jpmm-31-3-11

Lower semicontinuity and relaxation of signed functionals with linear growth in the context of A-quasiconvexity Margarida Ba´ıa†, Milena Chermisi‡, Jose´ Matias† and Pedro M. Santos† † Departamento de Matema´tica, Instituto Superior Te´cnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal ‡ Department of Mathematical Sciences, New Jersey Institute of Technology 323 Dr. M.L. King, Jr. Blvd., Newark NJ 07102, USA March 31, 2011 Abstract A lower semicontinuity and relaxation result with respect to weak-∗ convergence of measures is derived for functionals of the form µ ∈M(Ω;Rd)→ ∫ Ω f(µa(x)) dx+ ∫ Ω f∞ ( dµs d|µs| (x) ) d|µs|(x), where admissible sequences {µn} are such that {Aµn} converges to zero strongly in W−1,qloc (Ω) and A is a partial differential operator with constant rank. The integrand f has linear growth and L∞-bounds from below are not assumed. 1 Introduction In this work we start by deriving a lower semicontinuity result with respect to weak-∗ convergence of A-free measures for the functional F(µ) = ∫ Ω f(µa) dx+ ∫ Ω f∞ ( dµs d|µs| ) d|µs|, µ ∈M(Ω;Rd), (1.1) where Ω is an open bounded subset of RN , M(Ω;Rd) stands for the set of finite Rd-valued Radon measures over Ω, µ = µaLN + µs is the Radon-Nikody´m decomposition of µ with respect to the Lebesgue measure LN . Here and in what follows, the integrand f : Rd → R is assumed to be A- quasiconvex (see Section 2 for other notations and preliminary definitions), where A is a linear first order partial differential operator of the form A := N∑ i=1 A(i) ∂ ∂xi , A(i) ∈MM×d(R), M ∈ N, (1.2) that we assume throughout to satisfy Murat’s condition of constant rank (see Murat [15] and Fonseca & Mu¨ller [10]) i.e., there exists c ∈ N such that rank ( N∑ i=1 A(i)ξi ) = c for all ξ = (ξ1, ..., ξN ) ∈ SN−1. In addition we assume f to be Lipschitz continuous and we remark that this condition implies f to satisfy a linear growth condition at infinity of the type |f(v)| ≤ K(1 + |v|) (1.3) for all v ∈ Rd and for some K > 0. As usual (see Remark 3.1) we denote by f∞ the recession function of f , which for our problem is defined as f∞(ξ) := lim sup t→∞ f(tξ) t . (1.4) 1 As already proved by Fonseca & Mu¨ller [10] A-quasiconvexity with respect to the last variable turns out to be a necessary and sufficient condition for the lower semicontinuity of (u, v)→ ∫ Ω f(x, u(x), v(x)) dx for positive normal integrands f with linear growth among sequences (un, vn) such that un → u in measure, vn ⇀v in L 1 and Avn = 0. In Fonseca, Leoni & Mu¨ller [9] this result was partially extended by considering weak-∗ convergence in the sense of measures (in the variable v). Precisely the authors considered a funtional of the form v → ∫ Ω f(x, v(x)) dx and, in particular, it was proved that∫ Ω f (x, µa(x)) dx ≤ lim n→∞ ∫ Ω f(x, vn(x)) dx (1.5) for any sequence vn ⊂ L1(Ω;Rd) ∩ kerA and such that vn ⇀µ in the sense of measures, under the assumptions that f is a Borel measurable positive function with linear growth, Lipschitz continuous and A-quasiconvex in the last variable, and satisfying an appropriate continuity condition on the first variable (see Theorem 1.4 in [9]). Note that in (1.5) the term µs has not been considered. Here we extend this last result for a larger class of integrands where L∞-bounds from below are not assumed and to functionals taking into account the singular part of the limit measure µ. Namely, we prove the following theorem. Theorem 1.1. Let Ω ⊂ RN be a bounded open set and let f : Rd → R be A-quasiconvex and Lipschitz continuous. Let {µn} ⊂ M(Ω;Rd) be such that µn ∗⇀ µ ∈M(Ω;Rd), Aµn ∈W−1,qloc (Ω;RM ), 1 < q < NN−1 , Aµn W−1,qloc (Ω;R M )−→ 0 and |µn| ∗⇀ Λ ∈M(Ω) with Λ(∂Ω) = 0. Then F(µ) ≤ lim inf n→∞ F(µn) (1.6) where F is the functional in (1.1) with f∞ defined by (1.4). Note that lower semicontinuity may fail if Λ(∂Ω) 6= 0 (see Example 3.3). The proof of Theorem 1.1 is reduced to the case of sequences of C∞-functions by a regularization argument and an upper semicontinuous result based on Reshetnyak Continuity Theorem (see Section 3 and Proposition 3.2). To show Proposition 3.2 with a regular sequence of functions {un} we start, following ideas of Kristensen & Rindler [13], by estimating from below the limit of the sequence of local energies λn(A) := ∫ A f(un) dx. Contrary to the case for positive integrands, this step is essential to write the limit energy of λn, λ, exclusively in terms of µ. The result then follows from pointwise estimates on the Radon-Nikody´m Derivatives of λ obtained by the usual blow-up argument (introduced in Fonseca & Mu¨ller [11]). The main difficulty here arises in the treatment of the singular part dλd|µs| since we do not know how to characterize the blow-up limit. This difficulty is overcomed by an appropriate average process that allows us to get the estimate for this singular part. The motivation for this work relies on a characterization of Young measures generated by uniformly bounded and A-free sequences of measures through the duality with an appropriate set of functions with linear growth (work in progress). In the particular case where µ = Du for u ∈ BV (i.e. A = curl) Theorem 1.1 has been derived by Kristensen & Rindler [13]. In this context the notion of A-quasiconvexity reduces to that of quasiconvexity (which implies Lipchitz continuity). 2 The second objective of the present paper is to give a relaxation result for the functional (1.1) in the context of A-quasiconvexity. Namely, in the next theorem we show that the functional G defined by G(µ) := inf { lim inf n→∞ F(µn) : µn ∗ ⇀ µ, Aµn ∈W−1,qloc (Ω;RM ), Aµn W−1,qloc (Ω;R M )−→ 0, |µn| ∗⇀ Λ with Λ(∂Ω) = 0 } . admits an integral representation. Theorem 1.2. Let Ω ⊂ RN be a bounded open set and let f : Rd → R be Lipschitz continuous. Then for µ ∈M(Ω¯;Rd) ∩ kerA such that |µ|(∂Ω) = 0 we have that G(µ) = ∫ Ω QAf(µa(x)) dx+ ∫ Ω ( QAf )∞( dµs d|µs| ) d|µs|. where QAf denotes the quasiconvex envelope of f and ( QAf )∞ denotes its recession function. In the proof of Theorem 1.2 the lower bound is a immediate consequence of Theorem 1.1, while the upper bound is based on a regularization procedure together with an approximation by piecewise constant functions, that follows naturally from the definition of A-quasiconvexity. We finish this introduction by referring to Braides, Fonseca & Leoni [6] for other relaxation results in the context of A-quasiconvexity (for p > 1) and to Kristensen & Rindler [13] for relaxation for signed functionals in the context of gradients (i.e, as mentioned before µ = Du for some u ∈ BV ). The overall plan of this work in the ensuing sections will be as follows: Section 2 collects the main definitions and auxiliary results used in the proof of Theorem 1.1 that can be found in Section 3. In Section 4 we present the proof of Theorem 1.2. 2 Preliminary results In this section we recall the main results used in our analysis. We start by fixing some notations. 2.1 General Notations Throughout the text we will use the following notations: - Ω ⊂ RN , N ≥ 1, will denote an open bounded set; - LN and HN−1 denote, respectively, the N -dimensional Lebesgue measure and the (N − 1)- dimensional Hausdorff measure in RN ; - SN−1 stands for the unit sphere in RN ; - Q denotes the open unit cube centered at the origin with one side orthogonal to eN , where eN denotes the N th-element of the canonical basis of RN ; - Q(x, δ) denotes the open cube centered at x with side length δ > 0 and with one side orthogonal to eN ; - B stands for the unit open ball centered at the origin; - B(x, δ) denotes the ball centered at x with radius δ > 0; - MM×d(R) stand for the set of M × d real matrices; 3 - C∞per(Q;Rd) is the space of all Q-periodic functions in C∞(RN ;Rd); - Lqper(Q;Rd) is the space of all Q-periodic functions in L q loc(RN ;Rd); - D′(Ω;RM ) denotes the space of distributions in Ω with values in RM . - C represents a generic positive constant, which may vary from expression to expression; - lim n,m := lim n→∞ limm→∞ . 2.2 Measure Theory In this section we recall some notations and well known results in Measure Theory (see e.g Ambrosio, Fusco & Pallara [5], Evans & Gariepy [12] and Fonseca & Leoni [8], as well as the bibliography therein). Let X be a locally compact metric space and let Cc(X;Rd), d ≥ 1, denote the set of continuous functions with compact support on X. We denote by C0(X;Rd) the completion of Cc(X;Rd) with respect to the supremum norm. Let B(X) be the Borel σ-algebra of X. By the Riesz-Representation Theorem the dual of the Banach space C0(X;Rd), denoted by M(X;Rd), is the space of finite Rd- valued Radon measures µ : B(X)→ Rd under the pairing < µ,ϕ >:= ∫ X ϕdµ ≡ d∑ i=1 ∫ X ϕi dµi where ϕ = (ϕ1, ..., ϕd) and µ = (µ1, ..., µd). The space M(X;Rd) will be endowed with the weak∗- topology deriving from this duality. In particular a sequence {µn} ⊂ M(X;Rd) is said to weak∗- converge to µ ∈M(X;Rd) (indicated by µn ?⇀ µ) if for all ϕ ∈ C0(X;Rd) lim n→∞ ∫ X ϕdµn = ∫ X ϕdµ. If d = 1 we write by simplicity M(X) and we denote by M+(X) its subset of positive measures. Given µ ∈M(X;Rd) let |µ| denote its total variation and let supp µ denote its support. The following result can be found in Fonseca & Leoni [8, Corollary 1.204]. Proposition 2.1. Let µn ∈ M(X) such that µn ∗⇀µ in M(X) and |µn| ∗⇀ν in M(X). If A ⊂ X is open, A¯ compact and ν(∂A) = 0 then µn(A)→ µ(A). We recall that a measure µ is said to be absolutely continuous with respect to a positive measure ν, written µ << ν, if for every E ∈ B(X) the following implication holds: ν(E) = 0 ⇒ µ(E) = 0. Two positive measures µ and ν are said to be mutually singular, written µ ⊥ ν, if there exists E ∈ B(X) such that ν(E) = 0 and µ(X \E) = 0. For general vector-valued measures µ and ν we say that µ ⊥ ν if |µ| ⊥ |ν|. Theorem 2.2 (Lebesgue-Radon-Nikody´m Theorem). Let µ ∈M+(X) and ν ∈M(X;Rd). Then (i) there exists two Rd-valued measures νa and νs such that ν = νa + νs (2.1) with νa << µ and νs ⊥ µ. Moreover, the decomposition (2.1) is unique, that is, if ν = ν¯a + ν¯s for some measures ν¯a, ν¯s, with ν¯a << µ and ν¯s ⊥ µ, then νa = ν¯a and νs = ν¯s; 4 (ii) there is a µ-measurable function u ∈ L1(Ω;Rd) such that νa(E) = ∫ E u dµ for every E ∈ B(Ω). The function u is unique up to a set of µ measure zero. The decomposition ν = νa + νs is called the Lebesgue decomposition of ν with respect to µ (see [8, Theorem 1.115]) and the function u is called the Radon-Nikody´m derivative of ν with respect to µ, denoted by u = dν/dµ (see [8, Theorem 1.101]). The next result is a strong form of Besicovitch derivation Theorem due to Ambrosio and Dal Maso [4] (see also [5, Theorem 2.22 and Theorem 5.52] or [8, Theorem 1.155]). Theorem 2.3. Let µ ∈ M+(Ω) and ν ∈ M(Ω;Rd). Then there exists a Borel set N ⊂ Ω with µ(N) = 0 such that for every x ∈ (supp µ)\N dν dµ (x) = dνa dµ (x) = lim �→0 ν ( (x+ �D) ∩ Ω) µ ( (x+ �D) ∩ Ω) ∈ R and dνs dµ (x) = lim �→0 νs ( (x+ �D) ∩ Ω) µ ( (x+ �D) ∩ Ω) = 0, where D is any bounded, convex, open set D containing the origin (the exceptional set N is independent of the choice of D). In the sequel we denote by W−1,q(Ω;Rd) the dual space of W 1,q ′ 0 (Ω;Rd) where q′, the conjugate exponent of q, is given by the relation 1q + 1 q′ = 1. We finish this part by recalling that M(Ω;Rd) is compactly imbeded in W−1,q(Ω;Rd), 1 < q < NN−1 , since W 1,q′ 0 (Ω;Rd) ⊂⊂ C0(Ω) for q′ > N . 2.3 A corollary of Reshetnyak’s Theorem The objective of this part is to present a corollary of Reshetnyak Continuity Theorem useful for our main result in Section 3. Definition 2.4. (The space E(Ω;Rd)) Let E(Ω;Rd) denote the space of continuous functions f : Ω× Rd → R such that the mapping (x, ξ)→ (1− |ξ|)f ( x, ξ 1− |ξ| ) , x ∈ Ω, ξ ∈ B, (2.2) can be extended to a continuous function to the closure Ω×B. The recession function of an element f of E(Ω;Rd) is the continuous extension of (2.2) to the boundary of Ω×B. Namely we have the following definition. Definition 2.5. (Recession function) Let f be a function in E(Ω;Rd). Then recession function of f is defined by f∞(x, ξ) = lim x ′ → x ξ ′ → ξ t → ∞ f(x ′ , tξ ′ ) t . (2.3) for all (x, ξ) ∈ Ω×B. 5 The next lemma is an approximation result by functions in E(Ω;Rd) and is due to Alibert and Bouchitte´ ([3, Lemma 2.3]). Lemma 2.6. Let f : Ω× Rd → R be a lower semicontinuous function such that f(x, ξ) ≥ −C(1 + |ξ|). Then, there exists an nondecreasing sequence {fk} ⊂ E(Ω;Rd) such that supkfk(x, ξ) = f(x, ξ) and supkf ∞ k (x, ξ) = hf (x, ξ) where hf (x, ξ) := lim inf x ′ → x ξ ′ → ξ t → ∞ f(x ′ , tξ ′ ) t . The version of Reshetnyak’s Continuity Theorem we present here can be found in [13, Theorem 5] Theorem 2.7. (Reshetnyak’s Continuity Theorem) Let f ∈ E(Ω;Rd) and let µ, µn ∈ M(Ω;Rd) be such that µn ∗ ⇀µ in M(Ω;Rd) and 〈µn〉(Ω)→ 〈µ〉(Ω), where 〈ν〉 := √ 1 + |νa|2LN + |νs|, ν = νaLN + νs ∈M(Ω;Rd). Then lim n→∞ F˜(µn) = F˜(µ) where F˜(ν) := ∫ Ω f(x, νa(x)) dx+ ∫ Ω f∞ ( x, dνs d|νs| (x) ) d|νs|, ν ∈M(Ω;Rd). (2.4) As a corollary of Lemma 2.6 and Theorem 2.7 we derive an upper semicontinuity result useful in the proof of our main result Theorem 1.1. Corollary 2.8. Let f : Ω× Rd → R be a continuous function such that |f(x, ξ)| ≤ C(1 + |ξ|), for all x ∈ Ω, all ξ ∈ Rd, and some C > 0. Let µ, µn ∈M(Ω;Rd) be such that µn ∗⇀µ in M(Ω;Rd) and 〈µn〉(Ω)→ 〈µ〉(Ω). Then F˜(µ) ≥ lim sup n→∞ F˜(µn) (2.5) where F˜ is the functional defined in (2.4) and where the recession function of f is defined as follows f∞(x, ξ) := lim sup x ′ → x ξ ′ → ξ t → ∞ f(x ′ , tξ ′ ) t . Proof. By Lemma 2.6 we can find a nondecreasing sequence of continuous functions fh ∈ E(Ω;Rd), h ∈ N, such that for all (x, ξ) ∈ Ω× Rd sup h∈N fh(x, ξ) = −f(x, ξ) and sup h∈N f∞h (x, ξ) = h−f (x, ξ) = −f∞(x, ξ). 6 For each h ∈ N we have that lim sup n→∞ F˜(µn) = − lim inf n→∞ {−F˜(µn)} ≤ − lim n→∞ [∫ Ω fh(x, µ a n(x)) dx+ ∫ Ω f∞h ( x, dµsn d|µsn| (x) ) d|µsn| ] = − [∫ Ω fh(x, µ a(x)) dx+ ∫ Ω f∞h ( x, dµs d|µs| (x) ) d|µs| ] (2.6) by Theorem 2.7. Taking the infimum over h in (2.6), inequality (2.5) follows by the Monotone Convergence Theorem. 2.4 A-quasiconvexity We recall here the notion of A-quasiconvexity introduced by Dacorogna [7] and further devoloped by Fonseca & Mu¨ller [10], as well as some of its main properties. Let A : D′(Ω;Rd)→ D′(Ω;RM ) be the first order linear differential operator defined in (1.2). Definition 2.9. (A-quasiconvex function) A locally bounded Borel function f : Rd → R is said to be A-quasiconvex if f(v) ≤ ∫ Q f(v + w(x)) dx for all v ∈ Rd and for all w ∈ C∞per(Q;Rd) such that Aw = 0 in RN with ∫ Q w(x) dx = 0. Remark 2.10. If f has q-growth, i.e. |f(v)| ≤ C(1 + |v|q) for all v ∈ Rd, then the space of test functions C∞per(Q;Rd) in Definition 2.9 can be replaced by Lqper(Q,Rd) (see Remark 3.3.2 in [10]). Definition 2.11. (A-quasiconvex envelope) Let f : Rd → R be a continuous function. We define the A-quasiconvex envelope of f , QAf : Rd → R ∪ {−∞}, as QAf(v) := inf {∫ Q f(v + w(x)) dx : w ∈ C∞per(Q;Rd) such that Aw = 0 in RN and ∫ Q w(x) dx = 0 } . Remark 2.12. Let f : Rd → R be a continuous function. i) If f has linear growth at infinity and QAf(0) > −∞ then QAf(v) is finite for all v ∈ Rd. In addition QAf has also linear growth at infinity. ii) If f is Lipschitz continuous then QAf is also Lipschitz continuous. The next lemma is an adapted version of Lemma 4 in Kristensen & Rindler [13] for A-quasiconvex envelopes. Lemma 2.13. Let f : Rd → R be a continuous function with linear growth at infinity such that QAf(0) > −∞. Given γ > 0 define fγ(v) := f(v) + γ|v| for v ∈ Rd. Then QAfγ(v) ↓ QAf(v) and (QAfγ)∞(v) ↓ (QAf)∞(v) pointwise in v as γ → 0. The following proposition can be found in [10, Lemma 2.14]. Proposition 2.14. Given q > 1, there exists a linear bounded operator P : Lqper(Q;Rd)→ Lqper(Q;Rd) such that A(Pu) = 0. Moreover we have the following estimate ||u− Pu||Lq ≤ C‖Au‖W−1,q for every u ∈ Lqper(Q;Rd) with ∫ Q u = 0. 7 The following lower semicontinuity result is used in the proof of Theorem 1.1. Lemma 2.15. Let f : Rd → R be a A-quasiconvex and Lipschitz continuous function. Let a ∈ Rd and {un} ⊂ Lqper(Q;Rd) be a sequence such that un ∗⇀ aLN in M(Q;Rd) and |un| ∗⇀ Λ in M+(Q), with Λ(∂Q) = 0, and Aun → 0 in W−1,q(Q;RM ) for some 1 < q < NN−1 . Then lim inf n→∞ ∫ Q f(un) dx ≥ f(a). Proof. Choose ϕm ∈ C∞c (Q; [0, 1]) satisfying the condition ϕm = 1 on Q ( 0; 1− 1m ) and define {wm,n} ⊂ Lqper(Q;Rd) by wm,n = ϕm(un − a). Writting A(wm,n) = (Aϕm) (un − a) + ϕmAun we can conclude that lim n→+∞ ∫ Q wm,n(x) dx = 0 and A(wm,n) W −1,q(Q;RM )−→ n→∞ 0 (2.7) since un ∗ ⇀ a inM(Q;Rd) implies that un → a in W−1,q(Q;RM ). Define now the sequence {zm,n} ⊂ Lqper(Q;Rd) by zm,n := P ( wm,n − ∫ Q wm,n dx ) . Then, by Lipschitz continuity, A-quasiconvexity (see Remark 2.10) and Proposition 2.14 we have that∫ Q f(un) dx = ∫ Q f(un − a+ a) dx ≥ ∫ Q f(wm,n + a) dx− L ∫ Q |1− ϕm‖un − a| dx ≥ ∫ Q f ( wm,n − ∫ Q wm,n + a ) dx− L ∫ Q |1− ϕm‖un − a| dx −L ∣∣∣ ∫ Q wm,n dx ∣∣∣ ≥ ∫ Q f(zm,n + a)− L ∫ Q |1− ϕm‖un − a| dx− L ∣∣∣ ∫ Q wm,n dx ∣∣∣ −L ∫ Q ∣∣∣∣wm,n − ∫ Q wm,n dx− zm,n ∣∣∣∣ dx ≥ f(a)− L ∫ Q |1− ϕm‖un − a| dx− L ∣∣∣ ∫ Q wm,n dx ∣∣∣ −CL‖Awm,n‖W−1,q . Taking first the limit as n→∞ and using the definition of wm,n and (2.7), we have lim inf n→∞ ∫ Q f(un) dx ≥ f(a)− LΛ ( Q\Q ( 0, 1− 1 m )) −L|a| ( 1− ( 1− 1 m )N) . 8 The result now follows letting m→∞ since by hypothesis Λ(∂Q) = 0. Remark 2.16. Lemma 2.15 can also be applied to any cube P ⊂ RN . 2.5 Regularization of measures The aim of this part is to recall the definition of the regularization of a measure by means of its convolution with a standard mollifier as well as to gather its main properties. Let ρ ∈ C∞c ( RN ) with supp ρ ⊂ B and ∫RN ρ (x) dx = 1. For every ε > 0 let us define the mollifier ρε (x) := 1 εN ρ (x ε ) , x ∈ RN . (2.8) Note that supp ρε ⊂ B (0, ε). Given µ ∈ M ( Ω;Rd ) we may think µ as an element of M (RN ;Rd) with support contained in Ω. We define uε : RN → Rd by uε (x) := (µ ∗ ρε) (x) = ∫ RN ρε (x− y) dµ (y) , x ∈ RN (2.9) and for every Borel set E ⊂ Ω we denote Bε(E) := {x ∈ RN : dist (x,E) < ε}. Proposition 2.17. Let µ ∈ M (Ω;Rd) and uε be given as in (2.9). Then the following statements hold: (i) The function uε ∈ C∞ ( RN ;Rd ) and suppuε ⊂ Bε(Ω). Moreover Dα(µ ∗ ρε) = Dαµ ∗ ρε for α ∈ NN and the inequality ∫ E |µ ∗ ρε|(x) dx ≤ |µ|(Bε(E)) (2.10) holds whenever E ⊂ Ω is a Borel set. (ii) The measures µε := uεLN and |µε| weak*- converge in RN to µ and |µ|, respectively, as ε→ 0. (iii) If |µ|(∂Ω) = 0 then 〈µε〉(Ω)→ 〈µ〉(Ω) as ε→ 0. (iv) If Aµ ∈W−1,qloc (Ω;RM ), 1 ≤ q <∞, then Aun W−1,qloc (Ω;R M )−→ n→∞ Aµ. Proof. The assertions (i)-(ii) follow Theorem 2.2 in Ambrosio & Fusco & Pallara [5]. Proof of (iii). Let µˆ := (µ,LN ). As µˆ ∗ ρε ∗⇀ µˆ, we have lim inf |µˆ ∗ ρε|(Ω) ≥ |µˆ|(Ω). On the other hand as |µˆ ∗ ρε| ∗⇀ |µˆ| and |µ|(∂Ω) = 0 we have that lim sup |µˆ ∗ ρε|(Ω) ≤ lim sup |µˆ ∗ ρε|(Ω) ≤ |µˆ|(Ω) = |µˆ|(Ω). Now the result follows from the equalities 〈µε〉(Ω) = |µˆ ∗ ρε|(Ω) and 〈µ〉(Ω) = |µˆ|(Ω). Proof of (iv). We have that Aun = Aµ ∗ ρεn . Given U ⊂⊂ Ω let us see that Aun W −1,q(U ;RM )−→ n→∞ Aµ. Let V with U ⊂⊂ V ⊂⊂ Ω. As Aµ ∈ W−1,q(V ;RM ), there exist Ti ∈ Lq(V ;RM ), i = 0, ..., N , such that Aµ = T0 + N∑ i=1 ∂Ti ∂xi 9 (see Adams [1]). Given ϕ ∈ C∞c (U ;RM ) 〈Aun −Aµ, ϕ〉 = 〈 ρεn ∗ ( T0 + N∑ i=1 ∂Ti ∂xi ) − ( T0 + N∑ i=1 ∂Ti ∂xi ) , ϕ 〉 = 〈ρεn ∗ T0 − T0, ϕ〉 − N∑ i=1 〈 ρεn ∗ Ti − Ti, ∂ϕ ∂xi 〉 and consequently, by Ho¨lder inequality | 〈Aun −Aµ, ϕ〉 | ≤ N∑ i=0 ‖ρεn ∗ Ti − Ti‖Lq(U ;RM )‖ϕ‖W 1,q′ (U ;RM ). (2.11) By density (2.11) holds for any ϕ ∈W 1,q ′ 0 (U ;RM ) and then as N∑ i=0 ‖ρεn ∗ Ti − Ti‖Lq(U ;RM ) −→ n→∞ 0 we conclud