WEAK LOWER SEMICONTINUITY FOR NON COERCIVE POLYCONVEX INTEGRALS

WEAK LOWER SEMICONTINUITY FOR NON COERCIVE POLYCONVEX INTEGRALS

M. AMAR  V. DE CICCO  P. MARCELLINI  E. MASCOLO

Abstract. We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u : Ω ⊂ Rⁿ → R^min W^1,n(Ω; R^m)with n ≥ m ≥ 2, with respect to the weak W^1,p-convergence for p > m − 1, without assuming any coercivity condition.

Keywords: Vector-valued maps, Jacobian determinants, polyconvex integrals, lower semicontinuity, chain rule.

AMS-MSC: 49J45 - 49K20

1. Introduction

Direct methods of calculus of variations are a rigorous framework to treat mathematically interesting problems arising in science or engineering related to optimality conditions such as the minimization of the energy (or minimization of time, or of space, etc...), expressed in integral form such as, for instance,

(1.1) Z

Ω

g(x, u, Du) dx .

The mathematical theory of direct methods, rougly speaking, is based on the lower semi- continuity of the energy functional in (1.1) in a certain topology and on the relative compactness of bounded sets of functions u which compete for a minimum. These two properties are the main ingredients in order to assure the existence of equilibrium solu- tions. Compactness has been a main motivation for the introduction of theory of Func- tional Analysis in the last century. Lower semicontinuity, either in L^p or in the weak topology of W^1,p (the Sobolev space of L^p-functions with rst partial derivatives in L^p), classically is based on the convexity of the function g in (1.1) with respect to the gradi- ent variable Du. However, motivated by the applications to nonlinear elasticity, in 1977 J. Ball pointed out in [4] that convexity is completely unrealistic in the vectorial case.

Indeed, it conicts, for instance, with the natural requirement that the elastic energy is frame-indierent. Hence, it must be replaced by dierent and more general conditions, named quasiconvexity and polyconvexity, properties already introduced theoretically by Morrey in [25]. These new convexity conditions are now considered fundamental ones in nonlinear elasticity. In particular, polyconvexity takes into account a natural constitutive hypothesis which is the invariance of the energy under the transformation g 7→ g + φ, for every null Lagrangian φ.

1

The aim of this research is to consider the polyconvexity condition, to achieve new lower semicontinuity results useful for applications in this eld. We describe below in more details the mathematical problem.

Dealing with vector-valued maps, still it is not completely known a set of minimal as- sumptions for lower semicontinuity of integrals of the Calculus of Variations of the form

(1.2) I (u) =

Z

Ω

g(x, u, Du) dx ,

where u : Ω → R^m is a vector-valued map dened in an open set Ω ⊂ Rⁿ and Du is the m × n Jacobian matrix of its partial derivatives

u ≡ u¹, u², . . . , u^m
, Du = ∂u^α

∂x_i

α=1,2,...,m i=1,2,...,n

.

On the contrary, in the so-called scalar case (corresponding to m = 1) Serrin in 1961, in the pioneering paper [27], pointed out the convexity of g = g (x, s, ξ) with respect to the gradient variable ξ as a main (necessary and) sucient condition for the lower semicontinuity of the integral I (u) in (1.2). Serrin himself pointed out the special role of the x variable to obtain the lower semicontinuity of I (u) with respect to the L¹loc(Ω)- convergence. In fact, among his assumptions we mention either the coercivity of g , i.e.,

g (x, s, ξ) ≥ c |ξ|

for some positive constant c and for every (x, s, ξ) ∈ Ω × R × Rⁿ, or the continuity of the partial derivatives

(1.3) g_x(x, s, ξ) , g_ξ(x, s, ξ) , g_ξx(x, s, ξ) .

Some extensions of Serrin's results have been recently obtained by Gori-Marcellini [19]

and by several other authors, also in the context of BV -functions (see [18], [15], [9], [10], [3]).

In the vector-valued case m > 1 either the quasiconvexity or the polyconvexity of g with respect to the gradient variable ξ play a role. These convexity conditions are due to Morrey [25]; a reference paper is Ball [4]. In particular the function g(x, s, ξ) is said polyconvex with respect to the gradient variable ξ if it can be represented under the form (1.4) g (x, s, ξ) = f (x, s, M (ξ)) ,

where f is a convex function with respect to its last variable and, for every m × n matrix ξ ∈ M^m×n, M(ξ) denotes the vector

M (ξ) = ξ, adj₂ξ, . . . , adj_iξ, . . . , adj_min{m,n}ξ
.

Here adjiξ, for i = 2, . . . , min {m, n}, stands as the vector of the determinants of the i × i minors of the matrix ξ ∈ M^m×n. Thus M(ξ) is a vector in R^τ, with

τ = τ (n, m) =

min(m,n)

X

i=1

n i

 m i

 .

We assume that f : Ω × R^m× R^τ → [0, +∞)is a nonnegative function, convex in R^τ with respect to the last variable.

The lower semicontinuity for polyconvex integrals have been investigated by several au- thors in the past years, starting from the results by Morrey and Ball cited above. In the quoted papers the weak topology of W^1,p(Ω; R^m)is considered, for some p > min {m, n}.

A more recent counterexample by Maly [22] shows that the semicontinuity in the weak topology of W^1,p(Ω; R^m) is generically not true if p < min {m, n} − 1. Marcellini [23], [24] and Dacorogna-Marcellini [7] proved a lower semicontinuity result when p > n − 1, m = n and f = f (M(ξ)) or f = f (x, M(ξ)). The limit case p = n − 1 has been studied in the same context by Acerbi-Dal Maso [2], Dal Maso-Sbordone [8], Celada-Dal Maso [5]

and Fusco-Hutchinson [16]. See also Acerbi-Buttazzo-Fusco [1] for lower semicontinuity with respect to the L^∞-topology.

The situation changes considerably when a dependence on s is also allowed, since the presence of the (x, s)-variables cannot be treated as a simple perturbation. A result in this more general context is due to Gangbo [17] under the following structure assumptions:

f (x, s, M (ξ)) = a(x, s)h (x, M (ξ)). More recently Fonseca-Leoni [14], by using a blow-up argument, proved a semicontinuity result under the coercivity condition

f (x, s, M (ξ)) ≥ c |M (ξ)| ,

valid for some positive constant c and for every (x, s, ξ) ∈ Ω × R^m× R^m×n. They consider the weak topology of W^1,p(Ω; Rⁿ) in the limit case p = n − 1 too, with f depending only on the Jacobian determinant detξ = adjnξ of the n × n matrix ξ, i.e., f = f (x, s, detξ) and the coercivity condition f (x, s, detξ) ≥ c |detξ| holds.

In the spirit of Serrin's work, where either the coercivity or the continuity of the partial derivatives in (1.3) were assumed, in this paper we consider the vector-valued case m > 1 and we study the lower semicontinuity of the integral in (1.2) related to general polyconvex integrands; the lack of coercivity and the general form of the dependence on the (x, s)- variables force additional regularity assumptions on the integrand. In particular we prove the following result (for more general assumptions see Theorems 4.2 and 4.3).

Theorem Let us consider the vector-valued case m ≥ 2 and let n ≥ m. Let f be a nonnegative convex function with respect to the last variable, such that f and fx are continuous in Ω×R^m× R^τ. Then, for every uk, u ∈ W^1,m(Ω, R^m), such that uk converges to u in the weak topology of W^1,p(Ω, R^m) for some p > m − 1, we have

(1.5) lim inf

k→+∞

Z

Ω

f (x, u_k, M (Du_k)) dx ≥ Z

Ω

f (x, u, M (Du)) dx .

We explicitly note that the condition uk, u ∈ W^1,m(Ω; R^m) is necessary to make non ambiguous the denition of the integral functional. The conclusion (1.5) makes possible to extend the integral functional I (u) in (1.2), a-priori uniquely dened in W^1,m(Ω; R^m), to functions in W^1,p(Ω; R^m) for p > m − 1. See details for instance in [24].

We emphasize that in fact in this paper we give some lower semicontinuity results under more general assumptions than those stated above (see details in Theorems 3.6 and 4.5).

Relevant aspects, in particular, are: (i) we deal with integrands f depending on (x, u) and on all the minors of the matrix Du; (ii) coercivity of f is not assumed; (iii) we consider the weak convergence in W^1,p(Ω, R^m) for p > m − 1; i.e., below the natural exponent p = m which guarantee a non ambiguous denition of the integrals. We also notice that our results hold in the limit case p = m − 1 = 1, if m = n = 2 (see Proposition 3.7).

The proof of the theorem stated above is based on an interesting combination of two principal ingredients, which reect the basic role respectively of the convexity of f and the polyconvexity of g in (1.4). The rst one is the original approximation argument for a convex function f(x, s, ·) introduced by De Giorgi in [12]. In these notes f is approximated from below by a sequence of linear functions whose coecients, which depend on (x, s), are explicitly represented in terms of f, allowing us to deduce regularity properties of the coecients from the corresponding ones of f.

The second ingredient is a chain rule formula for determinants (see Lemmas 2.1 and 2.3 in Section 2). In the smooth case, for m = n, the chain rule has the following form: let B = B(x, s) be a function of class C¹ dened in Ω × Rⁿ, and let u ∈ W^1,n(Ω, Rⁿ); under the notations







v(x) = B(x, u(x)) b(x, s) = ∂B/∂s₁ D_xB(x, s) =

∂B

∂x1,_∂x^∂B

2, . . . ,_∂x^∂B

n

 with x ≡ (xi)_i=1,...,n and s ≡ (si)_i=1,...,n, we have

det Dv, Du², . . . , Duⁿ

= b(x, u) det Du + det D_xB(x, u), Du², . . . , Duⁿ
,

where (Dv, Du², . . . , Duⁿ) represents the n × n matrix whose columns are the n vectors Dv, Du², . . . , Duⁿ and similarly for the other matrix (DxB(x, u), Du², . . . , Duⁿ). The role and properties of determinants are crucial here. This formula (which admits a proper generalization to the case m < n) allows us to represent a term of the type b(x, u) det Du as an algebraic sum of Jacobian determinants, which are continuous in the weak topology of W^1,p(Ω, Rⁿ)for p > n−1. This allows us to pass to the limit as k → +∞ in expressions of the type

k→+∞lim Z

Ω

b(x, u_k) det Du_kdx = Z

Ω

b(x, u) det Du dx .

Let us nally mention that some recent lower semicontinuity results have been obtained, in the scalar case, using a chain rule formula for gradients (see [19], [18], [9], [10], [20], see also [21] for convex functionals in the vector-valued case).

Acknowledgments: The authors are grateful to Giovanni Leoni for helpful discussions during his visit in July 2006 to the Department of Mathematics of the University of Florence, Italy. The present work has been partially supported by GNAMPA-INdAM

(Gruppo Nazionale per l'Analisi Funzionale e Applicazioni - Istituto Nazionale di Alta Matematica), project Singolarità e semicontinuità nel Calcolo delle Variazioni.

2. Chain rule formulas for determinants

In this section we study some chain rule formulas involving determinants. The rst one is a pointwise formula and it is proved in a regular context, from which we derive an integral form under weaker assumptions.

In the following for a matrix ξ = (ξ^αi)α=1,2,...,n

i=1,2,...,n we also use the notation det ξ = det (ξ¹, ξ², . . . , ξⁿ), where, for α = 1, 2, . . . , n, the vector ξ^α = (ξ₁^α, ξ^α₂, . . . , ξ_n^α).

Lemma 2.1 (Pointwise chain rule). Let B = B(x, s) be a function belonging to C¹(Ω×Rⁿ). Then, for every u ∈ W^1,n(Ω; Rⁿ) ∩ L^∞_loc(Ω; Rⁿ), the function v : Ω → R, dened by

v(x) := B(x, u(x)) for a.e. x ∈ Ω belongs to Wloc^1,n(Ω) and, for almost every x ∈ Ω, we have (2.1) Ds1B x, u¹(x), . . . , uⁿ(x)
det Du(x) =

det Dv(x), Du²(x), . . . , Duⁿ(x)
−det D_xB(x, u(x)), Du²(x), . . . , Duⁿ(x)
. Proof. Let Ω⁰ ⊂⊂ Ω. First of all we note that by the usual chain rule formula, for j = 1, . . . , n and for almost every x ∈ Ω⁰, we obtain

D_xjv(x) = D_xjB(x, u(x)) +

n

X

i=1

D_siB(x, u(x)) D_xjuⁱ(x) . Hence

Dv(x) = DxB(x, u(x)) +

n

X

i=1

DsiB(x, u(x)) Duⁱ(x) for a.e. x ∈ Ω⁰. Therefore, by using the multilinearity properties of the determinants, it follows

det Dv(x), Du²(x), . . . , Duⁿ(x)

= det D_xB(x, u(x)) +

n

X

i=1

D_siB(x, u(x)) Duⁱ(x), Du²(x), . . . , Duⁿ(x)

!

= det D_xB(x, u(x)), Du²(x), . . . , Duⁿ(x)
+

n

X

i=1

D_siB(x, u(x)) det Duⁱ(x), Du²(x), . . . , Duⁿ(x)
.

Since det (Duⁱ(x), Du²(x), . . . , Duⁿ(x)) = 0, for i = 2, . . . , n, (2.1) follows for almost every x ∈ Ω⁰. Letting Ω⁰ → Ω, the thesis is accomplished.  Remark 2.2. Formula (2.1) can be proved also for a general Borel function B at every point (x, s), with s = u(x), in which B is dierentiable.

Moreover, we note that if B = B(x, s) is a locally Lipschitz continuous function on Ω × Rⁿ, then equality (2.1) holds provided that Ds1B x, u¹(x), . . . , uⁿ(x)
det Du(x) and

det (Dv(x), Du²(x), . . . , Duⁿ(x)) are interpreted to be zero whenever Du(x) = 0, irrespec- tive of whether Ds1B x, u¹(x), . . . , uⁿ(x)

and Dv(x) are dened.

Finally, observe that (2.1) holds also with s1 replaced with si, i = 1, . . . , n:

D_siB x, u¹(x), . . . , uⁿ(x)
det Du(x) =

det Du¹(x), . . . , Duⁱ⁻¹(x), Dv(x), Duⁱ⁺¹(x), . . . Duⁿ(x)
− det Du¹(x), . . . , Duⁱ⁻¹(x), D_xB(x, u(x)), Duⁱ⁺¹(x), . . . Duⁿ(x)
.

Lemma 2.3 (Integral chain rule). Let (x, s) ∈ Ω × Rⁿ and s = (s1, s⁰) ∈ R × Rⁿ⁻¹. Let B = B(x, s) be a continuous function such that, for every compact set K ⊂ Ω × Rⁿ there exists a proper positive constant CK, depending on K, such that

(2.2) |B(x, s1, s⁰) − B(y, t1, s⁰)| ≤ CK(|x − y| + |s1− t1|) ,

for every (x, s1, s⁰), (y, t₁, s⁰) ∈ K. Then, for every u ∈ W^1,n(Ω; Rⁿ) ∩ L^∞_loc(Ω; Rⁿ), we have

(2.3)

Z

Ω

ψ D_s1B x, u¹, . . . , uⁿ
det Du dx =

− Z

Ω

B x, u
det Dψ, Du², . . . , Duⁿ
dx

− Z

Ω

ψ det D_xB(x, u), Du², . . . , Duⁿ
dx .

for every ψ ∈ C0¹(Ω).

Remark 2.4. We observe that, in the previous lemma, the assumption u ∈ L^∞_loc(Ω; Rⁿ) can be removed if B(x, s) = 0 for |s| ≥ L, for some L > 0.

Proof. Let u ∈ W^1,n(Ω; Rⁿ) ∩ L^∞_loc(Ω; Rⁿ), the function dened by

Ω 3 x 7→ B(x, u(x))

belongs to W_loc^1,n(Ω). Let Ω⁰ ⊂⊂ Ωand ρε= ρ_ε(s⁰)be a standard C^∞-mollier in Rⁿ⁻¹. In what follows we will use the notation Bε to denote the convolution of B with ρε, i.e.

B_ε(x, s) = B_ε(x, s₁, s⁰) = Z

Rⁿ⁻¹

ρ_ε(s⁰− τ⁰) B(x, s₁, τ⁰) dτ⁰,

for x ∈ Ω⁰ and s ∈ Rⁿ, where 0 < ε < dist(Ω⁰, ∂Ω).

The function vε(x) = B_ε(x, u(x)) belongs to W^1,n(Ω⁰)and, by multiplying (2.1) for every ψ ∈ C₀¹(Ω⁰)and integrating by parts, we obtain

(2.4)

Z

Ω

ψD_s1B_ε x, u¹, . . . , uⁿ
det Du dx =

− Z

Ω

B_ε x, u
det Dψ, Du², . . . , Duⁿ
dx

− Z

Ω

ψ det D_xB_ε(x, u), Du², . . . , Duⁿ
dx .

Set kuk∞ = kuk_L^∞_(Ω⁰_;R^m₎. We observe that, by assumptions, for a proper compact set K ⊂ Ω × Rⁿ such that Ω⁰ × B(0, kuk∞+ 1) ⊂ K and (x, s) ∈ suppψ × B(0, kuk∞) and for every ε > 0 we have

|D_s1B_ε(x, s)| ≤ Z

Rⁿ⁻¹

ρ_ε(s⁰− τ⁰) |D_s1B(x, s₁, τ⁰)| dτ⁰ ≤ C_K,

|B_ε(x, s)| ≤ Z

Rⁿ⁻¹

ρ_ε(s⁰− τ⁰) |B(x, s₁, τ⁰)| dτ⁰ ≤ C_K,

|DxBε(x, s)| ≤ Z

Rⁿ⁻¹

ρε(s⁰ − τ⁰) |DxBε(x, s1, τ⁰)| dτ⁰ ≤ CK,

where we denote by CK also the positive constant such that |B(x, s)| ≤ CK, for (x, s) ∈ K.

The previuos estimates and standard properties of convolution imply

D_s1B_ε* D_s1B weakly^∗ in L^∞(suppψ × B(0, kuk∞+ 1)) ,

Bε* B weakly^∗ in L^∞(suppψ × B(0, kuk∞+ 1)) ,

D_xB_ε * D_xB weakly^∗ in L^∞(suppψ × B(0, kuk∞)) ,

and hence, since det Du belongs to L¹(Ω), the thesis follows, once we pass to the limit for ε → 0⁺ in (2.4).

 3. Lower semicontinuity in the model case m = n

In this section we prove the lower semicontinuity result in the model case m = n, when the function g (x, s, ξ) appearing in the integral functional in (1.2) explicitly depends only on (x, s) and the determinant det ξ = adjnξ of the n × n matrix ξ; i.e., g can be represented under the form

g (x, s, ξ) = f (x, s, det ξ) , the general case being considered in the next section.

In the following we use two results, which are classical tools in the framework of lower semicontinuity problems. Firstly, we recall the approximation theorem for convex func- tions due to De Giorgi (see [12]). This result states that any convex function f : R^ν → R, ν ≥ 1, is approximated by mean of a sequence of ane functions aj + b_j· ξ, aj ∈ R and b_j ∈ R^ν, where

aj :=

Z

R^ν

f (η) (ν + 1)αj(η) + ∇αj(η) · η
dη , (3.1)

b_j := − Z

R^ν

f (η)∇α_j(η)dη, (3.2)

with αj ∈ C₀¹(R^ν), j ∈ N, a nonnegative function such that R_R^να_j(η)dη = 1.

Lemma 3.1. Let f : R^ν → R be a convex function and aj, bj be dened as in (3.1) and (3.2). Then the following property holds:

f (ξ) = sup

j∈N

[a_j + b_j · ξ] ∀ξ ∈ R^ν.

Remark 3.2. The main feature of this approximation is that the coecients aj and bj

depend explicitly on f. In particular, when f depends also on (x, s) we have that (3.1) and (3.2) become

a_j(x, s) :=

Z

R^ν

f (x, s, η) (ν + 1)α_j(η) + ∇α_j(η) · η
dη , (3.3)

b_j(x, s) := − Z

R^ν

f (x, s, η)∇α_j(η)dη, (3.4)

which permit to deduce regularity properties with respect to (x, s) of the coecients aj, b_j from proper hypotheses satised by f. Hence, if f satises some continuity or Lipschitz continuity assumptions with respect to (x, s), then aj and bj inherit the same properties, too.

Now we recall the following localization lemma proven in [13].

Lemma 3.3. Let µ be a positive Radon measure dened on an open set Ω ⊂ Rⁿ. Consider a sequence {φj} of Borel positive functions dened on Ω. Then

Z

Ω

sup

j∈N

φ_j dµ = sup

j∈N





j

X

i=1

Z

Ai

φ_i dµ : A_i ⊂⊂ Ω open and pairwise disjoint



.

In order to approach our lower semicontinuity problem for polyconvex integrals, we rstly deal with functionals of the following type:

u ∈ W^1,n(Ω; Rⁿ) 7→

Z

Ω

a x, u
+ b x, u
det Du⁺dx ,

which, in view of the previous Lemma 3.1, will be the ane approximations of the general polyconvex functionals.

Lemma 3.4. Let a = a(x, s) and b = b(x, s) be two continuous functions. Assume that, for every compact set K ⊂ Ω × Rⁿ and for every (x, s), (y, s) ∈ K, b satises

(3.5) |b(x, s) − b(y, s)| ≤ C_K|x − y|

for a proper positive constant CK, depending on the compact set K. Moreover we assume that Dxb is a Carathéodory function. Then for every uk, u ∈ W^1,n(Ω; Rⁿ) with uk * u weakly in W^1,p(Ω; Rⁿ), p > n − 1, we have

Z

Ω

a x, u
+ b x, u
det Du⁺dx

≤ lim inf

k→+∞

Z

Ω

a x, u_k
+ b x, u_k
det Du_k+

dx .

Proof. We split the proof in two steps.

Step 1: First we assume that a(x, s) = 0 and b(x, s) = 0 for |s| ≥ L for some L > 0. By Lemma 3.3, it is easy to check that for every u ∈ W^1,n(Ω; Rⁿ) we have

Z

Ω

ψ [a(x, u) + b(x, u)det Du]⁺ dx

= sup





 Z

Ω

ψ[a(x, u) + b(x, u)det Du] dx : ψ ∈ C₀¹(Ω), 0 ≤ ψ ≤ 1





 .

Fix ψ ∈ C0¹(Ω) with 0 ≤ ψ ≤ 1 and let K := supp ψ × [−L, L]. By Fatou's Lemma, since a is locally bounded, we obtain that

(3.6) Z

Ω

ψa(x, u) dx ≤ lim inf

k→∞

Z

Ω

ψa(x, u_k) dx .

Dene now

(3.7) B(x, s) = B(x, s₁, s⁰) :=

s1

Z

0

b(x, τ, s⁰) dτ k ∈ N ,

so that Ds1B(x, s) = b(x, s). By applying Lemma 2.3 and Remark 2.4, it follows Z

Ω

ψb(x, u_k) det Du_kdx

= Z

Ω

ψDs1B x, u¹_k, . . . , uⁿ_k
det Dukdx

= − Z

Ω

B(x, uk) det Dψ, Du²_k, . . . , Duⁿ_k
dx

− Z

Ω

ψ det DxB(x, uk), Du²_k, . . . , Duⁿ_k
dx .

Since p > n − 1, for q = _p−n+1^p = _n−1^p
0

, we have that ( B(x, u_k) → B(x, u)

D_xB(x, u_k) → D_xB(x, u) strongly in L^q_loc(Ω) ,

and that all minors of order n − 1 of the n × (n − 1) matrix (Du²k, . . . , Duⁿ_k) weakly converges in L^n−1p (Ω) (see [26]), therefore we get

(3.8)

lim inf

k→+∞

Z

Ω

ψb(x, u_k) det Du_kdx

= − Z

Ω

B(x, u) det Dψ, Du², . . . , Duⁿ
dx

− Z

Ω

ψ det D_xB(x, u), Du², . . . , Duⁿ
dx

= Z

Ω

ψD_s1B x, u¹, . . . , uⁿ
det Du dx

= Z

Ω

ψb(x, u) det Du dx .

By collecting (3.6) and (3.8) we obtain

(3.9)

lim inf

n→∞

Z

Ω

ψ[a(x, u_k) + b(x, u_k)det Du_k] dx

≥ Z

Ω

ψ[a(x, u) + b(x, u)det Du] dx and by taking the supremum over all ψ, we get the desired result.

Step 2: We now remove the extra assumption that a (x, u) = 0 and b(x, u) = 0 for

|u| ≥ L.

For every j ∈ N dene

(3.10) σ_j(s) :=











1 |s| ≤ j − 1,

−|s| + j j − 1 < |s| ≤ j, 0 |s| > j.

Then, for every xed j ∈ N, we have

(3.11)

lim inf

k→∞

Z

Ω

[a(x, u_k) + b(x, u_k)det Du_k]⁺ dx

≥ lim inf

k→∞

Z

Ω

σ_j(u_k) [a(x, u_k) + b(x, u_k)det Du_k]⁺ dx.

Since the functions

a_j(x, s) := σ_j(s) a(x, s), b_j(x, s) := σ_j(s) b(x, s) satisfy the hypotheses of Step 1, (3.9) and (3.11) imply

lim inf

k→∞

Z

Ω

[a(x, u_k) + b(x, u_k)det Du_k]⁺ dx

≥ Z

Ω

σ_j(u) [a(x, u) + b(x, u)det Du]⁺ dx

≥

Z

{x∈Ω: |u(x)|≤j−1}

[a(x, u) + b(x, u)det Du]⁺ dx.

Letting j → ∞, the conclusion follows by Beppo Levi's Lemma.  Remark 3.5. Note that, if n = 2, the previous result continues to hold also in the limit case p = n − 1 = 1. Indeed, following the idea of Dacorogna and Marcellini [7, Lemma 2], we have the equality

lim

k→+∞



 Z

Ω

B(x, u_k) det Dψ, Du²_k
dx + Z

Ω

ψ det D_xB(x, u_k), Du²_k
dx





= Z

Ω

B(x, u) det Dψ, Du²
dx + Z

Ω

ψ det D_xB(x, u), Du²
dx , and then (3.8) still holds true for uk converging to u weakly in W^1,1(Ω; R²).

Now we are in the position to prove a lower semicontinuity inequality for polyconvex integrals of the form

Z

Ω

f x, u, det Du
dx

along sequences {uk} ⊂ W^1,n(Ω; Rⁿ) weakly converging in W^1,p(Ω; Rⁿ), for p > n − 1, without requiring any growth conditions on the integrand f.

Theorem 3.6. Let f : Ω × Rⁿ× R → [0, +∞) be a continuous function such that for a.e. x ∈ Ω and for all s ∈ Rⁿ f (x, s, ·) is convex ;

(3.12)

( ∀K ⊂⊂ Ω × Rⁿ× R ∃CK > 0 such that

|f (x, s, t) − f (y, s, t)| ≤ C_K|x − y| ∀(x, s, t), (y, s, t) ∈ K ; (3.13)

for every (x, t) ∈ Ω × R Dxf (x, ·, t) is a continuous function . (3.14)

Then for every uk, u ∈ W^1,n(Ω; Rⁿ) with uk * u weakly in W^1,p(Ω; Rⁿ), p > n − 1, we have

Z

Ω

f x, u, det Du
dx ≤ lim inf

k→∞

Z

Ω

f x, u_k, det Du_k
dx .

Proof. Let {uk}be a sequence in W^1,n(Ω; Rⁿ)converging to u ∈ W^1,n(Ω; Rⁿ), with respect to the w − W^1,p convergence, with p > n − 1. Since by Lemma 3.1 and Remark 3.2

f (x, s, t) = sup

j∈N

[a_j(x, s) + b_j(x, s)t]⁺,

where aj and bj are dened as in (3.3) and (3.4), for every xed j ∈ N, we have lim inf

k→+∞

Z

Ω

f x, uk, det Duk
dx ≥

lim inf

k→+∞

Z

Ω

aj(x, uk) + bj(x, uk)det Duk

+

dx .

By our assumptions, for every j ∈ N the functions aj and bj satisfy all the assumptions of Lemma 3.4. Therefore, we obtain

lim inf

k→+∞

Z

Ω

f x, u_k, det Du_k
dx

≥ Z

Ω

a_j(x, u) + b_j x, u
det Du)⁺dx .

Hence

lim inf

k→+∞

Z

Ω

f x, uk, det Duk
dx

≥ sup

j∈N

Z

Ω

a_j(x, u) + b_j x, u
det Du⁺dx .

The thesis follows by the localization Lemma 3.3. 

We note that, as in [7], in the special case n = 2 the previous result holds also in the limit case p = n − 1 = 1, as stated in the following proposition, which can be proved by using Remark 3.5.

Theorem 3.7. Let n = 2 and p = 1. Assume that Ω ⊂ R² is an open set and f : Ω × R²× R → [0, +∞) is a continuous function satisfying (3.12), (3.13) and (3.14). Then for every uk, u ∈ W^1,2(Ω; R²) with uk* u weakly in W^1,1(Ω; R²), we have

Z

Ω

f x, u, det Du
dx ≤ lim inf

k→+∞

Z

Ω

f x, uk, det Duk
dx .

4. Lower semicontinuity in more general cases In this section we extend Theorem 3.6, to more general contexts.

4.1. Dependence on all minors: the case m = n. In this section we extend Theorem 3.6 to the case generally considered in the literature, where f depends on all the minors of the Jacobian matrix Du.

Let

σ =

n−1

X

k=1

n k

2

and for any matrix ξ ∈ MI^n×n, we denote by Mⁿ⁻¹ ξ

the vector in R^σ, whose components are given by the determinants of all minors of ξ up to the order n − 1; i.e.,

M ξ
= (ξ, adj₂ξ, . . . , adj_iξ, . . . , adj_nξ) = (Mⁿ⁻¹ ξ
, det ξ) . As above, we rstly deal with functionals of the form:

u ∈ W^1,n(Ω; Rⁿ) 7→

Z

Ω

a x, u
+ b⁰ x, u
·Mⁿ⁻¹ Du
+ b_σ+1 x, u
det Du⁺dx ,

where a : Ω × Rⁿ→ R and b = (b⁰, b_σ+1) : Ω × Rⁿ→ R^σ× R.

Lemma 4.1. Let a : Ω × Rⁿ → R and b = (b⁰, b_σ+1) : Ω × Rⁿ → R^σ × R be continuous functions. Assume that, for every compact set K ⊂ Ω×Rⁿand for every (x, s), (y, s) ∈ K, b_σ+1 satisfy

(4.1) |b_σ+1(x, s) − b_σ+1(y, s)| ≤ C_K|x − y|

for a proper positive constant CK, depending on the compact set K. Moreover we assume that Dxbσ+1 is a Carathéodory function. Then for every uk, u ∈ W^1,n(Ω; Rⁿ) with uk * u weakly in W^1,p(Ω; Rⁿ), p > n − 1, we have

(4.2)

Z

Ω

a x, u
+ b⁰ x, u
·Mⁿ⁻¹ Du
+ b_σ+1 x, u
det Du⁺dx ≤

lim inf

k→+∞

Z

Ω

a x, u_k
+ b⁰ x, u_k
·Mⁿ⁻¹ Du_k
+ b_σ+1 x, u_k
det Du_k+

dx .

Proof. We can assume that a(x, s) = 0 and b(x, s) = 0 for |s| ≥ L for some L > 0, since the general case follows by using the same truncation arguments as in the proof of Lemma 3.4. Recalling that for every u ∈ W^1,n(Ω; Rⁿ) we have

Z

Ω

a(x, u) + b⁰(x, u) · Mⁿ⁻¹ Du
+ b_σ+1(x, u)det Du+

dx

= sup (Z

Ω

ψ[a(x, u) + b⁰(x, u) · Mⁿ⁻¹ Du
+ b_σ+1(x, u)det Du] dx :

ψ ∈ C₀¹(Ω), 0 ≤ ψ ≤ 1 )

,

and taking into account the localization Lemma 3.3, in order to achieve the conclusion, it is enough to prove that for all ψ ∈ C₀¹(Ω):

Z

Ω

ψ[a(x, u) + b⁰(x, u) · Mⁿ⁻¹ Du
+ bσ+1(x, u)det Du] dx

≤ lim inf

k→+∞

Z

Ω

ψ[a(x, uk) + b⁰(x, uk) · Mⁿ⁻¹ Duk
+ bσ+1(x, uk)det Duk] dx for uk, u ∈ W^1,n(Ω; Rⁿ) with uk* u weakly in W^1,p(Ω; Rⁿ).

By proceeding as in the proof of Lemma 3.4, since (3.9) holds with b replaced by bσ+1, we have only to check that

(4.3)

k→+∞lim Z

Ω

ψb⁰(x, u_k) · Mⁿ⁻¹ Du_k
dx

= Z

Ω

ψb⁰(x, u) · Mⁿ⁻¹ Du
dx .

Formula (4.3) is a direct consequence of the fact that b⁰(x, u_k) → b⁰(x, u) strongly in L^q(Ω, R^σ), for q = _p−n+1^p = _n−1^p
0

and all minors up to order n − 1 of the n × n matrix

Du_k weakly converges in L^n−1p (Ω) (see [26]). 

Theorem 4.2. Let f : Ω × Rⁿ× R^σ × R → [0, +∞) be a continuous function such that for a.e. x ∈ Ω and for all s ∈ Rⁿ f (x, s, ·, ·) is convex ;

(4.4)

( ∀K ⊂⊂ Ω × Rⁿ× R^σ × R ∃CK > 0 such that

|f (x, s, η, t) − f (y, s, η, t)| ≤ C_K|x − y| ∀(x, s, η, t), (y, s, η, t) ∈ K ; (4.5)

∀(x, η, t) ∈ Ω × R^σ × R Dxf (x, ·, η, t) is a continuous function . (4.6)

Then for every uk, u ∈ W^1,n(Ω; Rⁿ) with uk * u weakly in W^1,p(Ω; Rⁿ), p > n − 1, we have

Z

Ω

f

x, u, Mⁿ⁻¹ Du
, det Du

dx ≤ lim inf

k→+∞

Z

Ω

f

x, u_k, Mⁿ⁻¹ Du_k
, det Du_k dx .

Proof. Let {uk}be a sequence in W^1,n(Ω; Rⁿ)converging to u ∈ W^1,n(Ω; Rⁿ), with respect to the w − W^1,p convergence. By Lemma 3.1 with ν = σ + 1 and ξ = (η, t), we have

f (x, s, η, t) = sup

j∈N

[aj(x, s) + (b⁰)j(x, s) · η + (bσ+1)j(x, s)t]⁺

where, for every j ∈ N, aj and bj = ((b⁰)_j, (b_σ+1)_j) ∈ R^σ× R are dened as in (3.3) and (3.4). Hence, for every j ∈ N, we have

lim inf

k→+∞

Z

Ω

f x, u_k, Mⁿ⁻¹ Du_k
, det Du_k
dx ≥

lim inf

k→+∞

Z

Ω

a_j(x, u_k) + (b⁰)_j x, u_k
· Mⁿ⁻¹ Du_k
+ (b_σ+1)_j x, u_k
det Du_k+

dx . Since for every xed j ∈ N the functions aj and bj satisfy the assumptions of Lemma 4.1, we get that inequality (4.2) holds and the conclusion follows by proceeding as in the proof

of Theorem 3.6. 

4.2. Dependence on all minors: the case m < n. In this subsection we assume u : Ω ⊆ Rⁿ→ R^m with m < n and set

σ⁰ =

m−1

X

j=1

 n j

  m j



and d =n m

 . For any matrix ξ ∈ MI^m×n, we denote by M^m−1 ξ

the vector in R^σ0, whose components are given by the determinants of all minors of ξ up to the order m − 1 and by adjmξ the vector in R^d, whose components are given by the determinants of all minors of ξ of the maximum order m.

Theorem 4.3. Let f : Ω × R^m× R^σ0× R^d → [0, +∞)be a continuous function such that for a.e. x ∈ Ω and for all s ∈ R^m f (x, s, ·, ·) is convex ;

(4.7)

( ∀K ⊂⊂ Ω × R^m× R^σ0× R^d ∃C_K > 0 such that

|f (x, s, η, ζ) − f (y, s, η, ζ)| ≤ C_K|x − y| ∀(x, s, η, ζ), (y, s, η, ζ) ∈ K ; (4.8)

∀(x, η, ζ) ∈ Ω×R^σ0×R^d, D_xf (x, ·, η, ζ) is a continuous function . (4.9)

Then for every uk, u ∈ W^1,m(Ω; R^m) with uk * u weakly in W^1,p(Ω; R^m), p > m − 1, we have

Z

Ω

f

x, u, M^m−1 Du
, adj_mDu

dx ≤ lim inf

k→+∞

Z

Ω

f

x, u_k, M^m−1 Du_k
, adj_mDu_k dx .

Proof. Let {uk} be a sequence in W^1,m(Ω; R^m) converging to u ∈ W^1,m(Ω; R^m), with respect to the w − W^1,p convergence, p > m − 1. By Lemma 3.1, here ν = σ⁰ + d and ξ = (η, ζ), we have that

f (x, s, η, ζ) = sup

j∈N

[aj(x, s) + b⁰_j(x, s) · η + ¯bj(x, s) · ζ]⁺

where, for every j ∈ N, aj and bj = (b⁰_j, (b_σ+1)_j, . . . , (b_σ+d)_j) = (b⁰_j, ¯b_j) ∈ R^σ0 × R^d are dened in (3.3) and (3.4). Hence, for every xed j ∈ N, we have

lim inf

k→+∞

Z

Ω

f x, u_k, M^m−1 Du_k
, adj_mDu_k
dx ≥

lim inf

k→+∞

Z

Ω

aj(x, uk) + b⁰_j x, uk
· M^m−1 Duk
+ ¯bj x, uk
· adj_mDuk

+

dx .

As in Lemma 3.4 we can assume that aj(x, s) = 0 and bj(x, s) = 0 for |s| ≥ L for some L > 0, since the general case follows by the same truncation argument. Moreover, we recall that for every j ∈ N and every u ∈ W^1,m(Ω; R^m) we have

Z

Ω

a_j(x, u) + b⁰_j(x, u) · M^m−1 Du
+ ¯b_j(x, u) · adj_mDu+

dx

= sup (Z

Ω

ψ[a_j(x, u) + b⁰_j(x, u) · M^m−1 Du
+ ¯b_j(x, u) · adj_mDu] dx :

ψ ∈ C₀¹(Ω), 0 ≤ ψ ≤ 1 )

.

Taking into account the localization Lemma 3.3, in order to achieve the thesis, it is enough to prove that for every uk, u ∈ W^1,m(Ω; R^m) with uk* u weakly in W^1,p(Ω; R^m),

Z

Ω

ψ[a(x, u) + b⁰(x, u) · M^m−1 Du
+ ¯b(x, u) · adj_mDu] dx

≤ lim inf

k→+∞

Z

Ω

ψ[a(x, u_k) + b⁰(x, u_k) · M^m−1 Du_k
+ ¯b(x, u_k) · adj_mDu_k] dx

where for the sake of simplicity we omitted the subscript index j. By proceeding as in Lemmas 3.4 and 4.1 we get that (3.6) holds and it is easy to check that (4.3) continues to be true with Mⁿ⁻¹ replaced by M^m−1. Hence, we have only to check that

(4.10)

lim

k→+∞

Z

Ω

ψ¯b(x, u_k) · adj_mDu_k dx

=

d

X

i=1



 lim

k→+∞

Z

Ω

ψb_σ+i(x, u_k) (adj_mDu_k)_i dx





=

d

X

i=1

Z

Ω

ψb_σ+i(x, u) (adj_mDu)_i dx

= Z

Ω

ψ¯b(x, u) · adj_mDu dx ,

where ¯b = (bσ+1, . . . , b_σ+d) ∈ R^d. Formula (4.10) is a consequence of the chain rule formula applied to adjmDu

i, for every i = 1, . . . , d. Indeed, xed i = 1, . . . , d and dene the C¹-function

B¯_i(x, s) = ¯B_i(x, s₁, s⁰) =

s1

Z

0

b_σ+i(x, τ, s⁰) dτ

(here s⁰ ∈ R^m−1) and for u ∈ W^1,m(Ω; R^m) let ¯vi(x) = ¯B_i(x, u(x)). Taking into account that m < n, with a proof similar to that of Lemma 2.1, we obtain:

(4.11) b_σ+i(x, u¹, . . . , u^m) adj_mDu

i = D_s1B¯_i x, u¹, . . . , u^m

adj_mDu

i = adj_m(D¯v_i, Du², . . . , Du^m)

i− adj_m(D_xB¯_i(x, u), Du², . . . , Du^m)

i. The corresponding integral formulation is given by

(4.12)

Z

Ω

ψ D_s1B¯_i x, u¹, . . . , uⁿ

adj_mDu

idx =

− Z

Ω

B¯i x, u

adj_m(Dψ, Du², . . . , Duⁿ)

idx

− Z

Ω

ψ adj_m(DxB¯i(x, u), Du², . . . , Duⁿ)

idx ,

for every ψ ∈ C0¹(Ω). Finally, taking into account (4.12) and the fact that p > m−1, (4.10) can be obtained reasoning as in the proof of Lemma 3.4 (see in particular (3.8)).  4.3. Weakening of the regularity of the integrand. In this subsection we consider the case where the integrand admits a Sobolev dependence (with an exponent q >> 1) with respect to the x-variable instead of the Lipschitz continuity condition. To this purpose, we rstly need a proper new chain rule, under a Sobolev type dependence. For the sake of simplicity, we consider only the model case m = n and g(x, s, ξ) = f(x, s, det ξ).

Lemma 4.4. Let p > n − 1, q = _p−n+1^p = _n−1^p
0

and B = B(x, s) be a function such that B is a Carathéodory function s.t. B ∈ L^∞loc(Ω × Rⁿ) ;

(4.13)







∀s ∈ Rⁿ B(·, s) ∈ W^1,q(Ω) , D_xB is a Carathéodory function and

∀K ⊂⊂ Ω × Rⁿ ∃ aK ∈ L^q(Ω) s.t. for every s ∈ Rⁿ and a.e. x ∈ Ω

|D_xB(x, s)| ≤ a_K(x) ; (4.14)

for a.e. x ∈ Ω B(x, ·) ∈ C⁰(Rⁿ) ; (4.15)

( B(x, ·, s⁰) ∈ C¹(R) for a.e. x ∈ Ω and for every s⁰ ∈ Rⁿ⁻¹

and Ds1B is a Carathéodory function s.t. Ds1B ∈ L^∞_loc(Ω × Rⁿ) . (4.16)

Then, the same conclusions of Lemma 2.3 hold.

Proof. It is sucient to follows the arguments in the proof of Lemma 2.3, by remarking that assumption (4.14) permits to pass to the limit, as ε → 0⁺, in (2.4). 

By using previous lemma we obtain the following lower semicontinuity theorem.

Theorem 4.5. Let p > n − 1, q = _p−n+1^p = _n−1^p
0

and f : Ω × Rⁿ× R → [0, +∞) be a Borel function belonging to L^∞loc(Ω × Rⁿ× R) such that

for a.e. x ∈ Ω and for all s ∈ Rⁿ f (x, s, ·) is convex ; (4.17)

for a.e. x ∈ Ω and for all t ∈ R f(x, ·, t) ∈ C⁰(Rⁿ) ; (4.18)







∀(s, t) ∈ Rⁿ× R f(·, s, t) ∈ W^1,q(Ω) and

∀K ⊂⊂ Ω × Rⁿ× R ∃ aK ∈ L^q(Ω) s.t.

for a.e. x ∈ Ω and for every (s, t) ∈ Rⁿ× R |Dxf (x, s, t)| ≤ a_K(x) ; (4.19)

for a.e. x ∈ Ω and for every t ∈ R Dxf (x, ·, t) is a continuous function . (4.20)

Then for every uk, u ∈ W^1,n(Ω; Rⁿ) with uk → u weakly in W^1,p(Ω; Rⁿ), we have Z

Ω

f x, u, det Du
dx ≤ lim inf

k→+∞

Z

Ω

f x, u_k, det Du_k
dx .

Remark 4.6. Under the same assumptions of Theorem 4.5, when n = 2, the result still holds in the limit case p = 1, by proceeding as in Theorem 3.7.

Moreover, condition (4.20) can be dropped if

f (x, s, t) = g(x, s₁, t)h(s₂, . . . , s_n) , with h continuous and g satisfying (4.17)(4.19).

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M. Amar  V. De Cicco, Dipartimento di Metodi e Modelli Matematici, Università di Roma La Sapienza, Via A. Scarpa 16, 00161 Roma, Italy

P. Marcellini  E. Mascolo, Dipartimento di Matematica U. Dini, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy