( t ) := J ( tξ ) | t | n − and J ξε ( t ) := 1 ε J ξ (cid:18) tε (cid:19) . (2.20)By (1.6) and using spherical coordinates, we have Z R J ξ ( t )( | t | ∧ | t | ) dt < + ∞ (2.21)for H n − -a.e. ξ ∈ S n − , and in view of (2.2) we obtain Z β ( ξ ) α ( ξ ) J ξ ( t ) dt ≤ c J . (2.22)Moreover, J ξε ( t ) = 1 ε J ξ (cid:18) tε (cid:19) = 1 ε J (cid:18) tξε (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) tε (cid:12)(cid:12)(cid:12)(cid:12) n − = J ε ( tξ ) | t | n − . (2.23)For ξ ∈ S n − , A ⊂ R , and ε >
0, we define F ξε ( v, A ) := 1 σ n − ε Z A W ( v ( t )) dt + ε Z A Z A J ξε ( s − t )( v ′ ( s ) − v ′ ( t )) dsdt (2.24)for v ∈ W , ( A ) ∩ L ( A ), where σ n − := H n − ( S n − ).7 Compactness and interpolation in dimensionone
For a set A contained in R n and for η > A ) η := { x ∈ R n : dist( x, A ) < η } , ( A ) η := { x ∈ A : dist( x, ∂A ) > η } . (3.1)The main result of this section is the following theorem. Theorem 3.1
Let ξ ∈ S n − , let A ⊂ R be a bounded open set, and let { u ε } ⊂ W , ( A ) ∩ L ( A ) be such that M := sup ε F ξε ( u ε , A ) < + ∞ , (3.2) where F ξε is defined in (2.24). Then there exists a sequence ε j → + such that { u ε j } converges in L ( A ) to some function u ∈ BV ( A ; {− , } ) . Moreover,there exists a constant c J,W > , independent of ξ , A , and { u ε } , such that S u ≤ Mc J,W , (3.3) where S u denotes the number of jump points of u . Next we introduce some auxiliary lemmas that will be used in the proof ofTheorem 3.1.
Lemma 3.2
Let ξ ∈ S n − , let A ⊂ R be an open set, let ε > , let α < β , andlet u ∈ W , (( A ) εγ J ) , where γ J is the constant in (2.1). Then for a.e. t ∈ A , ε Z t − εαt − εβ J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) ds ≥ ε ( β − α ) Z βα J ξ ( z ) dz ! − (cid:18) u ′ ( t ) − u ( t − εα ) − u ( t − εβ ) ε ( β − α ) (cid:19) , (3.4) where J ξ and J ξε are defined in (2.20). Proof.
It is enough to show that for every λ ∈ R we have ε Z t − εαt − εβ J ξε ( t − s )( λ − u ′ ( s )) ds ≥ ε ( β − α ) Z βα J ξ ( z ) dz ! − (cid:18) λ − u ( t − εα ) − u ( t − εβ ) ε ( β − α ) (cid:19) . This inequality follows by considering the Euler–Lagrange equation of the min-imum problem min Z t − εαt − εβ J ξε ( t − s )( λ − v ′ ( s )) ds v ∈ W , (( t − εβ, t − εα )) satisfying v ( t − εβ ) = u ( t − εβ ) and v ( t − εα ) = u ( t − εα ). Remark 3.3
Under the same assumptions of Lemma 3.2, it follows from (2.1),(2.2), and (3.4) that ε ( u ′ ( t )) ≤ δ J ε (cid:0) u ( t − εα ( ξ )) − u ( t − εβ ( ξ )) (cid:1) + 2 c J ε Z t + εγ J t − εγ J J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) ds for a.e. t ∈ A . Lemma 3.4
Let γ J be the constant in (2.1). Then there exists a constant c J,W > such that ε Z τσ Z τ + εγ J σ − εγ J J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) dsdt + 1 ε Z τ + εγ J σ − εγ J W ( u ( t )) dt ≥ c J,W (3.5) for every ξ ∈ S n − , for every ε > , for every σ , τ , with σ < τ , and for every u ∈ W , (( σ − εγ J , τ + εγ J )) such that u ( t ) ∈ (cid:0) − , (cid:1) for every t ∈ ( σ, τ ) , (3.6) and either u ( σ ) = − and u ( τ ) = (3.7) or u ( σ ) = and u ( τ ) = − . (3.8) Proof.
Fix ξ , ε , σ , τ , and u as in the statement of the lemma, and let ˆ α and ˆ β be such that α ( ξ ) < ˆ α < ˆ β < β ( ξ ), and α ( ξ ) − ˆ α > δ J , ˆ β − ˆ α > δ J , β ( ξ ) − ˆ β > δ J , (3.9)where δ J is the constant in (2.1). By (2.4) and (3.6), we have W ( u ( t )) ≥ C W for every t ∈ ( σ, τ ). Therefore, if τ − σ > εδ J / , then1 ε Z τσ W ( u ε ( t )) dt > δ J C W . (3.10)If τ − σ ≤ εδ J / , define A := (cid:26) t ∈ ( σ, τ ) : | u ′ ( t ) | ≥
12 1 τ − σ (cid:27) . (3.11)We consider now two cases. 9 ase 1: Assume that for every t ∈ A there exist α ∈ [ α ( ξ ) , ˆ α ] and β ∈ [ ˆ β, β ( ξ )]such that | u ( t − εα ) − u ( t − εβ ) | ε ( β − α ) < | u ′ ( t ) | . Then (cid:18) u ′ ( t ) − u ( t − εα ) − u ( t − εβ ) ε ( β − α ) (cid:19) ≥
14 ( u ′ ( t )) . Therefore, by Lemma 3.2, ε Z t − εαt − εβ J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) ds ≥ ε ( β − α ) Z βα J ξ ( z ) dz ! − ( u ′ ( t )) , and integrating over A , using (2.22) and (3.9), we obtain ε Z A Z t − εα ( ξ ) t − εβ ( ξ ) J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) dsdt ≥ εδ J c J Z A ( u ′ ( t )) dt . (3.12)By (3.7), (3.8), and (3.11) using Jensen’s inequality and τ − σ ≤ δ J ε , we have Z A ( u ′ ( t )) dt = Z τσ ( u ′ ( t )) dt − Z ( σ,τ ) \ A ( u ′ ( t )) dt ≥ τ − σ −
14 1 τ − σ ≥ · εδ J . Hence, from (3.12) we deduce that ε Z τσ Z τ − εα ( ξ ) σ − εβ ( ξ ) J ξε ( t − s )( u ′ ( t ) − u ′ ( s )) dsdt ≥ δ J c J . (3.13) Case 2:
It remains to study the case in which there exists t ∈ A such that | u ( t − εα ) − u ( t − εβ ) | ε ( β − α ) ≥ | u ′ ε ( t ) | for every α ∈ [ α ( ξ ) , ˆ α ] and for every β ∈ [ ˆ β, β ( ξ )]. By (3.11) and the inequality τ − σ ≤ εδ J / , we have | u ( t − εα ) − u ( t − εβ ) | ε ( β − α ) ≥ τ − σ ) ≥ εδ J , hence by (3.9), | u ( t − εα ) − u ( t − εβ ) | ≥
16( ˆ β − ˆ α ) δ J ≥ . | u ( t − εα ) | ≥ α ∈ [ α ( ξ ) , ˆ α ], then by (2.4) we have W ( u ( t − εα )) ≥ c W for every α ∈ [ α ( ξ ) , ˆ α ]. This leads to W ( u ( t )) ≥ c W for every t ∈ [ t − ε ˆ α, t − εα ( ξ )], hence1 ε Z τ + εγ J σ − εγ J W ( u ( t )) dt ≥ ε Z t − εα ( ξ ) t − ε ˆ α W ( u ( t )) dt ≥ ˆ α − α ( ξ ) c W ≥ δ J c W , (3.14)where in the last inequality we used (3.9).If there exists α ∈ [ α ( ξ ) , ˆ α ] such that | u ( t − εα ) | <
2, then | u ( t − εβ ) | > β ∈ [ ˆ β, β J ] (if not, there exists β ∈ [ ˆ β, β ( ξ )] such that | u ( t − εβ ) | ≤ | u ( t − εα ) − u ( t − εβ ) | <
4, a contradiction). Consequently, forevery β ∈ [ ˆ β, β ( ξ )] we have W ( u ( t − εβ )) ≥ c W . This leads to W ( u ( t )) ≥ c W for every t ∈ [ t − εβ ( ξ ) , t − ε ˆ β ], hence1 ε Z τ + εγ J σ − εγ J W ( u ( t )) dt ≥ ε Z t − ε ˆ βt − εβ ( ξ ) W ( u ( t )) dt ≥ β ( ξ ) − ˆ βc W ≥ δ J c W , (3.15)where in the last inequality we used (3.9). The conclusion follows now from(3.10), (3.13), (3.14), and (3.15). Lemma 3.5 (Interpolation inequality in dimension one)
There exists aconstant c (1) J,W such that ε Z A ( u ′ ( t )) dt ≤ c (1) J,W F ξε ( u, ( A ) εγ J ) . (3.16) for every ξ ∈ S n − , for every ε > , for every open set A ⊂ R , and for every u ∈ W , (( A ) εγ J ) , where γ J is the constant in (2.1). Proof.
Fix ξ , ε , A , and u as in the statement of the lemma, and define U := { t ∈ A : u ( t − εα ( ξ )) , u ( t − εβ ( ξ )) / ∈ [ , ] } ,V := { t ∈ A : u ( t − εα ( ξ )) , u ( t − εβ ( ξ )) / ∈ [ − , − ] } . (3.17)If t ∈ V , then by (2.8),( u ( t − εα ( ξ )) − u ( t − εβ ( ξ ))) ≤ u ( t − εα ( ξ )) − + 2( u ( t − εβ ( ξ )) − ≤ c W (cid:0) W ( u ( t − εα ( ξ ))) + W ( u ( t − εβ ( ξ ))) (cid:1) . Using (2.9) we prove the same inequality for t ∈ U . Integrating and usingRemark 3.3, we obtain ε Z U ∪ V ( u ′ ( t )) dt ≤ (cid:0) c W δ J + 2 c J (cid:1) F ξε ( u, ( A ) εγ J ) . (3.18)If t ∈ A \ ( U ∪ V ), then either u ( t − εα ( ξ )) ∈ [ − , − ] and u ( t − εβ ( ξ )) ∈ [ , ]11r u ( t − εβ ( ξ )) ∈ [ − , − ] and u ( t − εα ( ξ )) ∈ [ , ] . Then ( u ( t − εα ( ξ )) − u ( t − εβ ( ξ ))) ≤ . (3.19)Moreover there exist σ and τ , satisfying t − εγ J ≤ t − εβ ( ξ ) ≤ σ < τ ≤ t − εα ( ξ ) ≤ t + εγ J (3.20)and such that u ( t ) ∈ (cid:0) − , (cid:1) for every t ∈ ( σ, τ )and either u ( σ ) = and u ( τ ) = − or u ( σ ) = − and u ( τ ) = . By Lemma 3.4 and by (3.20), there exists c J,W > c J,W ≤ ε Z t + εγ J t − εγ J Z t +2 εγ J t − εγ J J ξε ( r − s )( u ′ ε ( r ) − u ′ ε ( s )) dsdr + 1 ε Z t +2 εγ J t − εγ J W ( u ε ( r )) dr . Therefore by (3.19) we have1 ε Z A \ ( U ∪ V ) ( u ( t − εα ( ξ )) − u ( t − εβ ( ξ ))) dt ≤ c J,W Z A Z t + εγ J t − εγ J Z t +2 εγ J t − εγ J J ξε ( r − s )( u ′ ε ( r ) − u ′ ε ( s )) dsdrdt (3.21)+ 9 c J,W ε Z A Z t +2 εγ J t − εγ J W ( u ε ( r )) drdt . Since 12 η Z A Z t + ηt − η f ( r ) drdt ≤ Z ( A ) η f ( t ) dt for every η > f : A → [0 , + ∞ ], from (3.21)we obtain1 ε Z A \ ( U ∪ V ) ( u ( t − εα ( ξ )) − u ( t − εβ ( ξ ))) dt ≤ ˜ c J,W F ξε ( u, ( A ) εγ J ) . (3.22)for a suitable constant ˜ c J,W depending only on J and W . The conclusion followsfrom (3.18) and (3.22) using Remark 3.3. Proof of Theorem 3.1.
By (3.2) we have that Z A W ( u ε ( t )) dt ≤ M ε . (3.23)12y (2.3) and (2.4) this implies that { u ε } converges to 1 in L ( A ) and, up to asubsequence (not relabeled) pointwise a.e. in A .Let γ J > I ε ofall intervals ( σ − εγ J , y ε + εγ J ) such that ( σ, τ ) is contained in ( A ) εγ J , and u ε satisfies (3.6) and either (3.7) or (3.8) in ( σ, τ ). Note that by the intermediatevalue theorem for all ε > I ε are contained in A . It follows from (2.4) and(3.23) that M ε ≥ Z τσ W ( u ε ( t )) dt ≥ τ − σ c W , hence τ − σ ≤ c W M ε . (3.24)In particular, for every I ∈ I ε we havediam I ≤ (4 c W M + 2 γ J ) ε . (3.25)Moreover, by (3.2) and (3.5), if I , . . . , I k are pairwise disjoint intervals in I ε ,then k ≤ Mc J,W . (3.26)Let B ε be the union of all intervals in I ε and let C ε be the collection ofits connected components. Observe that distinct elements of C ε must containdisjoint intervals of I ε , and so by (3.26) the number of elements of C ε is uniformlybounded. To be precise, C ε ≤ Mc J,W . (3.27)Next we claim that if C ∈ C ε , thendiam C ≤ C W M + 2 γ J ) (cid:18) Mc J,W + 1 (cid:19) ε . (3.28)Assume by contradiction that (3.28) fails. Let k be the integer such that Mc J,W
Let { u ε } ⊂ L (Ω) be such that M := sup ε W ε ( u ε ) < + ∞ . (4.2) Then u ε − u (1) ε → strongly in L (Ω) . Proof.
By (2.11) and (4.2) we have that Z Ω W ( u ε ( x )) dx → ε → + . By (2.3) and (2.4) this implies that, up to a subsequence, | u ε ( x ) | → x ∈ Ω. Hence, u ε ( x ) − u (1) ε ( x ) → x ∈ Ω. On the other hand,by (2.4), ( u ε ( x ) − u (1) ε ( x )) ≤ ( u ε ( x )) ≤ c W W ( u ε ( x )) + 2 , so that the conclusion follows from (4.2) and the (generalized) Lebesgue domi-nated convergence theorem.In what follows, given a Borel set E ⊂ R n and a function u : E → R , forevery ξ ∈ S n − and for every y ∈ Π ξ (see (2.18)) we define the one-dimensionalfunction u ξy ( t ) := u ( y + tξ ) , t ∈ E ξy , (4.4)where E ξy is defined in (2.19). 14 emma 4.2 For every A ⊂ R n open, ε > , and u ∈ W , ( A ) ∩ L ( A ) , wehave F ε ( u, A ) ≥ Z S n − Z Π ξ F ξε ( u ξz , A ξz ) d H n − ( z ) d H n − ( ξ ) . Proof.
By Fubini’s theorem, Proposition 2.2, (2.15), (2.23), and (2.24), weobtain F ε ( u, A )= 1 σ n − ε Z S n − Z Π ξ Z A ξz W ( u ( z + tξ )) dtd H n − ( z ) d H n − ( ξ )+ ε Z S n − Z Π ξ Z A ξz Z A ξz J ξε ( t − s ) |∇ u ( z + tξ ) −∇ u ( z + sξ ) | dtdsd H n − ( z ) d H n − ( ξ ) ≥ σ n − ε Z S n − Z Π ξ Z A ξz W ( u ξz ( t )) dtd H n − ( z ) d H n − ( ξ )+ ε Z S n − Z Π ξ Z A ξz Z A ξz J ξε ( t − s )(( u ξz ) ′ ( t ) − ( u ξz ) ′ ( s )) dtdsd H n − ( z ) d H n − ( ξ )= Z S n − Z Π ξ F ξε ( u ξz , A ξz ) d H n − ( z ) d H n − ( ξ ) . Proof of Theorem 1.1.
Let ε j → + and, for simplicity, write u j := u ε j . ByLemma 4.2, Z S n − Z Π ξ F ξε j (( u j ) ξz , Ω ξz ) d H n − ( z ) d H n − ( ξ ) ≤ M . (4.5)We claim that there exist a collection ξ , . . . , ξ n ∈ S n − of linearly independentvectors and a subsequence (not relabeled) such thatlim j → + ∞ Z Π ξi F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) d H n − ( z ) =: M i < + ∞ , (4.6)for every i = 1, . . . , n .Indeed, using Fatou’s lemma by (4.5) we have that Z S n − lim inf j → + ∞ Z Π ξ F ξε j (( u j ) ξz , Ω ξz ) d H n − ( z ) d H n − ( ξ ) ≤ M . (4.7)Hence, there exists ξ ∈ S n − such thatlim inf j → + ∞ Z Π ξ F ξ ε j (( u j ) ξ z , Ω ξ z ) d H n − ( z ) =: M < + ∞ , (4.8)and we can extract a subsequence (not relabeled) such that (4.6) holds for i = 1.We proceed by induction. Assume that we found a collection ξ , . . . , ξ k ∈ S n − , 1 ≤ k < n , of linearly independent vectors and a subsequence (not rela-beled) such that (4.6) holds for every i = 1, . . . , k . Note that this subsequence15till satisfies (4.5), and hence (4.7). Therefore we can find ξ k +1 ∈ S n − , linearlyindependent of ξ , . . . , ξ k , such thatlim inf j → + ∞ Z Π ξk +1 F ξ k +1 ε j (( u j ) ξ k +1 z , Ω ξ k +1 z ) d H n − ( z ) =: M k +1 < + ∞ , and we can extract a subsequence (not relabeled) such that (4.6) holds also for i = k + 1. After n steps we obtain that (4.6) is satisfied for every i = 1, . . . , n .Given i = 1, . . . , n and δ >
0, for every j let A ij := n z ∈ Π ξ i : F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) > M i δ o , (4.9)and let v ij ∈ L (Ω) be defined by ( ( v ij ) ξ i z := ( u (1) j ) ξ i z if z ∈ Π ξ i \ A j , ( v ij ) ξ i z := 0 if z ∈ A j , (4.10)where u (1) j is the truncated function defined using (4.1). By (4.6) and (4.9) wehave lim sup j → + ∞ H n − ( A ij ) ≤ δ , hence (4.10) yields lim sup j → + ∞ k v ij − u (1) j k L (Ω) ≤ δ diam(Ω) . (4.11)By Theorem 3.1 for every z ∈ Π ξ i the set { ( u j ) ξ i z (1 − χ A ij ( z )) : j ∈ N } isrelatively compact in L (Ω ξ i z ), where χ A ij ( z ) = 1 for z ∈ A ij and χ A ij ( z ) = 0 for z A ij . Therefore the same property holds for the set of truncated functions { ( u (1) j ) ξ i z (1 − χ A ij ( z )) : j ∈ N } . It follows that for every z ∈ Π ξ i the set { ( v ij ) ξ i z : j ∈ N } is relatively compact in L (Ω ξ i z ). Since this property is valid for every i = 1, . . . , n , we can apply the characterization by slicing of precompact setsof L (Ω) given by [5, Theorem 6.6] and we obtain that the set { u (1) j : j ∈ N } is relatively compact in L (Ω). In turn, by Lemma 4.1 the set { u j : j ∈ N } isrelatively compact in L (Ω), hence there exist a subsequence (not relabeled) ,such that u j converges in L (Ω) to some function u . By (1.9),lim j → + ∞ Z Ω W ( u j ( x )) dx = 0 , which, together with (2.3) and (2.4), implies that u ( x ) ∈ {− , } for a.e. x ∈ Ω.It remains to show that u ∈ BV (Ω). Using Fubini’s theorem we find thatthere exists a subsequence (not relabeled) such that( u j ) ξ i z → u ξ i z in L (Ω ξ i z ) . (4.12)16oreover, Fatou’s lemma and (4.6) imply that Z Π ξi lim inf j → + ∞ F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) d H n − ( z ) ≤ M i , (4.13)hence lim inf j → + ∞ F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) < + ∞ (4.14)for H n − -a.e. z ∈ Π ξ i . Fix z ∈ Π ξ i satisfying (4.12) and (4.14), and extract asubsequence { ˆ u j } , depending on z , such thatlim j → + ∞ F ξ i ε j ((ˆ u j ) ξ i z , Ω ξ i z ) = lim inf j → + ∞ F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) . (4.15)By (3.3), (4.12), and (4.15) we have S u ξiz ≤ c J,W lim inf j → + ∞ F ξ i ε j (( u j ) ξ i z , Ω ξ i z ) . Since u ξ i z ( t ) ∈ {− , } for a.e. t ∈ Ω ξ i z , we deduce that | Du ξ i z | (Ω ξ i z ) ≤ c J,W lim inf j → + ∞ F ξ i ε j (( u j ) ξ i z , Ω ξ i z )for H n − -a.e. z ∈ Π ξ i . This property holds for every i = 1, . . . , n . Therefore, wecan apply the characterization by slicing of BV functions given by [7, Remark3.104] and we obtain from (4.13) that u ∈ BV (Ω).For A ⊂ R n and η > Lemma 4.3 (Interpolation inequality)
There exists a constant c ( n ) J,W suchthat ε Z A |∇ u ( x ) | dx ≤ c ( n ) J,W F ε ( u, ( A ) εγ J ) . (4.16) for every ε > , for every open set A ⊂ R n , and for every u ∈ W , (( A ) εγ J ) ,where γ J is the constant in (2.1). Proof.
Fix ε , A , and u as in the statement of the lemma, and define B :=( A ) εγ J . Given ξ ∈ S n − , for H n − a.e. z ∈ Π ξ we have that ( A ξz ) εγ J ⊂ B ξz and the sliced function u ξz (see (4.4)) belongs to W , ( B ξz ). Hence by Lemma3.5 we have ε Z A ξz (( u ξz ) ′ ( t )) dt ≤ c (1) J,W F ξε ( u ξz , B ξz ) . Integrating this inequality in z over Π ξ we obtain ε Z A ( ∇ u ( x ) · ξ ) dx ≤ c (1) J,W Z Π ξ F ξε ( u ξz , B ξz ) d H n − ( z ) . Integrating this inequality in ξ over S n − and using Lemma 4.2, together withthe identity R S n − | a · ξ | d H n − ( ξ ) = ω n | a | , we deduce ω n ε Z A |∇ u ( x ) | dx ≤ c (1) J,W F ε ( u, B ) . This concludes the proof. 17
The modification theorem
In this section we prove that we can modify an admissible sequence to match amollification of its limit in a neighborhood of the boundary, without increasingthe limit energy.Given ν ∈ S n − , let w ν ( x ) := (cid:26) x · ν > , − x · ν < . (5.1)When ν = e n , the superscript ν is omitted. Let θ ∈ C ∞ c ( R n ) be such thatsupp θ ⊂ B (0), R R n θ ( x ) dx = 1, and for every σ > θ σ ( x ) := 1 σ n θ (cid:16) xσ (cid:17) , x ∈ R n . (5.2)Note that supp θ σ ⊂ B σ (0). There exists a constant C θ >
1, independent of σ ,such thatsup R n | ( w ν ∗ θ σ ) − w ν | ≤ , (5.3)( w ν ∗ θ σ ) ( x ) = 1 if x · ν > σ, ( w ν ∗ θ σ ) ( x ) = − x · ν < − σ , (5.4) ∇ ( w ν ∗ θ σ ) ( x ) = 0 if | x · ν | > σ , (5.5)sup R n |∇ ( w ν ∗ θ σ ) | ≤ C θ σ and sup R n |∇ ( w ν ∗ θ σ ) | ≤ C θ σ . (5.6)Let P be a bounded polyhedron of dimension n − ν ∈ S n − be a normal to P . For every ρ > P ρ := { x + tν : x ∈ P , t ∈ ( − ρ/ , ρ/ } . (5.7) Theorem 5.1 (Modification Theorem)
Let P be a bounded polyhedron ofdimension n − containing , let ρ > , let ε j → + , and let { u j } be a sequencein W , ( P ρ ) ∩ L ( P ρ ) such that u j → w ν in L ( P ρ ) . Then there exists a constant δ P ρ > depending only on P ρ such that for every < δ < δ P ρ there exists asequence { v j } ⊂ W , ( P ρ ) ∩ L ( P ρ ) such that v j → w ν in L ( P ρ ) , v j = u j in ( P ρ ) δ , v j = w ν ∗ θ ε j on P ρ \ ( P ρ ) δ , and lim sup j → + ∞ F ε j ( v j , P ρ ) ≤ lim sup j → + ∞ F ε j ( u j , P ρ ) + κ δ , (5.8) where κ > is a constant independent of j , δ , and P ρ . Remark 5.2
By choosing a suitable subsequence, under the same assumptionsof Theorem 5.1 we obtain that lim inf j → + ∞ F ε j ( v j , P ρ ) ≤ lim inf j → + ∞ F ε j ( u j , P ρ ) + κ δ . (5.9)18o prove Theorem 5.1 we use the estimate of the following lemma. Lemma 5.3
Let ε > , let y ∈ R n , let A be a measurable subset of R n , and let g : A → R be a measurable function such that ≤ g ( x ) ≤ ( a | x − y | ) ∧ b for every x ∈ A , (5.10) for some constants a and b . Then Z A J ε ( x − y ) g ( x ) dx ≤ M J (cid:0) ( εa ) ∨ b (cid:1) , (5.11) where M J is the constant given in (1.6) and α ∨ β := max { α, β } . Proof.
Using (1.5) and the change of variables z = ( x − y ) /ε , we obtain Z A J ε ( x − y ) g ( x ) dx ≤ a Z A ∩ B ε ( y ) J ε ( x − y ) | x − y | dx + b Z A \ B ε ( y ) J ε ( x − y ) | x − y | ε dx ≤ ε a Z B (0) J ( z ) | z | dz + b Z R n \ B (0) J ( z ) | z | dz . The conclusion follows from (1.6).
Lemma 5.4
Let < ε < δ , let A and B be open sets in R n , with dist( A, B ) ≥ δ ,and let u ∈ W , ( A ∪ B ) . Then J ε ( u, A, B ) ≤ εω (cid:16) εδ (cid:17) Z A ∪ B |∇ u ( x ) | dx , (5.12) where ω ( t ) := 2 Z R n \ B /t (0) J ( z ) | z | dz → as t → + . Proof.
Using a change of variables we obtain J ε ( u, A, B ) = ε Z A Z B J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dxdy ≤ ε Z B (cid:16) Z A J ε ( x − y ) dy (cid:17) |∇ u ( x ) | dx + 2 ε Z A (cid:16) Z B J ε ( x − y ) dx (cid:17) |∇ u ( y ) | dy ≤ ε Z B (cid:16) Z R n \ B δ ( x ) J ε ( x − y ) dy (cid:17) |∇ u ( x ) | dx + 2 ε Z A (cid:16) Z R n \ B δ ( y ) J ε ( x − y ) dx (cid:17) |∇ u ( y ) | dy ε Z R n \ B δε (0) J ( z ) dz Z A ∪ B |∇ u ( x ) | dx ≤ ε Z R n \ B δε (0) J ( z ) | z | dz Z A ∪ B |∇ u ( x ) | dx . This leads to (5.12). The fact that ω ( t ) → + as t → + follows from (1.6). Proof of Theorem 5.1.
It is not restrictive to assume that δ < , ε j < δ ,and 8 ε j γ J < δ for every j . To simplify the notation, set e u j := w ν ∗ θ ε j . From(5.5) and (5.6) it follows that ε j Z P ρ |∇ e u j ( x ) | dx ≤ C θ,P for every j , (5.14)for some constant C θ,P > P and θ .If the right-hand side of (5.8) is infinite, then there is nothing to prove. Thus,by extracting a subsequence (not relabeled), without loss of generality we mayassume that F ε j ( u j , P ρ ) ≤ M < + ∞ for every j , (5.15)for a suitable constant M > v j will be constructed as v j := ϕ j u j + (1 − ϕ j ) e u j , (5.16)where ϕ j ∈ C ∞ c ( R n ) are suitable cut-off functions satisfying ϕ j ( x ) = 1 for x ∈ ( P ρ ) δ and ϕ j ( x ) = 0 for x / ∈ ( P ρ ) δ/ . Introduce the set S := n x ∈ P ρ : δ < dist (cid:0) x, ∂P ρ (cid:1) ≤ δ o . (5.17)To construct the cut-off functions we divide S into m j pairwise disjoint layersof width δ m j .Consider the sequence { η j } defined by η j := Z P ρ ( u j ( x ) − e u j ( x )) dx + Z P ρ Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x )) dxdy . (5.18)By Fubini’s theorem, a change of variables, (1.6), and (5.18), we obtain Z P ρ Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x )) dxdy = Z P ρ (cid:18)Z P ρ \ B εj ( x ) J ε j ( x − y ) dy (cid:19) ( u j ( x ) − e u j ( x )) dx ≤ Z P ρ ( u j ( x ) − e u j ( x )) dx Z R n \ B (0) J ( z ) dz ≤ M J Z P ρ ( u j ( x ) − e u j ( x )) dx . η j → + as j → + ∞ , because { u j } and { e u j } converge to w ν in L ( P ρ ).Without loss of generality, we assume that η j < for every j . Let m j be theunique integer such that √ ε j + √ η j ε j < m j ≤ √ ε j + √ η j ε j + 1 . (5.19)Since ε j < m j < √ ε j and m j < √ ε j + √ η j ε j (5.20)and η j m j ε j ≤ √ ε j + √ η j and m j ε j ≤ √ ε j + √ η j ) . (5.21)Divide S into m j pairwise disjoint layers of width δ m j , S ij := (cid:26) x ∈ P ρ : δ i − δ m j < dist (cid:0) x, ∂P ρ (cid:1) < δ iδ m j (cid:27) , (5.22) i = 1 , . . . , m j .For every open set A ⊂ R d define G j ( A ) := J ε j ( u j , A, P ρ ) + W ε j ( u j , A )+ ε j Z A |∇ u j ( x ) | dx + 1 ε j Z A ( u j ( x ) − e u j ( x )) dx (5.23)+ 1 ε j Z A Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x )) dxdy . Hence, using (5.15), (5.18), and Lemma 4.3, we obtain m j X i =1 G j ( S ij ) ≤ G j ( S ) ≤ K − η j ε j , where K := M + c ( n ) J,W M + 1, and so there exists i j ∈ { , . . . , m j } such that,setting S j := S i j j , we have G j ( S j ) ≤ K − m j + η j m j ε j ≤ K √ ε j + √ η j ≤ K , (5.24)where in the last inequalities we used (5.20), (5.21), and the fact that ε j < , η j < , and K ≥
1. Define A j := (cid:26) x ∈ P ρ : dist( x, ∂P ρ ) > δ i j δ m j (cid:27) ,A ∗ j := (cid:26) x ∈ P ρ : dist( x, ∂P ρ ) > δ i j δ m j − δ m j (cid:27) , (5.25) B j := (cid:26) x ∈ P ρ : dist( x, ∂P ρ ) < δ i j − δ m j (cid:27) , ϕ j ( x ) := Z A ∗ j θ δ mj ( x − y ) dy . Then ϕ j ∈ C ∞ c ( R n ) and the following properties hold, thanks to (5.6) and(5.20): ϕ j = 1 in A j , ≤ ϕ j ≤ S j , ϕ j = 0 in B j , (5.26)sup |∇ ϕ j | ≤ C θ δ √ ε j + √ η j ε j ≤ C θ δε j , sup |∇ ϕ j | ≤ C θ δ ε j + η j ε j , (5.27)where C θ is the constant given in (5.6).Let v j be the function defined by (5.16). Since ( P ρ ) δ ⊂ A j and P ρ \ ( P ρ ) δ/ ⊂ B j , we have that v j = u j in ( P ρ ) δ and v j = e u j on P ρ \ ( P ρ ) δ/ . Moreover, since u j and e u j converge to w ν in L ( P ρ ), we have that v j → w ν in L ( P ρ ). Notethat ∇ v j := ϕ j ∇ u j + (1 − ϕ j ) ∇ e u j + ( u j − e u j ) ∇ ϕ j . (5.28)Fix 0 < η < . Using the inequality | a + b | ≤ | a | − η + | b | η , we obtain |∇ v j ( x ) − ∇ v j ( y ) | ≤ − η (cid:12)(cid:12) ϕ j ( x ) ∇ u j ( x ) − ϕ j ( y ) ∇ u j ( y )+ (1 − ϕ j ( x )) ∇ e u j ( x ) − (1 − ϕ j ( y )) ∇ e u j ( y ) (cid:12)(cid:12) (5.29)+ 1 η (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) . In view of the same inequality and the convexity of | · | , we get (cid:12)(cid:12) ϕ j ( x ) ∇ u j ( x ) − ϕ j ( y ) ∇ u j ( y ) + (1 − ϕ j ( x )) ∇ e u j ( x ) − (1 − ϕ j ( y )) ∇ e u j ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) ϕ j ( x )( ∇ u j ( x ) − ∇ u j ( y )) + ( ϕ j ( x ) − ϕ j ( y )) ∇ u j ( y )+ (1 − ϕ j ( x ))( ∇ e u j ( x ) − ∇ e u j ( y )) − ( ϕ j ( x ) − ϕ j ( y )) ∇ e u j ( y ) (cid:12)(cid:12) ≤ − η (cid:12)(cid:12) ϕ j ( x )( ∇ u j ( x ) − ∇ u j ( y )) + (1 − ϕ j ( x ))( ∇ e u j ( x ) − ∇ e u j ( y )) (cid:12)(cid:12) + 1 η (cid:12)(cid:12) ( ϕ j ( x ) − ϕ j ( y ))( ∇ u j ( y ) − ∇ e u j ( y )) (cid:12)(cid:12) ≤ ϕ j ( x )1 − η (cid:12)(cid:12) ∇ u j ( x ) − ∇ u j ( y ) (cid:12)(cid:12) + 1 − ϕ j ( x )1 − η (cid:12)(cid:12) ∇ e u j ( x ) − ∇ e u j ( y ) (cid:12)(cid:12) + 1 η ( ϕ j ( x ) − ϕ j ( y )) (cid:12)(cid:12) ∇ u j ( y ) − ∇ e u j ( y ) (cid:12)(cid:12) . |∇ v j ( x ) − ∇ v j ( y ) | ≤ ϕ j ( x )(1 − η ) (cid:12)(cid:12) ∇ u j ( x ) − ∇ u j ( y ) (cid:12)(cid:12) + 1 − ϕ j ( x )(1 − η ) (cid:12)(cid:12) ∇ e u j ( x ) − ∇ e u j ( y ) (cid:12)(cid:12) + 2 η ( ϕ j ( x ) − ϕ j ( y )) (cid:12)(cid:12) ∇ u j ( y ) − ∇ e u j ( y ) (cid:12)(cid:12) + 1 η (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) , hence for every pair of open sets A , B ⊂ P ρ we obtain by (2.14) J ε j ( v j , A, B ) ≤ J ε j ( u j , A, B ∩ ( A j ∪ S j ))(1 − η ) + J ε j ( e u j , A, B ∩ ( S j ∪ B j ))(1 − η ) + 2 ε j η Z A (cid:16) Z B J ε j ( x − y )( ϕ j ( x ) − ϕ j ( y )) dx (cid:17) |∇ u j ( y ) − ∇ e u j ( y ) | dy (5.30)+ ε j η Z A (cid:16) Z B J ε j ( x − y ) (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) dxdy. By (2.17) we have J ε j ( v j , P ρ ) = J ε j ( u j , A j ) + J ε j ( v j , S j ) + J ε j ( e u j , B j )+ 2 J ε j ( v j , S j , A j ∪ B j ) + 2 J ε j ( v j , A j , B j ) . (5.31)We now estimate all the terms but the first on the right-hand side of (5.31).By (5.30), J ε j ( v j , S j ) ≤ J ε j ( u j , S j )(1 − η ) + J ε j ( e u j , S j )(1 − η ) (5.32)+ 2 ε j η Z S j (cid:16) Z S j J ε j ( x − y )( ϕ j ( x ) − ϕ j ( y )) dx (cid:17) |∇ u j ( y ) − ∇ e u j ( y ) | dy + ε j η Z S j (cid:16) Z S j J ε j ( x − y ) (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) dxdy. From (2.17) and (5.5) it follows that J ε j ( e u j , S j ∪ B j ) = J ε j ( e u j , ( S j ∪ B j ) ∩ P ε j )+ 2 J ε j ( e u j , ( S j ∪ B j ) ∩ P ε j , ( S j ∪ B j ) \ P ε j ) . (5.33)By the mean value theorem and by (5.6), for every y ∈ P ρ the function g ( x ) := |∇ e u j ( x ) −∇ e u j ( y ) | satisfies (5.10) with a = C θ ε j and b = C θ ε j , hence by Lemma 5.10we obtain Z P ρ J ε j ( x − y ) |∇ e u j ( x ) − ∇ e u j ( y ) | dx ≤ C θ M J ε j . J ε j ( e u j , S j , S j ∪ B j ) + J ε j ( e u j , B j ) ≤ J ε j ( e u j , S j ∪ B j ) ≤ L n (( S j ∪ B j ) ∩ P ε j ) 4 C θ M J ε j . We now use the fact that there exist two constants C P ρ > δ P ρ > P ρ , such that L n ((( P ρ ) δ \ ( P ρ ) δ ) ∩ P ε ) ≤ C P ρ ε ( δ − δ ) (5.34)for every 0 < ε < δ < δ < δ P ρ . Therefore J ε j ( e u j , S j , S j ∪ B j ) + J ε j ( e u j , B j ) ≤ C P ρ C θ M J δ . (5.35)By the mean value theorem, (5.20), and (5.27), for every y ∈ S j the function g ( x ) = ( ϕ j ( x ) − ϕ j ( y )) satisfies (5.10) with a = C θ δε j and b = 1 ≤ C θ δ , wherewe used the inequalities C θ ≥ δ ≤
1. Hence, by Lemma 5.3 we have Z P ρ J ε j ( x − y )( ϕ j ( x ) − ϕ j ( y )) dx ≤ C θ δ M J . In turn, by (5.5), (5.6), (5.23), and (5.24),2 ε j η Z S j (cid:16) Z P ρ J ε j ( x − y )( ϕ j ( x ) − ϕ j ( y )) dx (cid:17) |∇ u j ( y ) − ∇ e u j ( y ) | dy ≤ C θ M J ηδ ε j Z S j |∇ u j ( y ) | dy + 2 C θ M J ηδ ε j L n ( S j ∩ P ε j ) (5.36) ≤ C θ M J ηδ (cid:0) K √ ε j + √ η j (cid:1) + 2 C P ρ C θ M J ηδ √ ε j , where in the last inequality we used the estimate L n ( S j ∩ P ε j ) ≤ C P ρ δ ε j m j ≤ C P ρ δε j √ ε j , (5.37)which follows fron (5.20) and (5.34).To treat the last term on the right-hand side of (5.32) we observe that (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) ( u j ( x ) − e u j ( x ))( ∇ ϕ j ( x ) − ∇ ϕ j ( y ))++ ( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) ≤ u j ( x ) − e u j ( x )) (cid:12)(cid:12) ∇ ϕ j ( x ) − ∇ ϕ j ( y ) (cid:12)(cid:12) + 2( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) (cid:12)(cid:12) ∇ ϕ j ( y ) (cid:12)(cid:12) . J , we obtain ε j η Z S j (cid:16) Z S j J ε j ( x − y ) (cid:12)(cid:12) ( u j ( x ) − e u j ( x )) ∇ ϕ j ( x ) − ( u j ( y ) − e u j ( y )) ∇ ϕ j ( y ) (cid:12)(cid:12) dxdy ≤ ε j η Z S j (cid:16) Z S j J ε j ( x − y ) |∇ ϕ j ( x ) − ∇ ϕ j ( y ) | dx (cid:17) ( u j ( y ) − e u j ( y )) dy (5.38)+ 2 ε j η Z S j (cid:16) Z S j J ε j ( x − y )( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) dx (cid:17) |∇ ϕ j ( y ) | dy . By the mean value theorem and (5.27), for every y ∈ S j the function g ( x ) = |∇ ϕ j ( x ) − ∇ ϕ j ( y ) | satisfies (5.10) for every x ∈ R n , with a = C θ δ ε j + η j ε j ≤ C θ δ √ ε j + √ η j ε j and b = C θ δ √ ε j + √ η j ε j ≤ C θ δ √ ε j + √ η j ε j , where we used the in-equalities δ ≤ ε j ≤ , and η j ≤ . Hence, by Lemma 5.3 we have Z P ρ J ε j ( x − y ) |∇ ϕ j ( x ) − ∇ ϕ j ( y ) | dx ≤ C θ M J δ ε j + η j ε j . In turn, by (5.23) and (5.24),2 ε j η Z S j (cid:16) Z P ρ J ε j ( x − y ) |∇ ϕ j ( x ) − ∇ ϕ j ( y ) | dx (cid:17) ( u j ( y ) − e u j ( y )) dy ≤ C θ M J ηδ ( ε j + η j ) 1 ε j Z S j ( u j ( y ) − e u j ( y )) dy (5.39) ≤ C θ M J Kηδ ( ε j + η j ) . Since J is even, by Fubini’s theorem, a change of variables, and (5.27),2 ε j η Z S j (cid:16) Z P ρ J ε j ( x − y )( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) dx (cid:17) |∇ ϕ j ( y ) | dy ≤ C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ ∩ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) dx (cid:17) dy + 2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x ) − u j ( y ) + e u j ( y )) dx (cid:17) dy ≤ C θ ηδ ε j + η j ε j Z B εj (0) J ε j ( z ) (cid:16) Z S j ( u j ( y + z ) − e u j ( y + z ) − u j ( y ) + e u j ( y )) dy (cid:17) dz + 2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x )) dx (cid:17) dy (5.40)+ 2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y ) dx (cid:17) ( u j ( y ) − e u j ( y )) dy . ε j < δ/
4, by (5.20) and (5.22) for y ∈ S j and | z | ≤ ε j the segment joning y and y + z is contained in ( P ρ ) δ/ , and so by the mean value theorem for | z | ≤ ε j , Z S j ( u j ( y + z ) − e u j ( y + z ) − u j ( y ) + e u j ( y )) dy ≤ | z | Z ( P ρ ) δ/ |∇ u j ( y ) − ∇ e u j ( y ) | dy . Therefore, recalling that 2 ε j γ J < δ/
4, it follows from (1.5), (1.6), (5.14), andLemma 4.3, that2 C θ ηδ ε j + η j ε j Z B εj (0) J ε j ( z ) (cid:16) Z S j ( u j ( y + z ) − e u j ( y + z ) − u j ( y )+ e u j ( y )) dy (cid:17) dz ≤ C θ ηδ ε j + η j ε j Z B εj (0) J ε j ( z ) | z | dz Z ( P ρ ) δ/ |∇ u j ( y ) − ∇ e u j ( y ) | dy ≤ C θ ηδ ( ε j + η j ) ε j Z B (0) J ( z ) | z | dz Z ( P ρ ) δ/ |∇ u j ( y ) | dy (5.41)+ 2 C θ ηδ ( ε j + η j ) ε j Z B (0) J ( z ) | z | dz Z ( P ρ ) δ/ |∇ e u j ( y ) | dy ≤ C θ M J c ( n ) J,W
Mηδ ( ε j + η j ) + 2 C θ C θ,P M J ηδ ( ε j + η j ) . By (5.23) and (5.24)2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( x ) − e u j ( x )) dx (cid:17) dy (5.42) ≤ C θ Kηδ ( ε j + η j ) . Using (1.6), (5.23), and (5.24) we obtain2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y ) dx (cid:17) ( u j ( y ) − e u j ( y )) dy ≤ C θ M J ηδ ε j + η j ε j Z S j ( u j ( y ) − e u j ( y )) dy ≤ C θ M J Kηδ ( ε j + η j ) . (5.43)Combining (5.32), (5.35), (5.36), (5.38), (5.39), (5.40), (5.41), (5.42), and(5.43), we have J ε j ( v j , S j ) + J ε j ( e u j , B j ) ≤ J ε j ( u j , S j )(1 − η ) + 4 C P ρ C θ M J (1 − η ) δ + σ (1) j , (5.44)where σ (1) j → + as j → + ∞ . 26ext we consider the term J ε j ( v j , S j , A j ∪ B j ) in (5.31) . By (5.30), using(5.26), J ε j ( v j , S j , A j ∪ B j ) ≤ J ε j ( u j , S j , A j )(1 − η ) + J ε j ( e u j , S j , B j )(1 − η ) + 2 ε j η Z S j (cid:16) Z A j ∪ B j J ε j ( x − y )( ϕ j ( x ) − ϕ j ( y )) dx (cid:17) |∇ u j ( y ) − ∇ e u j ( y ) | dy + ε j η Z S j Z A j ∪ B j J ε j ( x − y )( u j ( y ) − e u j ( y )) |∇ ϕ j ( y ) | dxdy . (5.45)Since η < /
2, by (5.23) and (5.24) we have J ε j ( u j , S j , A j )(1 − η ) ≤ J ε j ( u j , S j , A j ) ≤ K √ ε j + 4 √ η j . (5.46)The second and third terms on the right-hand side of (5.45) can be estimatedusing (5.35) and (5.36). For the last term, we use the fact that ∇ ϕ j ( x ) = 0 if x ∈ A j ∪ B j . Hence, by a change of variables, from (1.6), (5.23), (5.24), (5.27)and from the inequalities δ ≤ ε j ≤
1, and η j ≤
1, we obtain ε j η Z S j Z A j ∪ B j J ε j ( x − y )( u j ( y ) − e u j ( y )) |∇ ϕ j ( y ) | dxdy ≤ ε j η Z S j Z B εj ( y ) J ε j ( x − y )( u j ( y ) − e u j ( y )) |∇ ϕ j ( y ) − ∇ ϕ j ( x ) | dxdy + ε j η Z S j Z P ρ \ B εj ( y ) J ε j ( x − y )( u j ( y ) − e u j ( y )) |∇ ϕ j ( y ) | dxdy ≤ C θ ηδ ( ε j + η j ) ε j Z S j (cid:16) Z B εj ( y ) J ε j ( x − y ) | x − y | dx (cid:17) ( u j ( y ) − e u j ( y )) dy + 2 C θ ηδ ε j + η j ε j Z S j (cid:16) Z P ρ \ B εj ( y ) J ε j ( x − y ) dx (cid:17) ( u j ( y ) − e u j ( y )) dxdy (5.47) ≤ C θ M J ηδ ε j + η j ε j Z S j ( u j ( y ) − e u j ( y )) dy ≤ C θ M J Kηδ ( ε j + η j ) . Therefore, by (5.35), (5.36), (5.45), (5.46), and (5.47) we get J ε j ( v j , S j , A j ∪ B j ) ≤ C P ρ C θ M J (1 − η ) δ + σ (2) j , (5.48)where σ (2) j → + as j → + ∞ .We now estimate the term J ε j ( v j , A j , B j ) in (5.31). Since v j = u j in A j , v j = e u j = 1 in B j , and dist( A j , B j ) = δ m j , by a change of variables and in view27f (5.14), (5.21), and Lemmas 4.3 and 5.4, for j large enough we obtain J ε j ( v j , A j , B j ) ≤ ω (cid:0) m j ε j δ (cid:1)(cid:18) ε j Z B j |∇ e u j ( x ) | dx + ε j Z A j |∇ u j ( y ) | dy (cid:19) ≤ ω (cid:16) √ ε j + √ η j δ (cid:17) ( C θ,P + c ( n ) J,W M ) . (5.49)Combining (5.31), (5.35), (5.44), (5.48), and (5.49) we deduce J ε j ( v j , P ρ ) ≤ J ε j ( u j , P ρ )(1 − η ) + 12 C P ρ C θ M J (1 − η ) δ + σ (3) j , (5.50)where σ (3) j → + as j → + ∞ .Next we consider the term W ε j ( v j , P ρ ). Fix x ∈ S j with x · ν > ε j , so that e u j ( x ) = 1. By (2.5) and (2.6) we have W ( v j ( x )) ≤ W ( u j ( x )) if u j ( x ) ≥ − a W .Let s < − W ( s ) = max [ − , W =: M W . (5.51)If u j ( x ) ≤ s , then either u j ( x ) ≤ v j ( x ) ≤ − − ≤ v j ( x ) ≤
1. In both caseswe get W ( v j ( x )) ≤ W ( u j ( x )), either by (2.6) or by (5.51). If s < u j ( x ) < − a W , then s < v j ( x ) < W ( v j ( x )) ≤ W ( s ) = M W by (2.6) and (5.51). We conclude that W ( v j ( x )) ≤ W ( u j ( x )) + M W for every x ∈ S j with x · ν > ε j . Integrating we obtain1 ε j Z S j ∩{ x · ν>ε j } W ( v j ( x )) dx ≤ ε j Z S j ∩{ x · ν>σ j } W ( u j ( x )) dx + M W ε j L n ( S j ∩ {| u j − | > a W } ∩ { x · ν > ε j } ) ≤ ε j Z S j ∩{ x · ν>ε j } W ( u j ( x )) dx + M W ε j a W Z S j ∩{ x · ν>ε j } ( u j ( x ) − dx A similar inequality can be obtained for S j ∩ { x · ν < − ε j } , and adding thesetwo inequalities we conclude that1 ε j Z S j \ P εj W ( v j ( x )) dx ≤ ε j Z S j \ P εj W ( u j ( x )) dx + M W a W ε j Z S j \ P εj ( u j ( x ) − e u j ( x )) dx , (5.52)28here in the last inequality we used the fact that e u j = w ν on P ρ \ P ε j .On the other hand, since W ( v j ( x )) ≤ W ( u j ( x )) + M W for every x ∈ P ρ ,integrating over S j ∩ P ε j and using (5.37), we obtain1 ε j Z S j ∩ P εj W ( v j ( x )) dx ≤ ε j Z S j ∩ P εj W ( u j ( x )) dx + M W ε j L n ( S j ∩ P ε j ) ≤ ε j Z S j ∩ P εj W ( u j ( x )) dx + C P ρ M W δ √ ε j . (5.53)Adding (5.52) and (5.53) gives1 ε j Z S j W ( v j ( x )) dx ≤ ε j Z S j W ( u j ( x )) dx + M W a W ε j Z S j ( u j ( x ) − e u j ( x )) dx + C P ρ M W δ √ ε j , hence by (5.23) and (5.24) we have1 ε j Z S j W ( v j ( x )) dx ≤ ε j Z S j W ( u j ( x )) dx + M W a W ( K √ ε j + √ η j ) + C P ρ M W δ √ ε j . (5.54)By (5.3), (5.4), (5.34), and (5.51) we get1 ε j Z B j W ( v j ( x )) dx = 1 ε j Z B j W ( e u j ( x )) dx ≤ M W ε j L n ( B j ∩ P ε j ) ≤ C P ρ M W δ . (5.55)From (5.54) and (5.55) it follows that1 ε j Z P ρ W ( v j ( x )) dx ≤ ε j Z P ρ W ( u ( x )) dx + C P ρ M W δ + σ (4) j , (5.56)where σ (4) j → + as j → + ∞ .Adding (5.50) and (5.56) we obtain F ε j ( v j , P ρ ) ≤ F ε j ( u j , P ρ )(1 − η ) + C P ρ (48 C θ M J + M W ) δ + σ (5) j where σ (5) j → + as j → + ∞ . This implies thatlim sup j → + ∞ F ε j ( v j , P ρ ) ≤ − η ) lim sup j → + ∞ F ε j ( u j , P ρ ) + κ δ , where κ is a constant independent of j , δ , and P ρ . Passing to the limit as η → + we obtain (5.8). 29 Gamma Liminf Inequality
In this section we prove the Γ-liminf inequality.
Theorem 6.1 ( Γ -Liminf ) Let ε j → + and let { u j } be a sequence in W , (Ω) ∩ L (Ω) such that u j → u in L (Ω) and lim inf j → + ∞ F ε j ( u j , Ω) < + ∞ . (6.1) Then u ∈ BV (Ω; {− , } ) and lim inf j → + ∞ F ε j ( u j , Ω) ≥ Z S u ψ ( ν u ) d H n − , (6.2) where ψ is defined by (1.13). Given ν ∈ S n − , let ν , . . . , ν n be an orthonormal basis in R n with ν n = ν ,let Q νρ := { x ∈ R n : | x · ν i | < ρ/ , i = 1 , . . . , n } , ˆ Q νρ := R n \ Q νρ , (6.3)and let S νρ := { x ∈ R n : | x · ν | < ρ/ } , ˆ S νρ := R n \ S νρ . When ν , . . . , ν n is the canonical basis e , . . . , e n in R n we omit the superscript ν in the above notation.We recall the definition of the sets V ν and X ν in (1.10) and in (1.12),respectively. We will use these sets in what follows. Further, as in Section 5, θ ε is the standard mollifier (see (5.2)), and we set˜ u ε := w ν ∗ θ ε , (6.4)where w ν is the function defined in (5.1), with ν ∈ S n − . Lemma 6.2
Let < ε < δ < / , let C δ := Q δ \ Q − δ , and let ˜ u ε be thefunction in (6.4), with ν = e n . Then J ε (˜ u ε , C δ ) ≤ κ δ for some constant κ > independent of ε and δ . Proof.
For every σ > C σδ := C δ ∩ {| x n | < σ } , ˆ C σδ := C δ ∩ {| x n | ≥ σ } ,and write C δ × C δ = ( C εδ × C εδ ) ∪ ( C εδ × ˆ C εδ ) ∪ ( ˆ C εδ × C εδ ) ∪ ( ˆ C εδ × ˆ C εδ ) . Since J is even, we have J ε (˜ u ε , C δ ) ≤ J ε (˜ u ε , C εδ ) + 2 J ε (˜ u ε , C εδ , ˆ C εδ ) + J ε (˜ u ε , ˆ C εδ ) . (6.5)30y (5.2) we have that ∇ ˜ u ε = 0 on ˆ C εδ and so J ε (˜ u ε , ˆ C εδ ) = 0 . (6.6)We now estimate the first term on the right-hand side of (6.5). Since ε ∇ ˜ u ε and ε ∇ ˜ u ε are bounded in L ∞ uniformly with respect to ε , there exists a constant c > |∇ ˜ u ε ( x ) − ∇ ˜ u ε ( y ) | ≤ cε (cid:16)(cid:12)(cid:12)(cid:12) x − yε (cid:12)(cid:12)(cid:12) ∧ (cid:12)(cid:12)(cid:12) x − yε (cid:12)(cid:12)(cid:12) (cid:17) for every x , y ∈ R n . Therefore, by the change of variables z = ( x − y ) /ε and(1.6) we get J ε (˜ u ε , C εδ ) ≤ cε Z C εδ Z C εδ J ε ( x − y ) (cid:16)(cid:12)(cid:12)(cid:12) x − yε (cid:12)(cid:12)(cid:12) ∧ (cid:12)(cid:12)(cid:12) x − yε (cid:12)(cid:12)(cid:12) (cid:17) dxdy (6.7) ≤ cM J ε L n ( C εδ ) ≤ n +1 cM J δ . Next we study the second term on the right-hand side of (6.5). Since ∇ ˜ u ε = 0on ˆ C εδ and ε ∇ ˜ u ε is bounded in L ∞ uniformly with respect to ε , there exists aconstant c > J ε (˜ u ε , C εδ , ˆ C εδ ) = ε Z C εδ (cid:16)Z ˆ C εδ J ε ( x − y ) dx (cid:17) |∇ ˜ u ε ( y ) | dy (6.8) ≤ cε L n ( C εδ ) Z R n \ B (0) J ( z ) dz ≤ n cM J δ , where we used again the change of variables z = ( x − y ) /ε and (1.6). Theconclusion follows by combining (6.5)–(6.8).The following result will be crucial in the proof of the Γ-liminf inequality. Lemma 6.3
Let < ε < δ < / , let u ∈ X ν be such u = ˜ u ε in Q ν \ Q ν − δ ,where ˜ u ε is the function defined in (6.4). Then there exist two constants κ and κ , depending only on the dimension n of the space, such that J ε ( u, V ν , R n ) − J ε ( u, Q ν ) ≤ κ δ + (cid:16) κ ω (cid:16) εδ (cid:17) + κ ω ( ε ) (cid:17) ε Z Q ν |∇ u ( x ) | dx , where κ is the constant in Lemma 6.2, and ω is the function defined in (5.13). Proof.
Without loss of generality, we may assume that ν = e n , the n -th vectorof the canonical basis. For simplicity we omit the superscript ν in the notationfor Q νρ , ˆ Q νρ , S νρ , ˆ S νρ , V ν , X ν , w ν , and the subscript ρ when ρ = 1. Write V × R n = (( V \ Q ) × Q ) ∪ (( V \ Q ) × ˆ Q ) ∪ ( Q × Q ) ∪ ( Q × ˆ Q ) (6.9) ⊂ ( ˆ S × Q ) ∪ (( V \ Q ) × S ) ∪ ( ˆ S × ˆ S ) ∪ ( Q × Q ) ∪ ( Q × ( S \ Q )) ∪ ( Q × ˆ S ) . J is even we have J ε ( u, V, R n ) − J ε ( u, Q ) ≤ ε Z ˆ S (cid:16)Z Q − δ J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy + ε Z V \ Q (cid:16)Z S − δ J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy (6.10)+ ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy , where we have used the equalities u = ± ∇ u = 0 in ˆ S − δ , which followfrom the facts that u ∈ X and u = ˜ u ε on Q \ Q − δ (see (5.4), (5.5), and theinequalities 0 < ε < δ < / ∇ u = 0 in ˆ S , we have ε Z ˆ S (cid:16)Z Q − δ J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ εω (cid:16) εδ (cid:17) Z Q − δ |∇ u ( x ) | dx . (6.11)To estimate the second term on the right-hand side of (6.10), we identify Z n with Z n − × Z so that for α = ( α , . . . , α n − ) ∈ Z n − and β ∈ Z we have( α, β ) = ( α , . . . , α n − , β ) ∈ Z n . Write S \ Q = [ α ∈ Z n − , | α | ∞ ≥ (( α,
0) + Q ) , V = [ β ∈ Z ((0 , β ) + Q ) , where | α | ∞ := max {| α | , . . . , | α n − |} . Then ε Z V \ Q (cid:16) Z S − δ J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ ε Z V \ Q (cid:16)Z S − δ ∩ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy (6.12)+ X α ∈ Z n − , | α | ∞ ≥ X β ∈ Z ε Z (0 ,β )+ Q (cid:16)Z ( α, Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy . By Lemma 5.4 and because ∇ u = 0 in V \ Q , we have ε Z V \ Q (cid:16)Z S − δ ∩ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ εω (cid:16) εδ (cid:17) Z S − δ ∩ Q |∇ u ( x ) | dx . To estimate the second term on the right-hand side of (6.12), we use the changeof variables ζ = x − y and observe that for x ∈ ( α,
0) + Q and y ∈ (0 , β ) + Q we32ave ζ ∈ ( α, − β ) + Q . Therefore, we obtain Z (0 ,β )+ Q (cid:16) Z ( α, Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy = Z ( α, Q |∇ u ( x ) | (cid:16)Z (0 ,β )+ Q J ε ( x − y ) dy (cid:17) dx ≤ Z ( α, Q |∇ u ( x ) | dx Z ( α, − β )+ Q J ε ( ζ ) dζ = Z Q |∇ u ( x ) | dx Z ( α, − β )+ Q J ε ( ζ ) dζ , where in the last equality we used the periodicity of u ∈ X . Hence X α ∈ Z n − , | α | ∞ ≥ X β ∈ Z ε Z (0 ,β )+ Q (cid:16)Z ( α, Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ ε Z Q |∇ u ( x ) | dx X α ∈ Z n − , | α | ∞ ≥ X β ∈ Z Z ( α, − β )+ Q J ε ( ζ ) dζ ≤ n ε Z Q |∇ u ( x ) | dx Z ˆ Q J ε ( ζ ) dζ . In the last inequality we used the fact that each point of ˆ Q belongs to at most2 n cubes of the form ( α, − β ) + Q for α ∈ Z n − , with | α | ∞ ≥
2, and β ∈ Z .After the change of variables z = ζ/ε we obtain (see (5.13)) Z ˆ Q J ε ( ζ ) dζ ≤ Z R n \ B /ε (0) J ( z ) dz ≤ ω ( ε ) . Combining the last five inequalities and using the periodicity of u , from (6.12)we obtain ε Z V \ Q (cid:16)Z S − δ J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy (6.13) ≤ (cid:16) ω (cid:16) εδ (cid:17) + 2 n ω ( ε ) (cid:17) ε Z S ∩ Q |∇ u ( x ) | dx = 3 n − (cid:16) ω (cid:16) εδ (cid:17) + 2 n ω ( ε ) (cid:17) ε Z Q |∇ u ( x ) | dx . Finally, to estimate the last term on the right-hand side of (6.10), we usethe inclusion Q × ( S \ Q ) ⊂ (cid:0) Q × ( S \ Q ) (cid:1) ∪ (cid:0) Q − δ × ( S ∩ ( Q \ Q ) (cid:1) ∪ (cid:0) ( Q \ Q − δ ) × ( Q δ \ Q ) (cid:1) ∪ (cid:0) ( Q \ Q − δ ) × ( S ∩ ( Q \ Q δ )) (cid:1) ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy ≤ ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy + ε Z Q − δ (cid:16)Z S ∩ ( Q \ Q ) J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy (6.14)+ ε Z Q \ Q − δ (cid:16)Z Q δ \ Q J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy + ε Z Q \ Q − δ (cid:16)Z S ∩ ( Q \ Q δ ) J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy . By Lemma 5.4, ε Z Q − δ (cid:16) Z S ∩ ( Q \ Q ) J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy + ε Z Q \ Q − δ (cid:16)Z S ∩ ( Q \ Q δ ) J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy (6.15) ≤ εω (cid:16) εδ (cid:17) Z S ∩ Q |∇ u ( x ) | dx = 2 · n − εω (cid:16) εδ (cid:17) Z Q |∇ u ( x ) | dx , where in the last equality we used the periodicity of u . On the other hand, byLemma 6.2 ε Z Q \ Q − δ (cid:16)Z Q δ \ Q ) J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy ≤ κ δ . (6.16)It remains to study the first term on the right-hand side of (6.14). We have ε Z Q (cid:16) Z S \ Q J ε ( x − y ) |∇ u ( x ) − ∇ u ( y ) | dx (cid:17) dy ≤ ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy (6.17)+ 2 ε Z Q (cid:16)Z S \ Q J ε ( x − y ) dx (cid:17) |∇ u ( y ) | dy . To estimate the first term on the right-hand side of (6.17) we write2 ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy = 2 ε X α ∈ Z n ∩ ( S \ Q ) Z Q (cid:16)Z α + Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy .
34y Fubini’s theorem and the change of variables ζ = x − y , we get Z Q (cid:16)Z α + Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy = Z α + Q (cid:16)Z Q J ε ( x − y ) dy (cid:17) |∇ u ( x ) | dx ≤ Z α + Q (cid:16)Z x − Q J ε ( ζ ) dζ (cid:17) |∇ u ( x ) | dx ≤ Z Q |∇ u ( x ) | dx Z α − Q J ε ( ζ ) dζ , where in the last inequality we have used the periodicity of u and the inclusion x − Q ⊂ α − Q for x ∈ α + Q . Hence,2 ε X α ∈ Z n ∩ ( S \ Q ) Z Q (cid:16)Z α + Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ ε Z Q |∇ u ( x ) | dx X α ∈ Z n ∩ ( S \ Q ) Z α − Q J ε ( ζ ) dζ ≤ n ε Z Q |∇ u ( x ) | dx Z ˆ Q J ε ( ζ ) dζ , where in the last inequality we used the fact that each point of ˆ Q belongs toat most 2 n − cubes of the form α − Q for α ∈ Z n ∩ ( S \ Q ). After the changeof variables z = ζ/ε , we obtain2 ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ n ε Z Q |∇ u ( x ) | dx Z R n \ B /ε (0) J ( z ) | z | dz . (6.18)We now estimate the second term on the right-hand side of (6.17). With thechange of variables z = ( x − y ) /ε , we have2 ε Z Q (cid:16)Z S \ Q J ε ( x − y ) dx (cid:17) |∇ u ( y ) | dy ≤ ε Z R n \ B /ε (0) J ( z ) | z | dz Z Q |∇ u ( y ) | dy . (6.19)Combining the inequalities (6.17)–(6.19), we obtain2 ε Z Q (cid:16)Z S \ Q J ε ( x − y ) |∇ u ( x ) | dx (cid:17) dy ≤ n εω ( ε ) Z Q |∇ u ( x ) | dx . (6.20)The conclusion follows from (6.11), (6.13), (6.14), (6.15), (6.16), and (6.20). Proof of Theorem 6.1.
By Theorem 1.1 we deduce that u ∈ BV (Ω; {− , } ).Let µ j be the nonnegative Radon measure on Ω defined by µ j ( B ) := 1 ε Z B W ( u j ( x )) dx + ε Z B Z Ω J ε ( x − y ) |∇ u j ( x ) − ∇ u j ( y ) | dxdy (6.21)for every Borel set B ⊂ Ω. Since µ j (Ω) = F ε j ( u j , Ω), by (6.1) µ j (Ω) is boundeduniformly with respect to j . Extracting a subsequence (not relabeled), we may35ssume that the liminf in (6.2) is a limit and that µ j ∗ ⇀ µ weakly ∗ in the space M b (Ω) of bounded Radon measures on Ω, considered, as usual, as the dual ofthe space C (Ω) of continuous functions on Ω vanishing on ∂ Ω. Let g be thedensity of the absolutely continuous part of µ with respect to H n − restrictedto S u . Then the inequality (6.2) will follow from g ( x ) ≥ ψ ( ν u ( x )) for H n − a.e. x ∈ S u . (6.22)To prove this inequality, fix x ∈ S u such that, setting ν := ν u ( x ), we havelim ρ → + ρ n Z Q νρ | u ( x + x ) − w ν ( x + x ) | dx = 0 , (6.23) g ( x ) = lim ρ → + µ ( x + Q νρ ) ρ n − < + ∞ . (6.24)It is well-known (see [21, Theorem 3 in Section 5.9]) that (6.23) and (6.24) holdfor H n − a.e. x ∈ S u . Since µ j ∗ ⇀ µ weakly ∗ in M b (Ω), by (2.15) and (6.21),using a change of variables, we get g ( x ) = lim ρ → + µ ( x + Q νρ ) ρ n − ≥ lim sup ρ → + lim sup j → + ∞ µ j ( x + Q νρ ) ρ n − ≥ lim sup ρ → + lim sup j → + ∞ F ε j ( u j , x + Q νρ ) ρ n − = lim sup ρ → + lim sup j → + ∞ F η j,ρ ( v j,ρ , Q ν ) , where η j,ρ := ε j /ρ and v j,ρ ( y ) := u j ( x + ρy ). On the other hand, since u j → u in L (Ω), by (6.23) we obtain0 = lim ρ → + lim j → + ∞ ρ n Z Q νρ | u j ( x + x ) − w ν ( x + x ) | dx = lim ρ → + lim j → + ∞ Z Q ν | v j,ρ ( x ) − w ν ( x ) | dx . Since for every ρ > j → + ∞ η j,ρ = 0 , by a diagonal argument we can choose ρ j → + such that, setting η j := η j,ρ j and v j := v j,ρ j , we have η j → + , v j → w ν in L ( Q ν ), and g ( x ) ≥ lim sup j → + ∞ F η j ( v j , Q ν ) . (6.25)The finiteness of g ( x ) and Theorem 1.1 yield that v j → w ν in L ( Q ν ). We cannow apply the modification Theorem 5.1: there exists δ ν > < δ < δ ν we obtain a sequence { w j } ⊂ W , ( Q ν ) ∩ L ( Q ν ) with w j → w ν in L ( Q ν ), w j = w ν ∗ θ ε j in Q ν \ Q ν − δ , andlim sup j → + ∞ F η j ( v j , Q ν ) ≥ lim sup j → + ∞ F η j ( w j , Q ν ) − κ δ , (6.26)36here, we recall, the constant κ is independent of δ . Extend w j to R n in sucha way that w j ( x ) = ± ± x · ν ≥ and w ( x + ν i ) = w ( x ) for all x ∈ R n andfor all i = 1, . . . , n −
1, where ν i are the vectors in (1.11). Then w j ∈ X ν andso we can apply Lemma 6.3 to obtainlim sup j → + ∞ F η j ( w j , Q ν ) ≥ lim sup j → + ∞ ( W η j ( w j , Q ν ) + J η j ( w j , V ν , R n )) + (6.27) − κ δ − lim sup j → + ∞ (cid:16) κ ω (cid:16) η j δ (cid:17) + κ ω ( η j ) (cid:17) η j Z Q ν |∇ w j ( x ) | dx , where we recall that W η j is defined in (2.13). By (1.13), W η j ( w j , Q ν ) + J η j ( w j , V ν , R n ) ≥ ψ ( ν ) (6.28)for every j with η j <
1. By (6.25) and (6.25) the finiteness of g ( x ) impliesthat F η j ( w j , Q ν ) is bounded uniformly with respect to j . Therefore Lemma4.3, together with the periodicity of w j , proves that the same property holdsfor η j R Q ν |∇ w j ( x ) | dx . Together with (5.13), (6.25), (6.26), (6.27), and (6.28),this shows that g ( x ) ≥ ψ ( ν ) − κ δ − κ δ for every 0 < δ < δ ν . Taking the limitas δ → + we obtain (6.22). This concludes the proof of the theorem. In this section we prove the Γ-limsup inequality. Fix ε j → + . For every u ∈ BV (Ω; {− , } ) we define F ′′ ( u, Ω) := inf (cid:8) lim sup j → + ∞ F ε j ( u j , Ω) : u j → u in L (Ω) (cid:9) . (7.1) Theorem 7.1 ( Γ -Limsup) For every u ∈ BV (Ω; {− , } ) we have F ′′ ( u, Ω) ≤ Z S u ψ ( ν u ) d H n − . (7.2)To prove the Γ-limsup inequality we need the results proved in the followinglemmas. Lemma 7.2
Let u ∈ BV loc ( R n ; {− , } ) and, for every ε > , let ˜ u ε be as in(6.4). Assume that there exists a bounded polyhedral set Σ of dimension n − such that S u = Σ , let Σ n − the union of all its n − dimensional faces, andlet (Σ n − ) δ be defined as in (3.1). Then there exists δ Σ > such that for < ε < δ < δ Σ we have J ε (˜ u ε , (Σ n − ) δ ) ≤ c δ H n − (Σ n − ) for some constant c > independent of ε , δ , and Σ . roof. It is enough to repeat the proof of Lemma 6.2 with C σδ and ˆ C σδ replacedby { x ∈ (Σ n − ) δ : dist( x, Σ) < ε } and { x ∈ (Σ n − ) δ : dist( x, Σ) ≥ ε } . Lemma 7.3
Let P be a bounded polyhedron of dimension n − containing with normal ν , let ρ > , and let P ρ be the n -dimensional prism defined in(5.7). Then for every η > there exists a sequence { u ε } ⊂ W , ( P ρ ) such that u ε → w ν in L ( P ρ ) and lim sup ε → + (cid:0) W ε ( u ε , P ρ ) + J ε ( u ε , P ρ , R n ) (cid:1) ≤ ( ψ ( ν ) + η ) H n − ( P ) . Proof.
Without loss of generality, we assume that ν = e n . For simplicity, weomit the superscript ν in the notation for w ν , X ν , V ν , Q ν , and the subscript ρ when ρ = 1. By the definition of ψ (see (1.13)), given η > ε ∗ ∈ (0 ,
1) and u ∗ ∈ X such that W ε ∗ ( u ∗ , Q ) + J ε ∗ ( u ∗ , V, R n ) ≤ ψ ( e n ) + η . (7.3)Define u ε ( x ) := u ∗ ( ε ∗ ε x ) for x ∈ R n . Since u ∗ ( x ) = ± ± x n ≥ /
2, thesequence { u ε } converges to w in L ( R n ).To estimate W ε ( u ε , P ρ ) and J ε ( u ε , P ρ , R n ), we consider the ( n − Q ( n − := Q ∩ { x n = 0 } and we set Z ε := n { α ∈ Z n : α n = 0 , ( α + Q ( n − ) ∩ (cid:16) ε ∗ ε P (cid:17) = Ø o . Observe that (cid:16) εε ∗ (cid:17) n − Z ε → H n − ( P ) as ε → + , (7.4)where Z ε is the number of elements of Z ε .Let S := { x ∈ R n : | x n | < / } . Since u ∗ ( x ) = ± ± x n ≥ /
2, by (2.3)we have W ( u ∗ ( x )) = 0 for x ∈ R n \ S . Therefore a change of variables and theperiodicity of u ∗ give W ε ( u ε , P ρ ) = (cid:16) εε ∗ (cid:17) n − W ε ∗ (cid:16) u ∗ , ε ∗ ε P ρ (cid:17) = (cid:16) εε ∗ (cid:17) n − W ε ∗ (cid:16) u ∗ , (cid:16) ε ∗ ε P ρ (cid:17) ∩ S (cid:17) ≤ (cid:16) εε ∗ (cid:17) n − X α ∈ Z ε W ε ∗ ( u ∗ , α + Q ) = (cid:16) εε ∗ (cid:17) n − Z ε W ε ∗ ( u ∗ , Q ) . (7.5)Similarly, J ε ( u ε , P ρ , R n ) = (cid:16) εε ∗ (cid:17) n − J ε ∗ (cid:16) u ∗ , ε ∗ ε P ρ , R n (cid:17) ≤ (cid:16) εε ∗ (cid:17) n − X α ∈ Z ε J ε ∗ ( u ∗ , α + V, R n ) = (cid:16) εε ∗ (cid:17) n − Z ε J ε ∗ ( u ∗ , V, R n ) . (7.6)The result now follows from (7.3)–(7.6).38 emma 7.4 Let u ∈ BV loc ( R n ; {− , } ) . Assume that there exists a boundedpolyhedral set Σ of dimension n − such that S u = Σ . For every ρ > let Σ ρ := { x ∈ R n : dist( x, Σ) < ρ/ } . Then for every σ > there exist ρ > and δ ∈ (0 , ρ ) with the following property: for every ε j → + there exists v j ∈ W , (Σ ρ ) such that v j = u on Σ ρ \ Σ ρ − δ and lim sup j → + ∞ F ε j ( v j , Σ ρ ) ≤ Z Σ ψ ( ν u ) d H n − + σ . Proof.
Let δ Σ > σ and ˆ σ with ˆ σ ∈ (0 , min { σ, δ Σ } ).There exist ρ ∈ (0 , ˆ σ ) and a finite number of bounded polyhedra P , . . . , P k of dimension n − n − P iρ ∩ P jρ = Ø for i = j and Σ ρ \ k [ i =1 P iρ ⊂ (Σ n − ) ˆ σ , (7.7)where P iρ and (Σ n − ) ˆ σ are defined as in (5.7) and Lemma 7.2, respectively.Find R , . . . , R k , bounded polyhedra of dimension n − n − P i ⋐ R i and R iρ ∩ R jρ = Ø for i = j .Fix η > η H n − (Σ) < σ/
2. By Lemma 7.3 for every i = 1, . . . , k , there exists a sequence { u ij } ⊂ W , ( R iρ ) such that u ij → u in L ( R iρ ), andlim sup j → + ∞ (cid:0) W ε j ( u ij , R iρ ) + J ε j ( u ij , R iρ , R n ) (cid:1) ≤ ( ψ ( ν i ) + η ) H n − ( R i ) . (7.8)By Theorem 5.1 there exist δ ∈ (0 , min { ˆ σ, ρ/ } ) and { v ij } ⊂ W , ( R iρ ) such that v ij → u in L ( R iρ ) as j → + ∞ , v ij = u ∗ θ ε j on R iρ \ ( R iρ ) δ , andlim sup j → + ∞ F ε j ( v ij , R iρ ) ≤ lim sup j → + ∞ F ε j ( u ij , R iρ ) + κ δ (7.9) ≤ ( ψ ( ν i ) + η ) H n − ( R i ) + κ ˆ σ , where, we recall, the costant κ > j , ˆ σ , and R iρ . Define v j := v ij on R iρ and v j := u ∗ θ ε j on A ρ := Σ ρ \ S ki =1 R iρ . Then v j ∈ W , (Σ ρ )and v j → u in L (Σ ρ ). Moreover v j = u on Σ ρ \ Σ ρ − δ for all j sufficiently large.By additivity we obtain W ε j ( v j , Σ ρ ) ≤ k X i =1 W ε j ( v j , R iρ ) + W ε j ( v j , A ρ ) . (7.10)Since ( u ∗ θ ε j )( x ) = ± x / ∈ Σ ε j and − ≤ ( u ∗ θ ε j )( x ) ≤
1, by (2.3) and(7.7) we have W ε j ( v j , A ρ ) ≤ W ε j ( u ∗ θ ε j , (Σ n − ) ˆ σ ∩ Σ ε j ) ≤ ε j M W L n ((Σ n − ) ˆ σ ∩ Σ ε j ) ≤ M W c Σ ˆ σ H n − (Σ n − ) , M W is the constant in (5.51) and c Σ > W ε j ( v j , Σ ρ ) ≤ k X i =1 W ε j ( v j , R iρ ) + M W c Σ ˆ σ H n − (Σ n − ) . (7.11)To estimate J ε j ( v j , Σ ρ ) we use the inclusionΣ ρ × Σ ρ ⊂ k [ i =1 ( R iρ × R iρ ) ∪ k [ i =1 ( P iρ × (Σ ρ \ R iρ )) ∪ k [ i =1 ((Σ ρ \ R iρ ) × P iρ ) ∪ (cid:18)(cid:16) Σ ρ \ k [ i =1 P iρ (cid:17) × (cid:16) Σ ρ \ k [ i =1 P iρ (cid:17)(cid:19) ∪ [ i = j ( R iρ × R jρ ) , which, together with (7.7), gives J ε j ( v j , Σ ρ ) ≤ k X i =1 J ε j ( v j , R iρ ) + k X i =1 J ε j ( v j , P iρ , Σ ρ \ R iρ ) (7.12)+ k X i =1 J ε j ( v j , Σ ρ \ R iρ , P iρ ) + J ε j ( v j , (Σ n − ) ˆ σ ) + X i = j J ε j ( v j , R iρ , R jρ ) . By Lemma 4.3 and (7.9) the sequence { ε j R R iρ |∇ v ij | dx } is uniformly boundedwith respect to j . Taking into account (5.5) and (5.6) we see that the sameproperty holds for { ε j R Σ ρ |∇ v j | dx } . Hence, by Lemma 5.4, the second, third,and fifth terms on the right-hand side of (7.12) tend to zero as j → + ∞ . ByLemma 7.2, J ε j ( v j , (Σ n − ) ˆ σ ) ≤ c ˆ σ H n − (Σ n − ) . (7.13)Combining (7.9), (7.11), (7.12), and (7.13) we getlim sup j → + ∞ F ε j ( v j , Σ ρ ) ≤ Z Σ ψ ( ν u ) d H n − + η H n − (Σ)+ κ ˆ σ + M W c Σ ˆ σ H n − (Σ n − ) + c ˆ σ H n − (Σ n − ) . Since η H n − (Σ) < σ/
2, the conclusion can be obtained by taking ˆ σ sufficientlysmall.We are now ready to prove Theorem 7.1. Proof of Theorem 7.1.
By [8, Lemma 3.1] for every u ∈ BV (Ω; {− , } )there exists a sequence { z k } in BV (Ω; {− , } ) converging to u in L (Ω) suchthat S z k is given by the intersection with Ω with a bounded polyhedral set Σ k of dimension n − H n − ( S z k ) → H n − ( S u ). By Reshetnyak’s convergencetheorem (see, e.g., [42]) this implies thatlim k → + ∞ Z S zk ψ ( ν z k ) d H n − = Z S u ψ ( ν u ) d H n − . F ′′ ( · , Ω) with respect to convergencein L (Ω) it suffices to prove (7.2) for u ∈ BV (Ω; {− , } ) such that S u = Ω ∩ Σwith Σ a bounded polyhedral set of dimension n − σ > < δ < ρ and v j ∈ W , (Σ ρ ) be as inLemma 7.4. Define u j := v j on Σ ρ and u j := u on Ω \ Σ ρ . The properties of v j imply that u j := u on Ω \ Σ ρ − δ for all j sufficiently large. Hence, by (2.3) wehave W ε j ( u j , Ω) ≤ W ε j ( u j , Σ ρ ) . (7.14)To estimate J ε j ( u j , Ω) we consider the inclusionΩ × Ω ⊂ (Σ ρ × Σ ρ ) ∪ (Σ ρ − δ × (Ω \ Σ ρ )) ∪ ((Ω \ Σ ρ ) × Σ ρ − δ ) (7.15) ∪ ((Ω \ Σ ρ − δ ) × (Ω \ Σ ρ − δ )) . Since ∇ u j = ∇ u = 0 on Ω \ Σ ρ − δ , in view of (7.15) we obtain J ε j ( u j , Ω) ≤ J ε j ( u j , Σ ρ )+ J ε j ( u j , Σ ρ − δ , Ω \ Σ ρ )+ J ε j ( u j , Ω \ Σ ρ , Σ ρ − δ ) . (7.16)By Lemmas 4.3 and 5.4 the last two terms tend to zero as j → ∞ , and byLemma 7.4 we deducelim sup j → + ∞ F ε j ( u j , Σ ρ ) ≤ Z Σ ψ ( ν u ) d H n − + σ . Together with (7.14) and (7.16) this shows that F ′′ ( u, Ω) ≤ lim sup j → + ∞ F ε j ( u j , Ω) ≤ Z Σ ψ ( ν u ) d H n − + σ . Letting σ tend to 0 we obtain (7.2). The authors wish to acknowledge the Center for Nonlinear Analysis (NSF PIREGrant No. OISE-0967140) where part of this work was carried out. The researchof G. Dal Maso was partially funded by the European Research Council underGrant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticityand Fracture”, the research of I. Fonseca and G. Leoni was partially fundedby the National Science Foundation under Grants No. DMS-1411646 and No.DMS-1412095, respectively.
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