M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE

Transcription

1 WEAK LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS IN THE LIIT CASE. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Abstract. We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps u : R n R m in W 1,m, 2 m n bounded in W 1,m 1 and convergent in L 1 under mild technical conditions but without any extra coercivity assumption on the integrand. 1. Introduction Let n, m be positive integers, let be an open set of R n and let u : R n R m be a map in the Sobolev space W 1,p, R m ) for some p 1. We denote by u the gradient of the map u, i.e., the m n matrix of the first derivatives of u. The energy functional associated to the map u is an integral of the type F u) = f l ux))) dx, 1.1) where l = m n and l A) denotes the vector with components are all the minors of order up to l of the gradient matrix A R m n, i.e., l A) = A, ad 2 A,..., ad i A,..., ad l A). For instance, 1 A) = A if l = 1, while if l = m = n the last component of l A) is the determinant det A of the matrix A. Energy functional as in 1.1) are considered in nonlinear elasticity, when l = m = n = 3; in particular det u taes into account the contribution to the energy given by changes of volume of the deformation u. The integrand f in 1.1) is assumed to be a convex function; this maes the integral F consistent with the theory of polyconvex and quasiconvex integrals see orrey [23], Ball [4], see also the boo by Dacorogna [6]). We assume that f is bounded below, say f l A)) 0 for all A R m n. To fix the ideas let us assume m = n 2. Then well-nown results by orrey [23] see also Acerbi-Fusco [2] and arcellini [21]) imply that the functional in 1.1) is lower semicontinuous with respect to the wea convergence in W 1,n, R n ). The modelling of cavitation phenomena forces then naturally to consider maps in the Sobolev class W 1,p, R m ) for p < n. For instance, deformation maps of the type ux) = v x ) x x, 1.2) with v : [0, 1] R an increasing function and v0) > 0, belong to W 1,p, R n ) for every p < n, but not to W 1,n, R n ). An extension of the energy functional outside the space W 1,n, R n ) is needed in this case. Different choices have been investigated. Referring to the prototype case of 2000 athematics Subect Classification. 49J45; 49K20. Key words and phrases. Polyconvex integrals, minors, lower semicontinuity. 1

2 2. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE the determinant of u in 1.2) see Ball [4], Fonseca-Fusco-arcellini [11], [12], Giaquinta-odica- Souče [19] and üller [24]), we recall the so called distributional determinant Det u as opposed to the total variation of the determinant, i.e., { } T V p u, ) := inf lim inf det u dx : u ) W 1,n, u u in W 1,p. T V p is an extension of the orginal integral, i.e., T V p u, ) = det u dx for u W 1,n, if and only if for every sequence u ) W 1,n converging wealy to u in W 1,p det u dx lim inf det u dx. 1.3) The lower semicontinuity inequality as in 1.3) for general integrands is the obect of investigation in Theorem 1.1 below under the wea convergence in W 1,p for p < n. arcellini observed in [22] that lower semicontinuity inequality still holds below the critical exponent n. Later on Dacorogna- arcellini [7] proved the lower semicontinuity for p > n 1 see also [15]), while alý [20] exhibited a counterexample in the case p < n 1. Finally, the limit case p = n 1 was addressed by Acerbi-Dal aso [1], Dal aso-sbordone [8], Celada-Dal aso [5] and Fusco-Hutchinson [17]. In particular, in [5] the integrand f can be any nonnegative convex function with no coercivity assumptions. The situation significantly changes when an explicit dependence either on x and/or on u is also allowed, since the presence of these variables cannot be treated as a simple perturbation. Results in this context are due to Gangbo [18], under a structure assumption, and to Fusco-Hutchinson [17] and Fonseca-Leoni [13], assuming the coercivity of the integrand. ore recently, Amar-De Cicco-arcellini-ascolo [3] studied the non-coercive case with u dependence on the integrand, assuming the strict inequality p > n 1. In this paper we deal with the limit case p = n 1 and consider integrals of the general form F u) = f x, ux), n ux))) dx. Other new issues here are that we do not assume coerciveness of the integrand and we allow the dependence on lower order variables and minors. Theorem 1.1. Let m = n 2 and f = fx, u, ξ) : R n R σ [0, ) be such that i) f C 0 R n R σ ) and fx, u, ) is convex for all x, u); ii) denoting ξ = z, t) R σ 1 R, if fx 0, u 0, z 0, ) is constant in t R for some x 0, u 0, z 0 ), then for all z R σ 1 fx 0, u 0, z, t) = inf {fy, v, z, s) : y, v, s) R n R}. 1.4) Then, for every sequence u ) W 1,n, R n ) satisfying we have u u in L 1 and sup u W 1,n 1 < 1.5) F u) lim inf F u ). Note that 1.5) assures that u W 1,n 1, R n ) for n 3, and u BV, R 2 ) if n = 2; in the latter case u denotes the approximate gradient. To highlight the main ideas of our strategy we choose to study first autonomous functionals as in 1.1) for which the proof simplifies. Semicontinuity actually holds in a more general setting, i.e.

3 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 3 under the assumption m n see Theorem 3.1). On the other hand, we are not able to remove the coerciveness assumption when m > n, a difficulty which was also present in the papers [5], [3] details are in Remar 2.8). Further results for m n are discussed for some special integrands see Section 3). The main tools used in the proof of Theorem 1.1 are: i) the De Giorgi s approximation theorem for convex integrands; ii) a suitable generalization of a truncation lemma by Fusco-Hutchinson [17] see Proposition 2.7); iii) a measure-theoretic lemma by Celada-Dal aso [5] see Lemma 2.4). All these tools are carefully combined in an argument which exploits assumption 1.4) in the statement above and the blow-up technique. Note though, that a less general version of the above theorem is proved in the final Section with a suitable new approximation technique, which we believe may be of interest herself. Let us now comment on assumption 1.4), that it is trivially satisfied when f is coercive in t. We consider a simple functional with integrand a positive piecewise affine function Iu) = ax, u) + bx, u) det u) + dx. In this particular case condition 1.4) becomes bx, u) = 0 = ax, u) ) We do not now whether 1.6) is ust a technical assumption or not. In fact, üller s isoperimetric inequality [25] implies that 1.6) is not needed when dealing with sequences with nonnegative determinants see Proposition 3.5). Thus, if one would show that 1.4) is a necessary condition for lower semicontinuity one should wor with sequences whose determinants change sign rapidly. However, all the more or less standard constructions of such sequences fail. The research presented in this paper too origin by the wor of two different groups, one in Firenze and the other one in Napoli. Before Summer 2012 the two groups independently reached quite similar results. Then our colleague Bernard Dacorogna, taling separately with some of us, pointed out the similarities. What to do: cooperation or competition? We decided to continue to study together the problem, and this the reason why the paper has six authors. 2. Definitions and preliminary results The aim of this section is to introduce some notations and to recall some basic definition and results which will be used in the sequel. We begin with some algebraic notation. Let n, m 2 and m n be the linear space of all m n real matrices. For A m n, we denote A = A i ), 1 i m, 1 n, where upper and lower indices correspond to rows and columns respectively. The euclidean norm of A will be denoted by A. The number of all minors up to the order m of any matrix in m n is given by m n m ) n ) σ :=. i i We shall also adopt the following notations. We set I l, = {α = α 1,..., α l ) N l : 1 α 1 < α 2 <... < α l }, where 1 l. If α I l,m and β I l,n, then α,β A) = deta α i β ). By l A) we denote the vector whose components are all the minors of order l, and by l A) the vector of all minors of order up to l.

4 4. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE As usual, Q r x), B r x) denote the open euclidean cube, ball in R n, n 2, with side r, radius r and center the point x, respectively. The center shall not be indicated explicitly if it coincides with the origin Approximation of convex functions. We survey now on an approximation theorem for convex functions, due to De Giorgi, that plays an important role in the framewor of lower semicontinuity problems see [9]). Given a convex function f : R R, 1, consider the affine functions ξ a + b, ξ, with a R and b R, given by a := fη) ν + 1)α η) + α η), η ) dη 2.1) R b := fη) α η)dη, 2.2) R where α C 1 0 R ), N, is a non negative function such that R α η)dη = 1. Lemma 2.1. Let f : R R be a convex function and a, b be defined as in 2.1). Then, fξ) = sup a + b, ξ ), for all ξ R. N The main feature of the approximation above is the explicit dependence of the coefficients a and b on f. In particular, if f depends on the lower order variables x, u) regularity properties of the coefficients a and b with respect to x, u) can be easily deduced from related hypotheses satisfied by f thans to formulas 2.1) and Lemma 2.1. In particular, the following approximation result holds. Theorem 2.2. Let f = fx, u, ξ) : R m R σ [0, ), be a continuous function, convex in the last variable ξ. Then, there exist two sequences of compactly supported continuous functions a i : R m R, b i : R m R σ such that, setting for every i N, f i x, u, ξ) := a i x, u) + b i x, u), ξ ) +, then fx, u, ξ) = sup f i x, u, ξ). i oreover, for every i N there exists a positive constant C i such that a) f i is continuous, convex in ξ and 0 f i x, u, ξ) C i 1 + ξ ) for all x, u, ξ) R m R σ ; 2.3) b) if γ i denotes a modulus of continuity of a i + b i we have f i x, u, ξ) f i y, v, η) C i ξ η + γ i x y + u v )1 + ξ η ) 2.4) for all x, u, ξ) and y, v, η) R m R σ. Let us finally recall the following localization lemma established in [10]. Lemma 2.3. Let µ be a positive Radon measure defined on an open set R n. Consider a sequence φ of Borel positive functions defined on. Then { } sup φ dµ = sup φ i dµ : U i open and pairwise disoint. N N U i

5 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS A truncation method for minors. We first recall a lemma proved in [5, Lemma 3.2]. Lemma 2.4. Let µ ) be a sequence of Radon measures on. Assume that a) there exists T D ) such that µ T in the sense of distributions on ; b) there exists a positive Radon measure ν such that µ + ν wealy in the sense of measures on. Then, there exists a Radon measure µ such that T = µ on and µ T wealy in the sense of measures on. An immediate consequence is the following corollary. Corollary 2.5. Let 2 l m n, u, u W 1,, R m ) be maps such that a) u ) is bounded in W 1,l 1, R m ); b) u ) converges to u in L, R m ); c) there exists c R τ, τ = ) m n ) l l, such that sup c, l u ) + dx <. Then, l 1 u ) l 1 u) and c, l u ) c, l u) wealy in the sense of measures on. Proof. Passing to a subsequence, not relabeled for convenience, we may suppose that for any 1 α = β = l 1 α,β u ) µ α,β We claim that for any h and α = β = h l 1 then wealy in the sense of measures. µ α,β = α,β u)l n. The case h = 1 is obvious. Then, by induction, suppose that the result is true for any α I h 1,m, β I h 1,n with 2 h l 1. Fix now α I h,m, β I h,n and a test function ϕ Cc ), then u α i ) ϕdµ α,β = lim ϕ det dx x β = lim u α ϕ, u α 2 1,..., uα h ) h = lim 1) 1 u α ϕ 1 x β1,..., x βh ) x ˆα1, ˆβ u ) dx, β where ˆα 1 = α 2,..., α h ), ˆβ = β 1,..., β 1, β +1,..., β h ). Since u u in L and by induction assumption ˆα1, ˆβ u ) ˆα1, ˆβ u) wealy in the sense of measures, we have ϕdµ α,β = lim h 1) 1 =1 u α 1 =1 ϕ x ˆα1, ˆβ u ) dx = β ϕ α,β u) dx, that gives the claim. To prove that c, l u ) ) converges to c, l u) wealy in the sense of measures, we first observe that arguing as before, assumptions in items a) and b) ensure that l u )) converges in D, R τ ) to some T. In particular, c, l u ) ) converges to c, T in D ). Then, using assumption c), Lemma 2.4 yields that up to a subsequence c, l u ) ) converges to c, T wealy in the sense of measures to some Radon measure µ = c, T. The equality µ = c, l u) follows by an integration by part argument as before. Since the limit is independent of the extracted subsequence the convergence for the whole sequence follows immediately.

6 6. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Remar 2.6. If condition c) above is strengthened to sup l u ) L 1 <, [17, Lemma 2.2] establishes the wea* convergence in the sense of measures of l u )) to l u) without assuming a). The next result is inspired to [17, Proposition 2.5]. Proposition 2.7. Let n m 2 and let u be W 1,m, R m ) functions and u in W 1,, R m ). Suppose that u u in L 1, R m ) and that sup m 1 u ) L 1 + c, m u ) L 1) <, for some c R τ. Then, there exists a sequence v ) W 1,m, R m ) converging to u in L, R m ), such that m 1 v ) m 1 u), c, m v ) c, m u) 2.5) wealy* in the sense of measures. oreover, an infinitesimal sequence of positive numbers s exists such that {x : u x) v x)} A := {x : u x) ux) > s } 2.6) and lim A 1 + m 1 v ) + c, m v ) ) dx = ) Proof. Let us first fix some notation that shall be used throughout this proof. If A, B are in m n and 1 m the components of A + B) can be written as certain linear combinations of products of components of the vectors i A) and i B). We denote these linear combinations writing A + B) = i A) i B), 2.8) with the convention that 0 A) = 0 B) = 1. Let 0 < s < t and denote by 1 r s t r ϕ s,t r) := s r t t s 0 r t, and by Φ s,t the function We set Note that i=0 Φ s,t y) := y ϕ s,t y ) for all y R m. v s,t := u + Φ s,tu u). L n {x : v s,t u }) L n {x : u u s}) s 1 u u L 1, v s,t u L t, 2.9) and in addition v s,t = u + DΦ s,tu u) u u) where DΦ s,t y) = ϕ s,t y )Id + ϕ s,t y ) y y y. Note that if 1 i m, and α, λ I i,m, since y y is a ran one matrix one easily infers that Finally, we define α,λ DΦ s,t y)) ϕ s,t y )) i + C y ϕ s,t y ) ϕ s,t y )) i ) C := {x : u u x)) = 0},

7 and it is easy to chec that LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 7 v s,t x) = ux) + ϕ s,t u x) ux) ) u x) ux)) for L n a.e. x in C. We now estimate the linear combination of the m-th order minors on the set E s,t := {x : s < u x) ux) < t} by taing into account 2.8) c, m v s,t ) dx = c, m v s,t ) dx + c, m v s,t ) dx 2.11) E s,t m i=0 C + C E s,t C E s,t + E s,t \C m i=0 E s,t C E s,t \C c, m i u) i ϕ s,t u u ) u u)) dx E s,t \C c, m i u) i DΦ s,t u u) u u)) dx 1 + m 1 u ) + c, m u ) ) dx m 1 DΦ s,t u u) u u)) + c, m DΦ s,t u u) u u)) ) dx, where C = Cm, n, u L ). An elementary but lengthy algebraic computation shows that if A m m and B m n then i A B) α,λ A) λ,β B) for 1 i m 1 β I i,n α,λ I i,m m A B) = det A) m B). By taing into account this estimate and 2.10) we conclude that m 1 DΦ s,t u u) u u)) dx and E s,t \C E s,t \C E s,t \C c, m DΦ s,t u u) u u)) dx Therefore, recalling 2.11) we get c, m v s,t ) dx C E s,t E s,t = E s,t \C 1 + C ) t s u u m 1 u u)) dx, detdφ s,t u u)) c, m u u) dx. 1 + m 1 u ) + c, m u ) ) dx + C t s E s,t \C m 1 u ) + c, m u ) ) u u dx. Repeating for the lower order minors the argument used to infer the previous estimate we get

8 8. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE E s,t ) m 1 v s,t ) + c, m v s,t ) C + C t s E s,t \C E s,t 1 + m 1 u ) + c, m u ) ) dx m 1 u ) + c, m u ) ) u u dx. 2.12) To deal with the last integral in 2.12) we recall that u u ) 0 on E s,t \ C, then we use the coarea formula to get for L 1 a.e. t > 0 1 lim m 1 u ) + c, m u ) ) u u dx s t t s {x \C : s< u u <t} 1 t = lim dr m 1 u ) + c, m u ) ) u u s t t s s {x \C : u u =r} u u ) dhn 1 m 1 u ) + c, m u ) = t dh n ) u u ) {x \C : u u =t} Let us now denote by C 0 a constant such that for all N m 1 u ) + c, m u ) dr dh n 1 0 {x \C : u u =r} u u ) = m 1 u ) + c, m u ) ) dx < C 0. \C We now recall the following elementary fact: If g is a nonnegative measurable function in [0, ) with 0 gr) dr C 0, then for every 0 < r 1 < r 2 there exists a set J of positive measure in r 1, r 2 ) such that for all r J C 0 rgr) lnr 2 /r 1 ). By applying for sufficiently large this inequality with 0 < r 1 = u u 1/2 L 1 < r 2 = u u 1/4 L 1 < 1 we find t u u 1/2, u L 1 u 1/4 ) so that L n {x : u L 1 u = t }) = 0 and m 1 u ) + c, m u ) t dh n 1 4C 0 u u ) ln u u L 1 {x \C : u u =t } The latter estimate and 2.12) and 2.13) imply that for every N there exist s u u 1/2, t L 1 ) such that ) m 1 v s,t ) + c, m v 5C 0 s,t ) dx. ln u u L 1 {x : s < u u <t } Therefore, we may conclude by choosing v := v s,t. Indeed, from 2.9), setting we have that A := {x : u u > s }, L n {x : v u }) L n A ) 1 s u u L 1 u u 1/2 L 1,. and v u L t u u 1/4 L 1,

9 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 9 m 1 v ) + c, m v ) ) dx A = m 1 v ) + c, m v ) ) dx {x : s < u u <t } + {x : u u t } m 1 u) + c, m u) ) dx 5C 0 + C m u) dx 0. ln u u L 1 A Finally, the wea* convergence stated in 2.5) follows from Corollary 2.5. Remar 2.8. Let us point out that the assumption m n plays a crucial role in the proof of Proposition 2.7. The reason is essentially of algebraic nature. In fact if m n and A m m and B m n from the algebraic equality m A B) = det A) m B) 2.14) we can trivially estimate c, m A B) with c, m B). Instead, when m > n equality 2.14) is replaced by a more complicate expression that involves suitable linear combinations of higher order minors and it is no longer true that c, m A B) C c, m B), for some positive constant C depending on A. We do not now if Proposition 2.7 fails to be true if m > n. Remar 2.9. For every m and n N, if the second condition in the statement of Proposition 2.7 is replaced with sup l u ) L 1 <, l = m n, then [17, Proposition 2.5] establishes a stronger result. ore precisely, if sup l u ) L 1 <, then the sequence v ) defined accordingly, converges to u in L and satisfies sup l v ) L 1 <, 2.6), and lim A 1 + l v ) ) dx = ) Furthermore, l v )) wealy* converges in the sense of measures to l u) A blow-up type lemma. Let f : R n R σ [0, ) be a continuous function, and consider F v, U) := fx, vx), l vx)))dx, U where U is any open subset in, v W 1,l 1, R m ), if l = m n 3, v BV, R m ) for l = 2. In the latter case v is the density of the absolutely continuous part of the distributional gradient of v. We shall show that to infer the lower semicontinuity inequality F u) lim inf along sequences u ) W 1,l, R m ) satisfying u u in L 1, R m ), F u ), 2.16) and sup u W 1,l 1 <, we can always reduce ourselves to affine target maps thans to the next lemma.

10 10. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Lemma Suppose that for L n a.e. x 0, and for all sequences ε 0 and v ) W 1,l, R m ) such that we have lim inf v v 0 := ux 0 ) y in L 1 Q 1, R m ), and sup v W 1,l 1 <, Q 1 fx 0 + ε y, ux 0 ) + ε v, l v )) dy fx 0, ux 0 ), l ux 0 ))), 2.17) then the lower semicontinuity inequality 2.16) holds. Proof. We employ the blow-up technique introduced by Fonseca & üller [16]. Without loss of generality we may assume that lim inf F u ) = lim F u ) < +, and define the traces of the) non negative Radon measures µ U) := F u, U), U open. Then, as sup µ ) < +, by passing to a subsequence if necessary, there exists a non negative Radon measure µ such that µ µ wealy in the sense of measures. In what follows, by using 2.17) we shall prove that dµ µ Q ε x)) x) = lim dln ε 0 L n Q ε x)) fx, ux), l ux))), 2.18) for L n a.e. x. Clearly, given 2.18) for granted, the conclusion easily follows as lim inf F u ) = lim inf µ ) µ) F u). To this aim we consider the Radon measures ν U) := u l 1 W 1,l 1 U,R n ) and suppose that ν ) converges wealy in the sense of measures to a Radon measure ν. The ensuing properties are satisfied for all x in a set 0 of full measure in dµ dl n x), dν dl n x) exist finite, and 1 lim ε 0 ε n+1 uy) ux) ux) y x) dy = ) Q εx) We shall establish 2.18) for all points in 0. Thus, with fixed x 0 0, let ε 0 be any sequence such that for every N we have µ Q ε x 0 )) = ν Q ε x 0 )) = ) By changing variables, 2.19) rewrites for x = x 0 as lim ux 0 + ε y) ux 0 ) ux 0 ) y ε dy = 0. Q 1 The choice of ε ) in 2.20) yields dµ dl n x µq ε x 0 )) 1 0) = lim ε n = lim lim ε n Q ε x 0 ) fx, u, l u ))dx = lim lim Q 1 fx 0 + ε y, ux 0 ) + ε v,, l v, ))dy, 2.21)

11 where we have set Clearly, v, W 1,l and by denoting we have that LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 11 v, y) := u x 0 + ε y) ux 0 ) ε. v 0 y) := ux 0 ) y 2.22) lim lim v, v 0 L 1 = 0. A standard diagonalization argument provides a subsequence for which 2.21) and 2.22) become v := v, v 0 L 1, and sup v W 1,l 1 < +, + > dµ dl n x 0) = lim fx 0 + ε y, ux 0 ) + ε v, l v )) dy, Q 1 and thus inequality 2.18) is implied by 2.17). 3. The model case In this section we discuss, for m n, the model case of autonomous functionals of the form F v) := f m vx))) dx. where v W 1,m 1, R n ), if n m 3, v BV, R 2 ) for n m = 2. In the latter case v is the density of the absolutely continuous part of the distributional gradient of v. Our result improves upon [5, Theorems 3.1 and 4.1] see also [14, Theorem 10]). Theorem 3.1. Let 2 m n and f : R σ [0, + ) be a convex function. Then, for every sequence u ) W 1,m, R m ) satisfying we have u u in L 1, and sup u W 1,m 1 < 3.1) F u) lim inf F u ). Proof. By Lemma 2.10 it is sufficient to show that lim inf f m v )) dy f m ux 0 ))), 3.2) Q 1 for all points x 0 of approximate differentiability of u and for all sequences v v 0 := ux 0 ) y in L 1, and sup v W 1,m 1 < +. Without loss of generality we may assume that the inferior limit in 3.2) is finite. Furthermore, to infer 3.2) we are left with proving for all i N lim inf f i m v )) dy f i m ux 0 ))), 3.3) Q 1 where f i ξ) = a i + b i, ξ ) +, a i R, b i R σ, are the functions in Lemma 2.1.

12 12. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Fix now v 0 L + 1 and set v x) v, x) := v x) v x) if v x) otherwise. The sequence v, ) is bounded in W 1,m 1 L and as f i 0, we have for all N f i m v, )) dy f i m v )) dy + a i L n {y Q 1 : v > }). 3.4) Q 1 Q 1 As 0 f i f and being the liminf in 3.2) finite by assumption, inequality 3.4) implies that sup b i, m v, ) ) + dy <. Q 1 Then, in view of Lemma 2.4, Proposition 2.7 provides a new sequence w ) satisfying all the conclusions there. In particular, we have f i m w )) dy f i m v, )) dy + a i + b i, m w ) )dy. 3.5) Q 1 Q 1 \A A Recalling the choice of, for sufficiently large we have {y Q 1 : v y) > } A = {y Q 1 : v y) v 0 y) > s }. Therefore, thans to 2.7), 3.4) and 3.5), we get lim inf f i m v )) dy lim inf Q 1 Q 1 f i m w )) dy. 3.6) From this inequality, 3.3) follows at once recalling the wea convergence of m 1 w )) to m 1 ux 0 )) and of b i, m w ) ) to b i, m ux 0 )). A simple variant of the proof of Theorem 3.1 allows us to treat also some special cases when a dependence on x and u appears cp. with [18] for m = n and p > n 1). To be precise, let us consider the functional F u) = hx, u)f m ux))) dx. 3.7) Proposition 3.2. Let m, n, f be as in Theorem 3.1 and F defined by 3.7). If h is a nonnegative continuous function in R m such that either h c 0 > 0 or h hx), then F u) lim inf for every sequence u ) W 1,m, R m ) satisfying u u in L 1, F u ) and sup u W 1,m 1 <. Proof. Let us first assume that h hx) is nonnegative and continuous. In this case the result follows from Theorem 3.1 by a simple continuity and localization argument. If instead h hx, u) c 0 > 0 we use Lemma 2.10 and reduce ourselves to show that lim inf hx 0 + ε y, ux 0 ) + ε v )f m v )) dy hx 0, ux 0 ))f m ux 0 ))), Q 1 where ε 0 and v v 0 = ux 0 ) y in L 1 Q 1 ), and the liminf on the left hand side can be taen finite.

13 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 13 Then the inequality above is proved if we show that for all i N we have lim inf hx 0 + ε y, ux 0 ) + ε v )f i m v )) dy hx 0, ux 0 ))f i m ux 0 ))), 3.8) Q 1 where f i ξ) = a i + b i, ξ ) +, a i R, b i R σ, are the functions in Lemma 2.1. Note that, since the above liminf is finite and h c 0, we infer sup b i, m v ) ) + dy <. 3.9) Q 1 For v 0 L + 1 consider the truncated functions v x) if v x) v, x) = v x) otherwise, v x) the sequence v, ) turns out to be bounded in W 1,m 1 L and it satisfies hx 0 + ε y, ux 0 ) + ε v, ) f i m v, )) dy Q 1 hx 0 +ε y, ux 0 )+ε v )f i m v )) dy+ a i hx 0 +ε y, ux 0 )+ε v, ) dy. Q 1 {y Q 1 : v >} Therefore, arguing as in Theorem 3.1 and taing into account 3.9) and Lemma 2.4, Proposition 2.7 gives a new sequence w ) satisfying all the conclusions there. Thus, using 2.6) and 2.7) as in the proof of 3.6), we get lim inf hx 0 + ε y, ux 0 ) + ε v )f i m v )) dy Q 1 lim inf hx 0 + ε y, ux 0 ) + ε w )f i m w )) dy. Q 1 From this inequality we easily get 3.8), since hx 0 + ε y, ux 0 ) + ε w )) converges uniformly in Q 1 to hx 0, ux 0 )), and since for any open subset U Q 1 L n U)f i m ux 0 ))) lim inf f i m w )) dy thans to the wea convergence of m 1 w )) to m 1 ux 0 )) and of b i, m w ) ) to b i, m ux 0 )). When trying to extend Theorem 3.1 to general integrands depending on lower order variables some difficulties arise. We have not been able neither to prove such a statement nor to find counterexamples in such a generality. Instead, we have established lower semicontinuity either strengthening condition 3.1) cp. with Proposition 3.3) or adding a further condition on the integrand cp. with Theorems 4.1 and 5.1). In Sections 4 and 5 we shall establish the lower semicontinuity property under condition 3.1) adding a mild technical assumption on the integrand f provided the domain and the target space have equal dimension, that is m = n. Instead, for all values of m and n, we shall prove below lower semicontinuity without any further technical assumption on the integrand along sequences that have all the set of minors bounded in L 1. Actually, the latter condition holding true, we need only to suppose convergence in L 1, no bound on any Sobolev norm for the relevant sequence is needed see [1, Theorem 3.5], [5, Theorem 2.2, U

14 14. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Corollary 2.3], [8, Theorem 3.1], [17, Theorem 3.3], and [13, Theorem 1.4, Corollary 1.5] for related results). Proposition 3.3. Let m and n 2, l = m n, and f = fx, u, ξ) : R m R σ [0, ), R n open, be in C 0 R m R σ ), and such that fx, u, ) is convex for all x, u) R m. Then, for every sequence u ) W 1,l, R m ) satisfying we have u u in L 1, and sup l u ) L 1 <, 3.10) F u) lim inf F u ). Proof. We can argue analogously to Lemma 2.10 and infer that to conclude we need to show lim inf fx 0 + ε y, ux 0 ) + ε v, l v )) dy fx 0, ux 0 ), l ux 0 ))), 3.11) Q 1 along sequences v ) W 1,l satisfying v v 0 = ux 0 ) y in L 1, and sup l v ) L 1 < +. Thus, we can apply [17, Proposition 2.5] cp. with Remars 2.6 and 2.9) that provides a sequence w ) W 1, converging to v 0 uniformly in Q 1, such that l w ) l ux 0 )) wealy* in the sense of measures and satisfying 2.15). In particular, by assuming the left hand side in 3.11) is finite, for all functions f i in Theorem 2.2, estimate 2.3) yields f i x 0 + ε y, ux 0 ) + ε w, l w )) dy Q 1 f i x 0 + ε y, ux 0 ) + ε v, l v )) dy + C i 1 + l w ) ) dy, Q 1 {v w } and moreover 2.4) gives f i x 0, ux 0 ), l w )) dy Q 1 f i x 0 + ε y, ux 0 ) + ε w, l w )) dy + γ i ε 1 + w L )) 1 + l w ) ) dy. Q 1 Q 1 Thus, the convexity of f i x 0, ux 0 ), ), the wea* convergence of the minors and 2.15) imply that lim inf f i x 0 + ε y, ux 0 ) + ε v, l v )) dy Q 1 lim inf f i x 0, ux 0 ), l w )) dy f i x 0, ux 0 ), l ux 0 ))). Q 1 Hence, inequality 3.11) follows. Remar 3.4. Note that the counterexample constructed in [18, Theorem 3.1] shows that a continuous, or better a lower-semicontinuous, dependence of the integrand f on the lower order variables is needed.

15 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 15 We point out that if we restrict to maps from R n to itself with positive determinants, assumption 3.1) implies that 3.10) holds locally thans to üller s isoperimetric inequality. Indeed, [25, Lemma 1.3] states that for every u W 1,n, R n ) and x 0 and for a.e. r 0, distx 0, )) one has B rx 0 ) det u dx n 1 n Cn) n 1 u) dh n 1. B rx 0 ) Thus, by integrating this inequality on 0, R), with R < distx 0, ), we get n 1 n det u dx Cn) n 1 u) dx. B R2 x 0 ) R B R x 0 ) Therefore, given a sequence u ) W 1,n, R n ) with positive determinants and bounded in W 1,n 1 we infer that for every sup n u ) L 1,R σ ) <. This observation leads to the following lower semicontinuity result for which we introduce some further notation: if ξ R σ we write ξ = z, t) R σ 1 R. Proposition 3.5. Let m = n 2, R n open, and f = fx, u, z, t) : R n R σ 1 0, ) [0, ) be in C 0 R n R σ 1 [0, )), and such that fx, u, ) is convex for all x, u) R n. Then, for every sequence u ) W 1,n, R n ) satisfying we have u u in L 1, det u > 0 L n a.e. in, and sup u W 1,n 1 <, F u) lim inf F u ). 4. The general case for m = n In this section we address the case of integrands depending on the full set of variables in case m = n. ore precisely, we consider the functional F v) := f x, vx), n vx))) dx, where v W 1,n 1, R n ), if n = m 3, v BV, R 2 ) for n = m = 2. In the latter case v is the density of the absolutely continuous part of the distributional gradient of v. The ensuing result improves upon [3, Theorem 4.2]. We recall the notation ξ = z, t) R σ 1 R if ξ R σ. Theorem 4.1. Let f = fx, u, ξ) : R n R σ [0, ) be such that i) f C 0 R n R σ ), and fx, u, ) is convex for all x, u), ii) if fx 0, u 0, z 0, ) is constant with respect to t R for some point x 0, u 0, z 0 ), then fx 0, u 0, z, t) = inf {fy, v, z, s) : y, v, s) R n R} := gz) for all z R σ ) Then, for every sequence u ) W 1,n, R n ) satisfying we have u u in L 1, and sup u W 1,n 1 < 4.2) F u) lim inf F u ). 4.3)

16 16. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Again, condition 4.2) assures that u BV, R 2 ) if n = 2, and u W 1,n 1, R n ) for n 3. Note that, by the very definition g is a convex function. In addition, we remar that if f is coercive with respect to the t variable, then condition ii) is trivially verified as no such point x 0, u 0, z 0 ) exists. Let us now prove Theorem 4.1. Proof of Theorem 4.1. We split the proof into several intermediate steps. Step 1. Reduction to affine target maps By Lemma 2.10 to infer 4.3) we are left with proving lim inf fx 0 + ε y, ux 0 ) + ε v, n v )) dy fx 0, ux 0 ), n ux 0 ))). 4.4) Q 1 along sequences satisfying v v 0 := ux 0 ) y L 1, and sup v W 1,n 1 < +, for all points x 0 of approximate differentiability of u. As usual we can assume that the left hand side in 4.4) is finite. Let us now distinguish two cases: a) there exists z 0 R σ 1 such that t fx 0, ux 0 ), z 0, t) is constant; b) no such a point exists. Step 2. Proof in case a) In this case we apply assumption ii) in the statement and use at once Theorem 3.1 to get lim fx 0 + ε y, ux 0 ) + ε v, Q n v )) dy 1 lim inf g n 1 v )) dy g n 1 ux 0 ))) = fx 0, ux 0 ), n ux 0 ))). Q 1 Step 3. Proof in case b) Let us now recall that by the approximation Theorem 2.2 there exist three sequences of continuous function with compact support, a i, γ i : R n R and β i : R n R σ 1 such that, setting for all i N and x, u, z, t) R n R σ 1 R, we have f i x, u, z, t) = a i x, u) + β i x, u), z + γ i x, u) t) +, fx, u, z, t) = sup f i x, u, z, t). i N Therefore, to prove 4.4) it is enough to show that for all i N lim Q 1 fx 0 + ε y, ux 0 ) + ε v, n v )) dy f i x 0, ux 0 ), n ux 0 ))). 4.5) To this aim note that there exists N such that γ x 0, ux 0 )) 0 since otherwise we would fall in case a). Without loss of generality we may assume γ x 0, ux 0 )) > 0. Otherwise, we replace the functions v = v 1,..., vn ) with v1, v2,..., vn ), the coefficient γ x, u) with γ x, u 1,..., u n ) and the remaining coefficients a and β accordingly.

17 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 17 Fix now > v 0 L + 1 and set v x) v, x) := v x) v x) if v x) otherwise. Then, as 0 f f, for all we have f x 0 + ε y, ux 0 ) + ε v,, n v, )) dy {y Q 1 : v } 4.6) fx 0 + ε y, ux 0 ) + ε v, n v )) dy. Q 1 4.7) Therefore, since the sequence v, ) is bounded in W 1,n 1 Q 1, R n ) we deduce that sup γ x 0 + ε y, ux 0 ) + ε v ) det v, ) + dy <. {y Q 1 : v } Recalling the choice γ x 0, ux 0 )) > 0, the continuity of γ yields for sufficiently large sup det v, ) + dy <, {y Q 1 : v } in turn implying sup det v, ) + dy <. Q 1 An application of Lemma 2.4 gives that, up to a subsequence not relabeled for convenience, the sequence det v, ) converges wealy* in the sense of measure in Q 1. In particular, det v, ) is bounded in L 1. Hence, Proposition 2.7 provides sequences s 0 and w ) in W 1,n, R n ) satisfying conclusions 2.5), 2.6) and 2.7) there. Note that, for sufficiently large, recalling the choice of, we have {y Q 1 : v y) > } A := {y Q 1 : v y) v 0 y) > s }. Therefore, estimate 2.3) and equation 4.7) imply for all i N f i x 0 + ε y, ux 0 ) + ε w, n w )) dy Q 1 f i x 0 + ε y, ux 0 ) + ε v,, n v, ) dy + C i 1 + n w ) ) dy Q 1 \A A f i x 0 + ε y, ux 0 ) + ε v,, n v, )) dy + C i 1 + A n w ) ) dy {y Q 1 : v } fx 0 + ε y, ux 0 ) + ε v, n v )) dy + C i Q n w ) ) dy. A The convergence of w ) to v 0 in L, the latter inequality, 2.7) and 2.4) imply lim inf fx 0 + ε y, ux 0 ) + ε v, n v )) dy lim inf f i x 0, ux 0 ), n w )) dy. Q 1 Q 1 In turn, from this and by taing into account that n w )) converges to n ux 0 )) wealy* in the sense of measures, by the convexity of f i x 0, ux 0 ), ) we get 4.5), thus concluding the proof.

18 18. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE In case the integrand depends only on the determinant, the same conclusion holds under a local version of the condition in 4.1). Proposition 4.2. Let f = fx, u, t) : R n R [0, ) be such that i) f C 0 R n R), and fx, u, ) is convex for all x, u), ii) if fx 0, u 0, ) is constant with respect to t R for some point x 0, u 0 ), then where and for all δ > 0 fx 0, u 0, t) = hx 0, u 0 ) 4.8) hx, u) := sup h δ x, u) 4.9) δ>0 h δ x, u) := inf {fy, v, t) : y, v, t) B δ x, u) R}. 4.10) Then, for every sequence u ) W 1,n, R n ) satisfying we have u u in L 1, and sup u W 1,n 1 < F u) lim inf F u ). Proof. We argue as in Theorem 4.1: first using the blow-up type Lemma 2.10 to reduce to inequality 4.4). At this point we distinguish as before the two cases a) and b). The latter is dealt with as before, while for case a) we argue as follows. Fix > 0 and consider the function v, in 4.6). Then, fx 0 + ε y, ux 0 ) + ε v,, det v, )dy Q 1 fx 0 + ε y, ux 0 ) + ε v, det v )dy + C L n {y Q 1 : v }). Q 1 In turn, for any δ > 0, by definition of h δ we get lim inf fx 0 + ε y, ux 0 ) + ε v,, det v, )dy h δ x 0, ux 0 )), Q 1 and the conclusion then follows as fx 0, ux 0 ), t) = sup δ h δ x 0, ux 0 )) for all t R. 5. An alternative approach In this section we give an alternative approach without using the blow-up technique exploited to prove Theorem 4.1. This approach relies on an approximation argument see also [18] for similar ideas) that wors in the simplified case treated in Proposition 4.2. ore precisely, we consider the case n = m and the functional F v) := f x, vx), det vx)) dx, where v W 1,n 1, R n ), if n 3, v BV, R 2 ) for n = 2. In the latter case v is the density of the absolutely continuous part of the distributional gradient of v. We state and prove the result under a slightly stronger technical assumption than the one used in Proposition 4.2 cp. with Lemma 5.3).

19 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 19 Theorem 5.1. Let f = fx, u, t) : R n R [0, ) be a continuous function such that fx, u, ) is convex for all x, u) R n. Assume also that the function x, u) R n inf fx, u, t) is continuous. 5.1) t R Then, for every sequence u ) W 1,n, R n ) satisfying we have u u in L 1, and sup u W 1,n 1 < 5.2) F u) lim inf F u ). 5.3) Note that condition 5.2) assures that u BV, R 2 ) if n = 2, and u W 1,n 1, R n ) for n 3. The proof of Theorem 5.1 hinges upon the following approximation result that will be proved at the end of the section. Lemma 5.2. Let g C c R ), 2, be nonnegative. Then for every ε > 0 there exists a function g ε of the type N ε g ε x) = b i,1 x 1 )... b i,n x n ) 5.4) where the b i, s are nonnegative functions in C c R) and for all x R. gx) ε g ε x) gx) Proof of Theorem 5.1. We divide the proof in two steps. Step 1. Algebraic simplifications of the integrand. Possibly replacing the integrand f with f inf t f,, t), without loss of generality, we may tae inf fx, u, t) = 0 for all x, u) t R Rn. 5.5) oreover, Theorem 2.2, a monotone approximation argument and the localization Lemma 2.3 show that we may reduce ourselves to prove 5.3) for integrands fx, u, t) = ax, u) + bx, u)t) +, where a and b are continuous functions with compact support in R n. Assumption 5.5) then yields that bx, u) = 0 = a + x, u) = ) Notice also that we may also assume that b 0. Otherwise, we observe that for all x, u, t) ax, u) + bx, u)t) + = ax, u) + b + x, u)t ) + + ax, u) b x, u)t ) + a + x, u), that the functional u L 1, R n ) a + x, ux)) dx is continuous with respect to the L 1 convergence and that for all u BV, R n ) ax, u) b x, u) det u ) + ãx, dx = ũ) + b x, ũ) det ũ ) + dx, where ũ = u 1, u 2,..., u n ), ãx, ũ) = ax, u) and bx, ũ) = bx, u). Finally, we claim that we may further reduce to the case where n bx, u) = β i x) γ i u ), 5.7) =1

20 20. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE for some continuous functions β i : [0, ), γ i : R [0, ) with compact supports. In fact, recall that from Lemma 5.2, there exists a sequence of functions s of the type of the functions appearing on the right hand side of 5.7), converging uniformly from below to b in R n. Then, if ε is a sequence of strictly positive numbers converging to zero, observe that for all x, u, t) ax, u) + bx, u)t) + a + s x, u) ) +. x, u) a x, u) + s x, u)t 5.8) bx, u) + ε In fact, if bx, u) = 0, then also s x, u) = 0 and the inequality above is trivial, while if bx, u) > 0 we only need to show the inequality when i.e. when s x, u) > 0 and To this aim, note that a + bt = a + s x, u) x, u) a x, u) + s x, u)t > 0, bx, u) + ε t > a x, u) s x, u) a+ x, u) bx, u) + ε. 5.9) a + s ) [ a + s t + b s )t + a+ b + ε s ) ]. b + ε b + ε So the proof of 5.8) will follow if we show that the quantity in square bracets is nonnegative. This is in turn a consequence of 5.9), since b s )t + a+ b + ε s ) b + ε a+ b s ) b + ε + a+ b + ε s ) b + ε 0. Then, the convergence of the right hand side of 5.8) to the left hand follows from the convergence of s x, u) to bx, u) and from 5.6). Finally, replacing β i with β i δ δ if necessary, δ > 0, in such a case note that 0 β i δ δ β i, and arguing as in the proof of the previous claim, we may also assume that for all i = 1,..., L n + i ) = 0, where + i = {x : β i x) > 0}. Step 2. The lower semicontinuity property. Let us now assume fx, u, t) = ax, u) + bx, u)t) + with b satisfying 5.6) and 5.7), and let us consider a sequence of functions u satisfying 5.2). We may also suppose that sup u <. Otherwise, set for > 0 u x) if u x) v, x) = u x) otherwise, u x) and define u similarly. Then, being the functions a and b with compact support in R n, we have F v, ) = F u ), F u ) = F u) if is large enough, and then it is sufficient to establish the lower semicontinuity result for the sequence v, ). As usual, in order to prove 5.3) we may assume that the inferior limit on the right hand side is a limit and that this limit is finite. Therefore, we have that n β i x) h=1 γ ih u h )) det u ) + dx C, for some positive constant C. Let us now set for any N and i = 1,..., v,i = B i1 u 1 ),..., B in u n )), where B ih r) = r 0 γ ih ϱ) dϱ.

21 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 21 The functions v i are defined similarly, by substituting u with u in the equality above. We then have that v,i ) converges to v i in L 1, R n ) and sup + i β i x) det v,i ) + dx <. 5.10) In addition, v,i ) is bounded in L and in W 1,n 1, so that, up to a subsequence not relabeled for convenience, det v,i ) converge, in the sense of distributions on Q 1. Therefore, in view of 5.10), Lemma 2.4 and a diagonal argument imply that det v,i ) locally wealy* converges in the sense of measures on + i. Then, [5, Lemma 1.2] see also [14, Theorem 1.2]) yields that in fact det v,i ) locally wealy* converge in + i to a Radon measure µ i whose absolutely continuous part with respect to the Lebesgue measure is det v i. Let us now fix a nonnegative continuous function ϕ, 0 ϕ 1, with compact support in and a nonnegative continuous function ψ, 0 ψ 1, with compact support in \ + i. Then, we get lim inf = lim = = fx, u, det u ) dx lim inf ax, u ) ϕ ψ dx + lim inf ax, u) ϕ ψ dx + + i ax, u) + bx, u) det u)ϕ ψ dx + ax, u ) + bx, u ) det u ) ϕ ψ dx + i β i x) ϕ ψ det v,i dx β i x) ϕ ψ det v i dx + + i ϕ ψ dµ s i, + i ϕ ψ dµ s i recalling the definition of v i. Thus, taing the supremum over all the possible positive continuous functions ϕ, 0 ϕ 1, we get lim inf fx, u, det u ) dx ν + ψ ) where for all Borel subset E in ν ψ E) := ax, u) + bx, u) det u) ψ dx + E E + i ψ dµ s i. Since the measure ψµs i is singular with respect to the Lebesgue measure, we have ν + ψ ) ax, u) + bx, u) det u) + ψ dx. Letting ψ 1 in \ + i lim inf fx, u, det u ) dx ax, u) + bx, u) det u) + dx. We next discuss the relationship between assumption 4.8) in Proposition 4.2 and assumption 5.1) in Theorem 5.1.

22 22. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Lemma 5.3. Let f C 0 R n R σ ), and fx, u, ) is convex for all x, u), and let g : R m [0, ) be defined by gx, u) := inf{fx, u, t) : t R} for all x, u), and assume that g is continuous. Then, f satisfies equality 4.8). Proof. Let h be the function defined in 4.9) and 4.10), then note that by the very definition of g and h, we have that hx, u) gx, u) fx, u, t) for all x, u, t). 5.11) Let x 0, u 0 ) be as in condition ii) of Proposition 4.2, then by definition gx 0, u 0 ) = fx 0, u 0, t) for all t. 5.12) Note that by the continuity assumption, for every ε > 0 there exists δ > 0 such that For such points y, v) we have that gives gy, v) gx 0, u 0 ) ε for all y, v) B δ x 0, u 0 ). fy, v, t) gy, v) gx 0, u 0 ) ε, for all t, h δ x 0, u 0 ) gx 0, u 0 ) ε, and then hx 0, u 0 ) = gx 0, u 0 ) by 5.11). The conclusion then follows from equality 5.12). To prove Lemma 5.2 we need first some technical results. We start with an elementary algebraic lemma. Lemma 5.4. For any integer 2 there exist N N, q r > 0, p i,r > 0, σ i,r {0, 1, 1}, 1 r N and 1 i, such that N 1 x i = p i,r σ i,r x i ). 5.13) q r r=1 Note that from 5.13) a similar formula holds for 1 + x i with the same q r s, p i,r s and changing the sign of only one of the σ i,r s. Proof. For = 2 we have 1 x 1 x 2 = 1 3 x x 2) + ) x 1 1 x 2 ), 5.14) thus proving formula 5.13). Suppose inductively that 5.13) holds up to some, then by 5.14) we have ) 1 x i )x +1 = 1 3 x i x 1 +1) x i 1 x +1 ), the result follows by applying 5.13) to the products of the first variables. Lemma 5.5. For every ε > 0 there exist N ε nonnegative functions g i C c R ) such that diamsupp g i ) < ε, N ε 1 ε g i x) 1, for all x Q 1, and each function g i is the finite sum of products of continuous, nonnegative functions of one variable with compact supports.

23 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 23 Proof. Fix an integer p 3 and divide Q 1 into p subcubes Q 1/p x h ) where x h = x h,1,..., x h, ). Fix ϕ Cc 1 R) such that ϕ = 1 on t 1/2, 0 ϕ 1, and supp ϕ 1/2 1/p, 1/2 + 1/p). Define, for 1 h p, ϕ h x) := ϕx i x h,i )p). Note that supp ϕ h is contained in the cube with center x h and side 1/p + 2/p 2 and intersects at most 3 1 supports of the remaining functions. Therefore, setting ψx) := p h=1 ϕ h x) we have that 1 ψx) 3 for every x R. Tae now a large integer > 3 to be chosen later and set for every x R and 1 h p ψ h x) := ϕ hx) ψx) = ϕhx) 1 1 ψx) ) The family ψ h ) h provides a partition of unity in Q 1, though each of its members is not a finite sum of products of functions depending on one variable. To this aim, observe that ψ h x) = ϕ hx) i=0 1 ψx) ) i, therefore if we choose 3 2 ψ L, we can find an integer q such that setting g h x) := ϕ hx) q 1 ψx) ) i, we have that 1 ε i=0 p h=1 g h x) 1 for all x Q 1. To conclude we are left with proving that the functions x 1 ψx) as finite sum of products of functions of one variable. In fact, observe that 1 ψx) = 1 1 p ϕ h x) = 1 1 p ϕx i x h,i )p) h=1 h=1 [ ] = 1 p p p 1 ϕx i x h,i )p). h=1 By applying Lemma 5.4 we get for every h p N 1 ϕx i x h,i )p) = q r r=1 [ p i,r and the result is proved if we choose satisfying in addition p min{p i,r : 1 i, 1 r N}. p ] σ i,r ϕx i x h,i )p) can be written

24 24. FOCARDI N. FUSCO C. LEONE P. ARCELLINI E. ASCOLO A.VERDE Proof of Lemma 5.2. Without loss of generality we may assume that g has support contained in Q 1. With fixed ε > 0 choose δ > 0 so that if x y < δ then gx) gy) < ε. Consider a finite family Lemma 5.4 provides a finite family of functions g i, 1 i N δ, with diamsupp g i ) < δ, 1 δ N δ g ix) 1. Then the required approximation is obtained setting N δ g ε x) := m i g i x) with m i = min g. x suppg i Acnowledgements The research of N. Fusco, C. Leone and A. Verde was supported by the 2008 ERC Advanced Grant N Analytic Techniques for Geometric and Functional Inequalities. Part of this wor was conceived when. Focardi visited the University of Naples. He thans N. Fusco, C. Leone, and A. Verde for providing a warm hospitality and for creating a stimulating scientific atmosphere. References [1] E. Acerbi, G. Dal aso, New lower semicontinuity result for polyconvex integrals, Calc. Var. Partial Differential Equations, ), [2] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational ech. Anal ), [3]. Amar, V. De Cicco, P. arcellini and E. ascolo, Wea lower semicontinuity for non coercive polyconvex integrals, Adv. Calc. Var., ), 2, [4] J.. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. ech. Anal., ), [5] P. Celada, G. Dal aso, Further remars on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire, ), [6] B. Dacorogna, Direct methods in the Calculus of Variations, Appl. ath. Sci. 78, Springer, Berlin, [7] B. Dacorogna, P. arcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C.R. ath. Acad. Sci. Paris, ), [8] G. Dal aso, C. Sbordone, Wea lower semicontinuity of polyconvex integrals: a borderline case, ath. Z., ), [9] E. De Giorgi, Teoremi di semicontinuità nel Calcolo delle Variazioni, Corso INDA, 1968/1969). [10] E. De Giorgi, G. Buttazzo, G. Dal aso, On the lower semicontinuity of certain integral functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. at. Natur. Rend. Lincei 9) at. Appl., ), [11] I. Fonseca, N. Fusco, P. arcellini, On the total variation of the Jacobian, J. Funct. Anal ), [12] I. Fonseca, N. Fusco, P. arcellini, Topological degree, Jacobian determinants and relaxation, Boll. Unione at. Ital. Sez. B ), [13] I. Fonseca, G. Leoni, Some remars on lower semicontinuity, Indiana Univ. ath. J., ), [14] I. Fonseca, G. Leoni, J. alý, Wea continuity and lower semicontinuity results for determinants, Arch. Rational ech. Anal ), [15] I. Fonseca, P. arcellini, Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geometric Analysis ), [16] I. Fonseca, S. üller, Quasi-convex integrands and lower semicontinuity in L 1 SIA J. ath. Anal., ), [17] N. Fusco, J.E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, anuscripta ath., ), [18] W. Gangbo, On the wea lower semicontinuty of energies with polyconvex integrands, J. ath. Pures Appl., ),

25 LOWER SEICONTINUITY FOR POLYCONVEX INTEGRALS 25 [19]. Giaquinta, G. odica, J. Souče, Cartesian currents in the calculus of variations, voll. I and II, Springer- Verlag, Berlin, [20] J. alý, Wea lower semicontinuity of polyconvex integrals, Proc. Edinb. ath. Soc., ), [21] P. arcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, anuscripta ath., ), [22] P. arcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, ), [23] C.B. orrey, ultiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, 130 Springer-Verlag, New Yor [24] S. üller, Det = det. A remar on the distributional determinant, C. R. Acad. Sci. Paris Sr. I ath ), [25] S. üller, Higher integrability of determinants and wea convergence in L 1, J. Reine Angew. ath ), Università di Firenze address: focardi@math.unifi.it address: marcellini@math.unifi.it address: mascolo@math.unifi.it Università di Napoli Federico II address: n.fusco@unina.it address: chileone@unina.it address: anverde@unina.it Dipartimento di atematica e Informatica U. Dini, Università degli Studi di Firenze, Viale orgagni 67/A, Firenze, Italy Dipartimento di atematica R. Caccioppoli, Università degli Studi Federico II di Napoli, Via Cinthia - onte S. Angelo, Napoli, Italy