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

Transcription

1 WEAK LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS IN THE LIMIT CASE M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO 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) = fm l ux))) dx, 1.1) where l = m n and M l A) denotes the vector whose components are all the minors of order up to l of the gradient matrix A R m n, i.e., M l A) = A, ad 2 A,..., ad i A,..., ad l A). For instance, M 1 A) = A if l = 1, while if l = m = n the last component of M l A) is the determinant det A of the matrix A. Energy functionals 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 Morrey [24], Ball [5], see also the boo by Dacorogna [7]). We assume that f is bounded below, say fm l A)) 0 for all A R m n. To fix the ideas let us assume m = n 2. Then well-nown results by Morrey [24] see also Acerbi-Fusco [2] and Marcellini [22]) 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, ) R an increasing smooth 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 Mathematics Subect Classification. 49J45; 49K20. Key words and phrases. Polyconvex integrals, minors, lower semicontinuity. 1

2 2 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE the determinant of u in 1.2) see Ball [5], Fonseca-Fusco-Marcellini [11], [12], Giaquinta-Modica- Souče [20] and Müller [25]), 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. Marcellini observed in [23] that the lower semicontinuity inequality still holds below the critical exponent n. Later on Dacorogna-Marcellini [8] proved the lower semicontinuity for p > n 1 see also [15]), while Malý [21] exhibited a counterexample in the case p < n 1. Finally, the limit case p = n 1 was addressed by Acerbi-Dal Maso [1], Dal Maso-Sbordone [9], Celada-Dal Maso [6] and Fusco-Hutchinson [18]. In particular, in [6] 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 [19], under a structure assumption, and to Fusco-Hutchinson [18] and Fonseca-Leoni [13], assuming the coercivity of the integrand. More recently, Amar-De Cicco- Marcellini-Mascolo [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), M 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 for some point x 0, u 0, z 0 ) R n R σ 1, then for all z, t) R σ 1 R 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 ). 1.6)

3 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 3 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 of u see [4, Theorem 3.83]). Let us point out that assumption ii) is automatically satisfied both in the autonomous case f = fξ) and in the following coercive case f = fx, u, ξ) c t, with c > 0 and ξ = z, t). This is the content of Corollary 2.3 in the former case, while in the second ust note that no such point x 0, u 0, z 0 ) exists. 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. 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 see Propositions 3.3 and 3.5), a difficulty which was also present in the papers [6], [3] details are in Remar 2.9). 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 [18] see Proposition 2.8); iii) a measure-theoretic lemma by Celada-Dal Maso [6] see Lemma 2.5). All these tools are carefully combined in an argument which exploits assumption 1.4) in the statement above and the blow-up technique. A resume of the contents of the paper is described next briefly. Section 2 is devoted to some technical results instrumental in the rest of the paper: we state a version of De Giorgi s celebrated approximation theorem, we prove some results concerning truncation for minors and a blow-up type lemma that will be repeatedly employed to reduce ourselves to target affine functions. In Section 3 we deal with the model case of autonomomus integrands for dimensions m n and related results. Finally, in Section 4, we provide the proof of Theorem 1.1. 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 is 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 m n be the linear space of all m n real matrices. For A M 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 of any matrix in M m n is given by σ := m n i=1 m i ) n 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 M α,β A) = deta α i β ). By M l A) we denote the vector whose components are all the minors of order l, and by M l A) the vector of all minors of order up to l. 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.

4 4 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE We shall often deal in what follows with convergences of measures. As usual, we shall name local wea convergence of Radon measures the one defined by duality with C c ), and wea convergence the one defined by duality with C 0 ) on the subset of finite Radon measures 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 [10]). Given a convex function f : R σ R, σ 1 a natural number, consider the affine functions ξ a i + b i, ξ, with a i R and b i R σ, given by a i := fη) σ + 1)α i η) + α i η), η ) dη 2.1) R σ b i := fη) α i η)dη, 2.2) R σ where, for all i N, α i ξ) := i σ αiq i ξ)), q i ) i = Q σ and α C0 1Rσ ) is a non negative function such that R αη)dη = 1. σ Lemma 2.1. Let f : R σ R be a convex function and a i, b i be defined as in 2.1)-2.2). Then, fξ) = sup a i + b i, ξ ), for all ξ R σ. i N The main feature of the approximation in Lemma 2.1 above is the explicit dependence of the coefficients a i and b i on f. In particular, if f depends on the lower order variables x, u) regularity properties of the coefficients a i and b i with respect to x, u) can be easily deduced from related hypotheses satisfied by f thans to formulas 2.1) and 2.2) 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 Moreover, 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 ω 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 + ω i x y + u v )1 + ξ η ) 2.4) for all x, u, ξ) and y, v, η) R m R σ. The compactness of the supports of a i and b i is obtained by first approximating f with a monotone sequence f x, u, ξ) := m x, u)fx, u, ξ), where m C c R m ), m = 1 on { u < } and m = 0 on R m \ +1 { u < + 1}), +1 a family of open sets exhausting ; and then applying to each f De Giorgi s approximation result in Lemma 2.1. Finally, by virtue of Lemma 2.1 we can show that the technical assumption stated in 1.4) of Theorem 1.1 is satisfied by autonomous integrands. Corollary 2.3. Let f = fξ) : R σ R be a convex function, and denote ξ = z, t) R σ 1 R. If R t fz 0, t) is bounded from above for some z 0 R σ 1, then fz, t) = fz) for all z, t) R σ 1 R, i.e. f does not depend on t.

5 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 5 Proof. By Lemma 2.1 f = sup i N f i, where f i ξ) := a i + β i, z + γ i t, for constants a i, γ i R and a vector β i R σ 1. Then, if fz 0, ) is bounded from above, necessarily γ i = 0 for all i N. In turn, this implies that f i z, t) = f i z), the conclusion then follows at once. Remar 2.4. More generally, the thesis of Corollary 2.3 holds for lower semicontinuous convex functions f : X Y R, X and Y two real locally convex topological vector spaces. Indeed, such a function f is the supremum of all continuous affine functionals below f itself A truncation method for minors. We first recall a lemma proved in [6, Lemma 3.2]. Lemma 2.5. Let µ ) be a sequence of signed 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 locally) wealy in the sense of measures on. Then, there exists a Radon measure µ such that T = µ on and µ T locally wealy in the sense of measures on. An immediate consequence is the following corollary see [18, Lemma 2.2], and also [6, Corollary 3.3] for m = n = l). Corollary 2.6. Let 2 l m n, u, u W 1,, R m ) be maps such that a) u ) converges to u in L, R m ); b) sup M l 1 u ) L 1 < ; c) there exists c R τ, τ = ) m n ) l l, such that sup c, M l u ) + dx <. Then, M l 1 u ) M l 1 u) wealy and c, M l u ) c, M l u) locally wealy in the sense of measures on. Proof. Passing to a subsequence, not relabeled for convenience, we may suppose that for any α I h,m, β I h,n, h l 1 M α,β u ) µ α,β We claim that for every α I h,m, β I h,n, h l 1 then wealy in the sense of measures. µ α,β = M α,β u)l n. To establish the case h = 1 fix a test function ϕ Cc ) and α I 1,m, β I 1,n, then in view of a) and being u a Sobolev function we have ϕ dµ α,β = lim ϕ uα x β dx = lim u α ϕ x β dx = u α ϕ x β dx = ϕ uα x β dx. 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 ) ϕ dµ α,β = lim ϕ det dx x β = lim u α 1 det ϕ, uα 2,..., uα h ) h = lim 1) 1 u α ϕ 1 x β1,..., x βh ) M x ˆα1, ˆβ u ) dx, β =1

6 6 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE 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 M ˆα1, ˆβ u ) M ˆα1, ˆβ u) wealy in the sense of measures, we have ϕ dµ α,β = lim h 1) 1 =1 u α 1 ϕ M x ˆα1, ˆβ u ) dx = β ϕm α,β u) dx, that gives the claim. To prove that c, M l u ) ) converges to c, M l u) locally wealy in the sense of measures, we first observe that arguing as before, the induction assumption and conditions in items a) and b) ensure that M l u )) converge in D, R τ ) to M l u). In particular, c, M l u ) ) converges to c, M l u) in D ). Then, using assumption c), Lemma 2.5 yields that up to a subsequence c, M l u ) ) converges locally wealy in the sense of measures to some Radon measure µ with µ = c, M l u). Since the limit is independent of the extracted subsequence the convergence for the whole sequence follows immediately. Remar 2.7. If condition c) above is strengthened to sup M l u ) L 1 <, [18, Lemma 2.2] establishes the wea* convergence in the sense of measures of M l u )) to M l u). The next result is inspired by [18, Proposition 2.5]. Proposition 2.8. 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 for some c R σ sup M m 1 ) u ) L 1 + c, M m u ) L 1 <. Then, there exists a sequence v ) W 1,m, R m ) converging to u in L, R m ), such that M m 1 v ) M m 1 u), c, M m v ) c, M m u) 2.5) wealy in the sense of measures. Moreover, 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 m 1 v ) + c, M m v ) ) dx = ) Proof. Let us first fix some notation that shall be used throughout this proof. If A, B are in M m n and 1 m the components of M A + B) can be written as certain linear combinations of products of components of the vectors M i A) and M i B). We denote these linear combinations writing M A + B) = M i A) M i B), 2.8) with the convention that M 0 A) = M 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 i=0 Φ s,t y) := y ϕ s,t y ) for all y R m.

7 We set Note that LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 7 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 and it is easy to chec that M 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}, 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 m v s,t ) dx = c, M m v s,t ) dx + c, M 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 m i u) M i ϕ s,t u u ) u u)) dx E s,t \C c, M m i u) M i DΦ s,t u u) u u)) dx 1 + M m 1 u ) + c, M m u ) ) dx M m 1 DΦ s,t u u) u u)) + c, M 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 m and B M m n then M i A B) M α,λ A) M λ,β B) for 1 i m 1 β I i,n α,λ I i,m M m A B) = det A)M m B). By taing into account this estimate and 2.10) we conclude that M m 1 DΦ s,t u u) u u)) dx and E s,t \C E s,t \C 1 + C ) t s u u M m 1 u u)) dx,

8 8 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE E s,t \C c, M m DΦ s,t u u) u u)) dx Therefore, recalling 2.11) we get c, M m v s,t ) dx C E s,t E s,t = E s,t \C detdφ s,t u u)) c, M m u u) dx. 1 + M m 1 u ) + c, M m u ) ) dx + C t s E s,t \C M m 1 u ) + c, M m u ) ) u u dx. Repeating for the lower order minors the argument used to infer the previous estimate we get ) M m 1 v s,t ) + c, M m v s,t ) C 1 + M m 1 u ) + c, M m u ) ) dx E s,t + C t s E s,t \C E s,t M m 1 u ) + c, M 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 m 1 u ) + c, M m u ) ) u u dx s t t s {x \C : s< u u <t} 1 t = lim dr M m 1 u ) + c, M m u ) ) u u s t t s s {x \C : u u =r} u u ) dhn 1 M m 1 u ) + c, M 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 m 1 u ) + c, M m u ) dr dh n 1 0 {x \C : u u =r} u u ) = M m 1 u ) + c, M 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 {x \C : u u =t } 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 m 1 u ) + c, M m u ) t dh n 1 4C 0. u u ) ln u u L 1

9 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 9 The latter estimate and 2.12) and 2.13) imply that for every N sufficiently large there exists s u u 1/2, t L 1 ) such that ) M m 1 v s,t ) + c, M 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 and 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, v u L t u u 1/4 L 1, M m 1 v ) + c, M m v ) ) dx A = M m 1 v ) + c, M m v ) ) dx {x : s < u u <t } + {x : u u t } M m 1 u) + c, M m u) ) dx 5C 0 + C M m u) dx 0. ln u u L 1 A Finally, the wea* convergence stated in 2.5) follows from Corollary 2.6 and the boundedness of c, M m v ) L 1), in turn implied by 2.6). Remar 2.9. Let us point out that the assumption m n plays a crucial role in the proof of Proposition 2.8. The reason is essentially of algebraic nature. In fact if m n and A M m m and B M m n from the algebraic equality M m A B) = det A)M m B) 2.14) we can trivially estimate c, M m A B) with c, M 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 m A B) C c, M m B), for some positive constant C depending on A. We do not now if Proposition 2.8 fails to be true if m > n. Remar For every m and n N, if the second condition in the statement of Proposition 2.8 is replaced with sup M l u ) L 1 <, l = m n, then [18, Proposition 2.5] establishes a stronger result. More precisely, if sup M l u ) L 1 <, then the sequence v ) defined accordingly, converges to u in L and satisfies sup M l v ) L 1 <, 2.6), and lim A 1 + M l v ) ) dx = ) Furthermore, M l v )) wealy* converges in the sense of measures to M l u).

10 10 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE 2.3. A blow-up type lemma. Let f : R m R σ [0, ) be a continuous function, and consider F v, U) := fx, vx), M 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 see, for instance, [4, Theorem 3.83]). 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. Lemma Suppose that for L n a.e. x 0, and for all sequences ε 0 and v ) W 1,l Q 1, 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, M l v )) dy fx 0, ux 0 ), M l ux 0 ))), 2.17) then the lower semicontinuity inequality 2.16) holds. Proof. We employ the blow-up technique introduced by Fonseca & Müller [17]. 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), Ml 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 m ) 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 and dµ dl n x), dν dl n x) exist finite, 1 lim ε 0 ε n+1 uy) ux) ux) y x) dy = ) Q εx)

11 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 11 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 fx, u, M l u ))dx Q ε x 0 ) = lim lim Q 1 fx 0 + ε y, ux 0 ) + ε v,, M l v, ))dy, 2.21) and dν dl n x νq ε x 0 )) 1 0) = lim ε n = lim lim ε n u l 1 W 1,l 1 Q ε x 0 ),R m ) = lim where we have set Clearly, v, W 1,l, and by denoting v, y) := u x 0 + ε y) ux 0 ) ε. v 0 y) := ux 0 ) y lim v, l 1 W 1,l 1 Q 1,R m ), 2.22) we have that lim lim v, v 0 L 1 = ) A standard diagonalization argument provides a subsequence for which 2.21), 2.22) and 2.23) become v := v, v 0 L 1, sup v W 1,l 1 <, and > dµ dl n x 0) = lim fx 0 + ε y, ux 0 ) + ε v, M 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 m vx))) dx. where v W 1,m 1, R m ), if n m 3, v BV, R 2 ) for n m = 2. Our result improves upon [6, Theorems 3.1 and 4.1] see also [16, Theorem 4.1] and [14, Theorem 10]).

12 12 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE 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 2.11 it is sufficient to show that lim inf f M m v )) dy f M 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 m v )) dy f i M 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. As 0 f i f and being the inferior limit in 3.2) finite, we infer that sup b i, M m v ) ) + dy <. 3.4) Q 1 Fix now M v 0 L + 1 and set v x) v,m x) := M v x) v x) if v x) M otherwise. Note that the sequence v,m ) is bounded in W 1,m 1 L, then b i, M m v,m )) has a limit in the sense of distributions. Moreover, 3.1) and 3.4) yield for every M as above sup b i, M m v,m ) ) + dy <. Q 1 Hence, with fixed ρ 0, 1), by the latter estimate and by Lemma 2.5, Proposition 2.8 applied on Q ρ provides a new sequence w ) satisfying all the conclusions there. We do not highlight the dependence of the various quantities on ρ for the sae of simplicity. In particular, we have f i M m w )) dy f i M m v,m )) dy + a i + b i, M m w ) )dy Q ρ Q ρ\a A f i M m v )) dy + a i + b i, M m w ) )dy. 3.5) Q 1 A The latter estimates follow from the positivity of f i and recalling that, by the choice of M, for sufficiently large we have {y Q ρ : v y) > M} A = {y Q ρ : v y) v 0 y) > s }. Therefore, thans to 2.7) and 3.5), we get lim inf f i M m v )) dy lim inf Q 1 Q ρ f i M m w )) dy. 3.6)

13 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 13 From this inequality, 3.3) follows at once recalling the wea convergence of M m 1 w )) to M m 1 ux 0 )) and of b i, M m w ) ) to b i, M m ux 0 )) on Q ρ, and finally by letting ρ 1. 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 [19] for m = n and p > n 1, and with [16, Remar 4.3]). To be precise, let us consider the functional F u) = hx, u)fm 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 2.11 and reduce ourselves to show that lim inf hx 0 + ε y, ux 0 ) + ε v )fm m v )) dy hx 0, ux 0 ))fm 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. 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 m v )) dy hx 0, ux 0 ))f i M 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 m v ) ) + dy <. 3.9) Q 1 For M v 0 L + 1 consider the truncated functions v x) if v x) M v,m x) = M x) v x) otherwise, the sequence v,m ) turns out to be bounded in W 1,m 1 L. Therefore, arguing as in Theorem 3.1 and taing into account 3.9) and Lemma 2.5, Proposition 2.8 gives a new sequence w ) satisfying all the conclusions there on Q ρ. 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 m v )) dy Q 1 lim inf hx 0 + ε y, ux 0 ) + ε w )f i M m w )) dy. Q ρ

14 14 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE From this inequality we easily get 3.8), since hx 0 + ε y, ux 0 ) + ε w )) converges uniformly in Q ρ to hx 0, ux 0 )), and since for any open subset U Q ρ L n U)f i M m ux 0 ))) lim inf f i M m w )) dy thans to the wea convergence of M m 1 w )) to M m 1 ux 0 )) and of b i, M m w ) ) to b i, M m ux 0 )) on Q ρ, and finally letting ρ 1. 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 Theorem 1.1). In Section 4 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 for a comparison with classical statements in literature see [1, Theorem 3.5], [6, Theorem 2.2, Corollary 2.3], [9, Theorem 3.1], [18, 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 u in L 1, and sup M l u ) L 1 <, 3.10) F u) lim inf F u ). Proof. We can argue analogously to Lemma 2.11 and infer that to conclude we need to show lim inf fx 0 + ε y, ux 0 ) + ε v, M l v )) dy fx 0, ux 0 ), M 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 M l v ) L 1 <. Thus, we can apply [18, Proposition 2.5] cp. with Remars 2.7 and 2.10) that provides a sequence w ) W 1, converging to v 0 uniformly in Q 1, such that M l w ) M l ux 0 )) wealy* in the sense of measures and satisfying 2.15). In particular, by assuming that 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, M l w )) dy Q 1 f i x 0 + ε y, ux 0 ) + ε v, M l v )) dy + C i 1 + M l w ) ) dy, Q 1 {v w }

15 and moreover 2.4) gives LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 15 f i x 0, ux 0 ), M l w )) dy Q 1 f i x 0 + ε y, ux 0 ) + ε w, M l w )) dy + ω i ε 1 + w L )) 1 + M 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, M l v )) dy Q 1 lim inf f i x 0, ux 0 ), M l w )) dy f i x 0, ux 0 ), M l ux 0 ))). Q 1 Hence, inequality 3.11) follows. Remar 3.4. Note that the counterexample constructed in [19, Theorem 3.1] shows that a continuous, or better a lower-semicontinuous, dependence of the integrand f on the lower order variables is needed. We point out that if we restrict to maps from R n to itself with positive determinants, assumption 3.1) implies the local boundedness in L 1 of all the minors up to order n thans to Müller s isoperimetric inequality cp. with 3.10)). Indeed, [26, 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) M n 1 u) dh n 1. B rx 0 ) Thus, if det u 0 L n a.e. on, by integrating this inequality on 0, R), with R < distx 0, ), we get n 1 n det u dx Cn) M 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 M 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 ). Note that the lower semicontinuity property established in Proposition 3.5 above is also enoyed by functionals with densities relevant in applications to elasticity, i.e. satisfying in addition fx, u, z, t) as t 0.

16 16 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE 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. More precisely, we consider the functional F v) := f x, vx), M n vx))) dx, where v W 1,n 1, R n ), if n = m 3, v BV, R 2 ) for n = m = 2. Below we give the proof of Theorem 1.1, a result that improves upon [3, Theorem 4.2] and [6, Theorems 3.1 and 4.1]. To begin with, we comment on assumption 1.4) by considering 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) 0. Moreover, we recall that assumption 1.4) is satisfied if f is coercive in t, or if f is autonomus, i.e. f = fξ) see Corollary 2.3). Before proving Theorem 1.1 we recall the notation ξ = z, t) R σ 1 R if ξ R σ. Note also that condition 1.5) in Theorem 1.1 assures that the limit function u BV, R 2 ) if n = 2, and u W 1,n 1, R n ) for n 3. Proof of Theorem 1.1. We split the proof into several intermediate steps. Step 1. Reduction to affine target maps By Lemma 2.11 to infer 1.6) we are left with proving lim inf fx 0 + ε y, ux 0 ) + ε v, M n v )) dy fx 0, ux 0 ), M n ux 0 ))). 4.1) 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 4.1) 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, M Q n v )) dy 1 lim inf gm n 1 v )) dy gm n 1 ux 0 ))) = fx 0, ux 0 ), M n ux 0 ))), Q 1 where we have denoted by g the convex function gz) := fx 0, ux 0 ), z, t) cp. with 1.4)). Step 3. Proof in case b)

17 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 17 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, f i x, u, z, t) = a i x, u) + β i x, u), z + γ i x, u) t) +, we have fx, u, z, t) = sup f i x, u, z, t). i N Therefore, to prove 4.1) it is enough to show that for all i N lim Q 1 fx 0 + ε y, ux 0 ) + ε v, M n v )) dy f i x 0, ux 0 ), M n ux 0 ))). 4.2) 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. Fix now M > v 0 L + 1 and set v x) if v x) M v,m x) := M v x) 4.3) otherwise. v x) Then, as 0 f f, for all we have f x 0 + ε y, ux 0 ) + ε v,m, M n v,m )) dy {y Q 1 : v M} fx 0 + ε y, ux 0 ) + ε v, M n v )) dy. Q 1 4.4) Therefore, since the sequence v,m ) 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,m ) + dy <. {y Q 1 : v M} Recalling the choice γ x 0, ux 0 )) > 0, the continuity of γ yields for sufficiently large sup det v,m ) + dy <, {y Q 1 : v M} in turn implying sup det v,m ) + dy <. Q 1 An application of Lemma 2.5 gives that, up to a subsequence not relabeled for convenience, the sequence det v,m ) converges locally wealy* in the sense of measures in Q 1. In particular, det v,m ) is bounded in L 1 loc Q 1). Hence, with fixed ρ 0, 1), Proposition 2.8 provides sequences s 0 and w ) in W 1,n Q ρ, R n ) satisfying conclusions 2.5), 2.6) and 2.7) there. Note that, for sufficiently large, recalling the choice of M, we have {y Q ρ : v y) > M} A = {y Q ρ : v y) v 0 y) > s }. Therefore, estimate 2.3) and equation 4.4) imply for all i N Q ρ f i x 0 + ε y, ux 0 ) + ε w, M n w )) dy

18 18 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE f i x 0 + ε y, ux 0 ) + ε v,m, M n v,m ) dy + C i 1 + M n w ) ) dy Q ρ\a A f i x 0 + ε y, ux 0 ) + ε v,m, M n v,m )) dy + C i 1 + M A n w ) ) dy {y Q ρ: v M} fx 0 + ε y, ux 0 ) + ε v, M n v )) dy + C i Q M 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, M n v )) dy lim inf f i x 0, ux 0 ), M n w )) dy. Q 1 Q ρ In turn, from this and by taing into account that M n w )) converges to M n ux 0 )) wealy* in the sense of measures on Q ρ, by the convexity of f i x 0, ux 0 ), ) we get lim Q 1 fx 0 + ε y, ux 0 ) + ε v, M n v )) dy ρ n f i x 0, ux 0 ), M n ux 0 ))), from which 4.2) follows straightforwardly as ρ 1. In case the integrand depends only on the determinant, the same conclusion holds under a local version of the condition in 1.4). Proposition 4.1. 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 ) hx, u) := sup h δ x, u) δ>0 h δ x, u) := inf {fy, v, t) : y, v, t) B δ x, u) 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 < F u) lim inf F u ). Proof. We argue as in Theorem 1.1: first using the blow-up type Lemma 2.11 to reduce to inequality 4.1). 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 M > 0 and consider the function v,m in 4.3). Then, fx 0 + ε y, ux 0 ) + ε v,m, det v,m )dy Q 1 fx 0 + ε y, ux 0 ) + ε v, det v )dy + C L n {y Q 1 : v M}). Q 1

19 LOWER SEMICONTINUITY FOR POLYCONVEX INTEGRALS 19 In turn, for any δ > 0, by definition of h δ we get lim inf fx 0 + ε y, ux 0 ) + ε v,m, det v,m )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. 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 M. 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. The authors wish to than the referee for the careful reading of the manuscript. References [1] E. Acerbi, G. Dal Maso, 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 Mech. Anal ), [3] M. Amar, V. De Cicco, P. Marcellini and E. Mascolo, Wea lower semicontinuity for non coercive polyconvex integrals, Adv. Calc. Var., ), 2, [4] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, in the Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New Yor, [5] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., ), [6] P. Celada, G. Dal Maso, Further remars on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire, ), [7] B. Dacorogna, Direct methods in the Calculus of Variations, Appl. Math. Sci. 78, Springer, Berlin, [8] B. Dacorogna, P. Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C.R. Math. Acad. Sci. Paris, ), [9] G. Dal Maso, C. Sbordone, Wea lower semicontinuity of polyconvex integrals: a borderline case, Math. Z., ), [10] E. De Giorgi, Teoremi di semicontinuità nel Calcolo delle Variazioni, Corso INDAM, 1968/1969). [11] I. Fonseca, N. Fusco, P. Marcellini, On the total variation of the Jacobian, J. Funct. Anal ), [12] I. Fonseca, N. Fusco, P. Marcellini, Topological degree, Jacobian determinants and relaxation, Boll. Unione Mat. Ital. Sez. B ), [13] I. Fonseca, G. Leoni, Some remars on lower semicontinuity, Indiana Univ. Math. J., ), [14] I. Fonseca, G. Leoni, J. Malý, Wea continuity and lower semicontinuity results for determinants, Arch. Rational Mech. Anal ), [15] I. Fonseca, P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geometric Analysis ), [16] I. Fonseca, J. Malý, Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire ), no. 3, [17] I. Fonseca, S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 SIAM J. Math. Anal., ), [18] N. Fusco, J.E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., ), [19] W. Gangbo, On the wea lower semicontinuty of energies with polyconvex integrands, J. Math. Pures Appl., ), [20] M. Giaquinta, G. Modica, J. Souče, Cartesian currents in the calculus of variations, voll. I and II, Springer- Verlag, Berlin, [21] J. Malý, Wea lower semicontinuity of polyconvex integrals, Proc. Edinb. Math. Soc., ), [22] P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., ), 1 28.

20 20 M. FOCARDI N. FUSCO C. LEONE P. MARCELLINI E. MASCOLO A.VERDE [23] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, ), [24] C.B. Morrey, Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, 130 Springer-Verlag, New Yor [25] S. Müller, Det = det. A remar on the distributional determinant, C. R. Acad. Sci. Paris Sr. I Math ), [26] S. Müller, Higher integrability of determinants and wea convergence in L 1, J. Reine Angew. Math ), 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 Matematica e Informatica U. Dini, Università degli Studi di Firenze, Viale Morgagni 67/A, Firenze, Italy Dipartimento di Matematica R. Caccioppoli, Università degli Studi Federico II di Napoli, Via Cinthia - Monte S. Angelo, Napoli, Italy