On some sharp conditions for lower semicontinuity in L^{1}

Michele Gori

Dipartimento di Matematica “L. Tonelli”, Università di Pisa

Via Buonarroti 2, 56127 Pisa, Italy gori@mail.dm.unipi.it

Francesco Maggi

Dipartimento di Matematica “U. Dini”, Università di Firenze

Viale Morgagni 67/A, 50134 Firenze, Italy maggi@math.unifi.it

Paolo Marcellini

Dipartimento di Matematica “U. Dini”, Università di Firenze

Viale Morgagni 67/A, 50134 Firenze, Italy marcell@math.unifi.it

Abstract

Let Ω be an open set of ℝ^{n} and let f : Ω × ℝ × ℝ^{n} be a nonnegative continuous function, convex with respect to ξ ∈ ℝ^{n}. Following the well known theory originated by Serrin [14] in 1961, we deal with the lower semicontinuity of the integral F (u, Ω) = ∫_{Ω} f (x, u(x), Du(x)) dx with respect to the L_{loc}^{1} (Ω) strong convergence. Only recently it has been discovered that dependence of f (x, s, ξ) on the x variable plays a crucial role in the lower semicontinuity. In this paper we propose a mild assumption on x that allows us to consider discontinuous integrands too. More precisely, we assume that f (x, s, ξ) is a nonnegative Carathéodory function, convex with respect to ξ, continuous in (s, ξ) and such that f(·, s, ξ) ∈ W_{loc}^{1,1} (Ω) for every s ∈ ℝ and ξ ∈ ℝ^{n}, with the L^{1} norm of f_{x}(·, s, ξ) locally bounded. We also discuss some other conditions on x; in particular we prove that Hölder continuity of f with respect to x is not sufficient for lower semicontinuity, even in the one dimensional case, thus giving an answer to a problem posed by the authors in [12]. Finally we investigate the lower semicontinuity of the integral F (u, Ω), with respect to the strong norm topology of L_{loc}^{1} (Ω), in the vector-valued case, i.e., when f : Ω × ℝ^{m} × ℝ^{m×n} → ℝ for some n ≥ 1 and m > 1.

Keywords: Lower semicontinuity, strong convergence in L^{1}, convex functions, local Lipschitz continuity, local Hölder continuity, calculus of variations.

1991 Mathematics Subject Classification: Primary 49J45. Secondary 35D05.

1 Introduction

Let us consider a functional F of integral type

F (u, Ω) = ∫_{Ω} f (x, u(x), Du(x)) dx ,  (1)

where Ω is an open set of ℝ^{n}, u varies in the Sobolev class W_{loc}^{1,1}(Ω), Du denotes the gradient of u and the function f = f(x, s, ξ) is defined for x ∈ Ω, s ∈ ℝ and ξ ∈ ℝ^{n}.

A classical problem in the calculus of variations is the study of the sequential lower semicontinuity of F on W^{1,1} in the norm topology of L_{loc}^{1} (Ω), i.e., to find out conditions on f sufficient to ensure

\{ \begin{matrix}u_{h}, u ∈ W_{loc}^{1,1}(Ω) \\ u_{h} → u in L_{loc}^{1}(Ω) \end{matrix} ⟹ F (u, Ω) ≤ lim inf_{h→∞} F (u_{h}, Ω) .  (2)

A necessary condition for (2) is the convexity of f(x, s, ·) for every (x, s) fixed. Furthermore is possible to give counterexamples to (2) if f is not bounded from below. Hence a reasonable (and classical) set of hypotheses on f to obtain (2) is

\{ \begin{matrix}f is continuous in Ω × ℝ × ℝ^{n}, \\ f is nonnegative in Ω × ℝ × ℝ^{n}, \\ f(x, s, ·) is convex in ℝ^{n} for all (x, s) ∈ Ω × ℝ. \end{matrix}  (3)

However, Aronszajn’s counterexample (see [13]) shows that in general (3) does not imply (2). Thus further assumption then (3) on f are necessary to obtain the lower semicontinuity in (2).

The first relevant and well known result in this direction has been given by Serrin [14]. Some extensions of Serrin’s Theorem are due to Dal Maso [5] and Ambrosio [1]. De Giorgi, Buttazzo and Dal Maso [7] considered integrands f independent of x ∈ ℝ^{n} and measurable with respect to s ∈ ℝ. Recently Fonseca and Leoni [10], [11] studied the vector-valued case too.

Serrin’s Theorem states that the lower semicontinuity (2) holds if f satisfies (3) and one of the following (disjoint) hypotheses

(a) f(x, s, ξ) → +∞, as|ξ| → +∞, for every (x, s) fixed,

(b) f(x, s, ·) is strictly convex in ℝ^{n}, for every (x, s) ∈ Ω × ℝ,

(c) the derivatives f_{x}, f_{ξ} and f_{xξ} exist and are continuous.

Conditions (a) and (b) go in the direction of a strengthening of the convex structure of f in (3). In condition (c) the convexity of f is untouched; this is useful, just to make a simple but instructive example, to treat integrands f that for some values of (x, s) are constant, say equal to zero, with respect to ξ.

Recently Gori and Marcellini in [12] were able to improve the case (c) above with the following result.

Theorem 1 Let us assume that f(x, s, ξ) satisfies (3) and that, for every compact set Ω′ × H × K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L_{Ω′×H×K} such that

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}| ,  (4)

whenever (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ Ω′ × H × K. Then the lower semicontinuity inequality in (2) holds.

Theorem 1 outlines that, what seems to be really relevant in condition (c) of Serrin’s Theorem, is a regularity assumption of local type and only on the x variable. Thus the following question arises: there exists an optimal assumption of regularity in the x variable to be added to (3) in order to get (2)? Some answers to this question can be found in this work.

We start discussing the sharpness of Theorem 1. In [12] the assumption (4) of local Lipschitz continuity of f with respect to x is compared with the weaker assumption of local Hölder continuity of f with respect to x, i.e., it is compared with the following property: for a real number α ∈ (0, 1) and for every fixed compact set Ω′ × H × K contained in Ω × ℝ × ℝ^{n} there exists a constant L = L_{Ω′×H×K} such that

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}|^{α}  (5)

for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ Ω′ ×H ×K. Through examples it is shown in [12] that, in general, for every exponent α ∈ (0, 1), there exists an n-dimensional integral F (u, Ω) (the dimension n depends on α, precisely n > 4α/(1 − α)) which is not lower semicontinuous in L^{1} and whose integrand f satisfies (3) and (5). This family of counterexamples leaves open the question to know if, for every n ∈ ℕ, there exists a critical exponent α(n) such that (2) is implied by conditions (3) and (5), when the Hölder condition (5) holds for α ∈ [α(n), 1).

In this paper we give a negative answer to this question showing that, for every α ∈ (0, 1) , there exists a one dimensional integral F (depending on α) as in (1) with the required properties. The counterexample is given in Section 2.

This counterexample seems to suggest that, in a sense, assumption (4) in Theorem 1 is sharp. However we have found that it can be significantly improved, once we change our point of view on the kind of regularity in the x variable that we are dealing with. To explain this idea, we first remark that (4) can be formulated in the following equivalent way:

f(·, s, ξ) ∈ W_{loc}^{1,∞} (Ω) , ∀ (s, ξ) ∈ ℝ × ℝ^{n},

and, for every Ω′ ×H ×K ⊂⊂ Ω×ℝ×ℝ^{n} there exists a constant L = L_{Ω′×H×K} such that

esssup_{x∈Ω′} |\frac{∂f}{∂x} (x, s, ξ)| ≤ L,  (6)

for every (s, ξ) ∈ H × K. Thus we try a comparison of (4) with summability conditions on the weak derivatives ∂f/∂x, rather than with qualified continuity conditions like Hölder continuity in (5). This position reveals very fruitful as the following theorem, which is a direct generalization of Theorem 1, shows. It is not more necessary to assume that f(x, s, ξ) is continuous with respect to all its arguments, but we can assume that f is a Carathéodory function, that is f is measurable with respect to x ∈ Ω and continuous with respect to (s, ξ) ∈ ℝ×ℝ^{n}.

Theorem 2 Let us assume that

\{ \begin{matrix}f is a locally bounded, Carathéodory function in Ω × ℝ × ℝ^{n}, \\ f is nonnegative in Ω × ℝ × ℝ^{n}, \\ f(x, s, ξ) is convex in ξ ∈ ℝ^{n} for all (x, s) ∈ Ω × ℝ. \end{matrix}  (7)

Moreover, let us suppose that f(·, s, ξ) ∈ W_{loc}^{1,1} (Ω) for every (s, ξ) ∈ ℝ × ℝ^{n} and that, for every compact set Ω′× H × K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L_{Ω′×H×K} such that

∫_{Ω′} |\frac{∂f}{∂x} (x, s, ξ)| dx ≤ L,  (8)

for every (s, ξ) ∈ H × K. Then the lower semicontinuity condition in (2) holds.

Theorem 2 is proved in Sections 3 and 4.

Finally, we turn our attention to the vector-valued case, i.e., we consider an open set Ω ⊂ ℝ^{n}, integers n, m ≥ 1, functions u : Ω → ℝ^{m} (so that Du(x) ∈ ℝ^{m×n} for x ∈ Ω), integrals F (u, Ω) as in (1) whose integrands f are defined in Ω × ℝ^{m} × ℝ^{m×n}. This case has been recently exploited by Fonseca and Leoni [10], [11]. In this paper we give some semicontinuity results, presenting new aspects when compared with the theory by Fonseca and Leoni. For a detailed discussion, statements, and proofs we refer to Section 5.

Acknowledgment. This research has been partially supported by the Italian Ministero dell’Università e della Ricerca Scientifica e Tecnologica (MURST). Some results proved in this paper have been announced in a meeting hold in Gaeta, Italy, in September 2001, to celebrate the 60th birthday of David Kinderlehrer. This paper is dedicated to David, with friendship and estimation.

2 Hölder continuity versus Lipschitz continuity

In this section we propose an example that originates from one in [12]. This kind of construction is inspired by an old example given by Aronszajn in 1941 (see Pauc [13], page 54 and following), and later exploited by Dal Maso (see Section 4 in [5]) in 1980.

As explained in the introduction, for every fixed α ∈ (0, 1), we seek for a one dimensional integrand f : ℝ×ℝ×ℝ → [0, +∞), that satisfies (3) and hypothesis (5) for the chosen α, but such that lower semicontinuity inequality (2) doesn’t hold for some sequences u_{h}.

Theorem 3 For every α ∈ (0, 1) there exists a sequence (u_{h})_{h∈ℕ} ∈ W^{1,∞}(0, 1), which uniformly converges to u ≡ 0, and there exists a function a : \overline{Ω} × ℝ → ℝ with the following properties:

(i) a(x, s) is bounded and uniformly continuous for (x, s) ∈ \overline{Ω} × ℝ;

(ii) for every s ∈ ℝ, a(x, s) is Hölder continuous (of exponent α) with respect to x ∈ \overline{Ω}; more precisely, there exists a constant L such that

|a(x_{1}, s) − a(x_{2}, s)| ≤ L |x_{1} − x_{2}|^{α} , ∀ x_{1}, x_{2} ∈ \overline{Ω}, ∀ s ∈ ℝ;

(iii) if we define

f(x, s, ξ) = |a(x, s)ξ − 1| , (x, s, ξ) ∈ \overline{Ω} × ℝ × ℝ,

then f(x, s, ξ) satisfies (3), the Hölder continuity property (5) and

lim_{h→+∞} ∫_{−1}^{1} f (x, u_{h}, u′_{h}) dx = 0 ; ∫_{−1}^{1} f (x, 0, 0) dx = 2 .  (9)

The proof is carried over into several steps.

Step 1 (definition of u_{h}): let Ω = (−1, 1). For h ∈ ℕ let v_{h} = v_{h}(x), x ∈ [0, 1], be defined by

v_{h}(x) = \{ \begin{matrix}0  for 0 ≤ x ≤ λ_{h} \\ x − λ_{h} for λ_{h} < x < 1 − λ_{h} \\ 1 − 2λ_{h}  for 1 − λ_{h} ≤ x ≤ 1 \end{matrix}  (10)

where λ_{h} is a sequence of real numbers converging to zero, such that λ_{h} ∈ (0, 1/2) for every h ∈ ℕ. We extend v_{h} as an even function in the interval [−1, 0). We have

{x ∈ Ω : v′_{h}(x) = 0} = (−1, − (1 − λ_{h})) ∪ (−λ_{h}, λ_{h}) ∪ (1 − λ_{h}, 1)

and thus

meas {x ∈ Ω : v′_{h}(x) = 0} = 4λ_{h} , ∀ h ∈ ℕ.  (11)

Since v_{h}(−1) = v_{h}(1), we again extend v_{h}(x) by periodicity from [−1, 1] to all ℝ. For every h ∈ ℕ let u_{h}(x) be defined by

u_{h}(x) = c_{h} + \frac{1}{a_{h}} v_{h} (b_{h}x) ,  (12)

with {a_{h}}, {b_{h}} sequences of integer numbers diverging to +∞ as h → +∞ and c_{h} converging to zero. Therefore u_{h} converges to u ≡ 0 in L^{∞}(Ω).

Step 2 (definition of a): passing to a subsequence if necessary, we can assume that the graphs of the functions u_{h} are disjoint. Let us define

E_{h} = {x ∈ (−1, 1) : u′_{h}(x) = 0} = {x ∈ (−1, 1) : v′_{h}(b_{h}x) = 0} .

We have

u′_{h}(x) = \frac{b_{h}}{a_{h}} v′_{h}(b_{h}x) = \{ \begin{matrix}±\frac{b_{h}}{a_{h}} if x ∈ [−1, 1] − E_{h} \\ 0 if x ∈ E_{h} \end{matrix}

and thus by periodicity reasons and by (11) we have

meas (E_{h}) = meas {x ∈ Ω : u′_{h}(x) = 0} = 4λ_{h} , ∀ h ∈ ℕ.  (13)

For every h ∈ ℕ and for x ∈ [−1, 1] − E_{h} and s = u_{h}(x), we define

a(x, s) = \frac{1}{u′_{h}(x)} = ±\frac{a_{h}}{b_{h}} , x ∈ [−1, 1] − E_{h} .  (14)

We also define a(x, 0) by continuity:

a(x, s) = 0, if x ∈ [−1, 1] and s = 0;

to this aim, since u_{h}(x) → 0 as h → +∞, we impose the condition

\{|a(x, s)| : x ∈ [−1, 1] − E_{h} , s = u_{h}(x)\} = \frac{a_{h}}{b_{h}} → 0 , as h → +∞. (15)

At this stage a(x, s) has been defined as a continuous function on a closed subset of \overline{Ω} × ℝ. In the next Step 3 we will extend it to the full \overline{Ω} × ℝ.

Step 3 (extension of a to \overline{Ω} × ℝ): for every fixed h ∈ ℕ, a(x, s) has been defined in Step 2 at the points (x, s) ∈ S_{h}, where S_{h} is the subset of the graph of u_{h} given by

S_{h} = {(x, s) : x ∈ [−1, 1] − E_{h}, s = u_{h}(x)} .

By the analytic expression of v_{h} and u_{h} in (10), (12), we have

0 ≤ v_{h} ≤ 1 − 2λ_{h} , c_{h} ≤ u_{h}(x) = c_{h} + \frac{1}{a_{h}} v_{h} (b_{h}x) ≤ c_{h} + \frac{1 − 2λ_{h}}{a_{h}} ;

therefore the set S_{h} is contained in the rectangle

R_{h} = \{(x, s) : x ∈ [−1, 1] , c_{h} ≤ s < c_{h} + \frac{1}{a_{h}} \} .

We will extend a(x, s) to the larger rectangle R′_{h} ⊂ [−1, 1] × ℝ, given by

R′_{h} = \{(x, s) : x ∈ [−1, 1] , \frac{1}{2} c_{h} ≤ s ≤ \frac{3}{2} c_{h} + \frac{1}{a_{h}} \} .

Passing possibly to a subsequence, we can assume that R′_{h} ∩ R′_{k} = ∅ if h ≠ k. Then we first extend a equal to zero out of the union ⋃_{h} R′_{h} (and in particular, by continuity, a(x, s) = 0 when s = c_{h}/2 and s = 3c_{h}/2 + 1/a_{h}).

In order to estimate the oscillation |a(x_{1}, s_{1}) − a(x_{2}, s_{2})| when (x_{1}, s_{1}), (x_{2}, s_{2}) vary in R′_{h}, we first consider (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ S_{h} and we prove the following Lipschitz estimate (with constant depending on h)

|a(x_{1}, s_{1}) − a(x_{2}, s_{2})| ≤ \frac{a_{h}}{λ_{h}} · |x_{1} − x_{2}| .  (16)

In fact, under the conditions x_{1}, x_{2} ∈ [−1, 1] − E_{h}, s_{1} = u_{h}(x_{1}), s_{2} = u_{h}(x_{2}), when (x_{1}, s_{1}), (x_{2}, s_{2}) are in the same connected component of S_{h} then a(x_{1}, s_{1}) = a(x_{2}, s_{2}); otherwise b_{h} · |x_{1} − x_{2}| ≥ 2λ_{h} . By the definition of a(x, s) in (14) we have

|a(x_{1}, s_{1}) − a(x_{2}, s_{2})| ≤ 2 \frac{a_{h}}{b_{h}} ≤ a_{h} · \frac{|x_{1} − x_{2}|}{λ_{h}} .

Of course this estimate also gives

|a(x_{1}, s_{1}) − a(x_{2}, s_{2})| ≤ \frac{a_{h}}{λ_{h}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|) ,  (17)

for every h ∈ ℕ and for every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ S_{h} with s_{1} = u_{h}(x_{1}), s_{2} = u_{h}(x_{2}).

Again, by the definition of a(x, s) in (14), we have

max \{|a(x, s)| : (x, s) ∈ S_{h}\} = \frac{a_{h}}{b_{h}} ;

if (x_{1}, s_{1}) ∈ S_{h}, (x_{2}, s_{2}) ∈ R′_{h} with either s_{2} = c_{h}/2 and s_{2} = 3c_{h}/2 + 1/a_{h}, then a(x_{2}, s_{2}) = 0 and |s_{1} − s_{2}| ≥ c_{h}/2. Therefore we obtain

|a(x_{1}, s_{1}) − a(x_{2}, s_{2})| = |a(x_{1}, s_{1})| ≤ \frac{a_{h}}{b_{h}}

≤ \frac{a_{h}}{b_{h}} · \frac{|s_{1} − s_{2}|}{c_{h}/2} ≤ \frac{a_{h}}{λ_{h}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|)

if we pose the condition

c_{h} ≥ 2\frac{λ_{h}}{b_{h}} .  (18)

This proves that the Lipschitz estimate (17) holds at every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R′_{h} where a(x, s) has been already defined. By using Mac Shane lemma, we can extend it to the rectangle R′_{h} with the same Lipschitz constant as in (17). That is we have

|a(x_{1}, s_{1}) − a(x_{2}, s_{2})| ≤ \frac{a_{h}}{λ_{h}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|) ,  (19)

for every h ∈ ℕ and for every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R′_{h}. Moreover we can assume that a is bounded in R′_{h} and that the following bound holds

max \{|a(x, s)| : (x, s) ∈ R′_{h}\} = \frac{a_{h}}{b_{h}} ,  (20)

for every h ∈ ℕ and for every (x, s) ∈ R′_{h}.

Step 4 (Hölder continuity of a): to test Hölder continuity of a(x, s) with respect to x we fix h and s and we estimate the quantity

sup \{\frac{|a(x_{1}, s) − a(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h}\} .  (21)

Let t > 0 be a new real parameter that we will choose later. We estimate the supremum in (21) separately for |x_{1} − x_{2}| ≥ t and for |x_{1} − x_{2}| ≤ t.

Under the further condition |x_{1} − x_{2}| ≥ t, the supremum in (21) can be estimate by computing separately the maximum value of the numerator and the minimum value of the denominator. By (20) we have

max {|a(x_{1}, s) − a(x_{2}, s)| : (x_{1}, s), (x_{2}, s) ∈ R′_{h}}

≤ 2 max \{|a(x, s)| : (x, s) ∈ R′_{h}\} = \frac{2a_{h}}{b_{h}} .

For the same s-values, since |x_{1} − x_{2}| ≥ t, we obtain

sup \{\frac{|a(x_{1}, s) − a(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h} |x_{1} − x_{2}| ≥ t\} ≤ \frac{2a_{h}}{b_{h} t^{α}} . (22)

While, if |x_{1} − x_{2}| ≤ t, we use the Lipschitz estimate (19) (with constant depending on h) with s_{1} = s_{2} ≡ s

|a(x_{1}, s) − a(x_{2}, s)| ≤ \frac{a_{h}}{λ_{h}} · |x_{1} − x_{2}|

and we obtain

sup \{\frac{|a(x_{1}, s) − a(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h} |x_{1} − x_{2}| ≤ t\} ≤ \frac{a_{h}}{λ_{h}} · t^{1−α} .

(23)

From (22) and (23) we deduce that

sup \{\frac{|a(x_{1}, s) − a(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h}\}

≤ max \{\frac{2a_{h}}{b_{h}} t^{−α}; \frac{a_{h}}{λ_{h}} · t^{1−α}\} ≤ 2a_{h} · max \{ \frac{t^{−α}}{b_{h}} ; \frac{t^{1−α}}{λ_{h}} \} .  (24)

The above inequality is valid for every t > 0. We consider the minimum of the right hand side with respect to t > 0, which is assumed when \frac{t^{−α}}{b_{h}} = \frac{t^{1−α}}{λ_{h}} , i.e., when t = \frac{λ_{h}}{b_{h}}.We obtain that the Hölder quotient in the left hand side of (24) is less than or equal to

2 \frac{a_{h}}{(λ_{h})^{α} (b_{h})^{1−α}} .  (25)

Previously we estimated the Hölder continuity with respect to x of a(x, s) in R′_{h}, for every fixed h ∈ ℕ. Thus, to obtain the Hölder continuity of a(x, s) with respect to x, with (x, s) ∈ \overline{Ω} × ℝ, we impose the further condition that the sequence in (25) remains bounded, i.e., that there exists L > 0 such that

\frac{a_{h}}{(λ_{h})^{α} (b_{h})^{1−α}} ≤ L , ∀ h ∈ ℕ.  (26)

Step 5 (lower semicontinuity test): let us prove that

lim_{h→+∞} ∫_{−1}^{1} f (x, u_{h}, u′_{h}) dx = 0 .  (27)

By the definition (14), for every h ∈ ℕ we obtain

f (x, u_{h}, Du_{h}) = |a(x, u_{h}) · u′_{h} − 1| = 0 , ∀ x ∈ [−1, 1] − E_{h} .

Thus, since u′_{h} = 0 on E_{h}, we have

∫_{−1}^{1} f (x, u_{h}, u′_{h}) dx = ∫_{E_{h}} f (x, u_{h}, u′_{h}) dx

≤ ∫_{E_{h}} {max_{Ω×ℝ} |a(x, s)| · |u′_{h}| + 1} dx = 4 λ_{h} ,  (28)

which converges to zero as h → +∞, since λ_{h} → 0 as h → +∞.

Step 6 (sufficient conditions): looking above, we required the following limit relations (see in particular (15), (26), (28) and (18))

\{ \begin{matrix}a_{h} → +∞ , b_{h} → +∞ , c_{h} → 0 , λ_{h} → 0 , \\ \frac{a_{h}}{b_{h}} → 0 , \\ \frac{a_{h}}{(λ_{h})^{α}(b_{h})^{1−α}} ≤ L , c_{h} ≥ 2 \frac{λ_{h}}{b_{h}} , ∀ h ∈ ℕ . \end{matrix} .  (29)

It remains to exhibit sequences of real parameters which satisfy the limit relations (29). To this aim, we consider for example, a parameter q > 1/ (1 − α), q ∈ ℕ, q > 1 and

a_{h} = h, b_{h} = h^{q}, \frac{a_{h}}{(λ_{h})^{α} (b_{h})^{1−α}} = 1.

Since q > 1 then \frac{a_{h}}{b_{h}} → 0. Moreover the last condition gives

(λ_{h})^{α} = \frac{a_{h}}{(b_{h})^{1−α}} = h^{1−q(1−α)}

and we obtain that λ_{h} → 0, since q > 1/ (1 − α). Finally we pose

\frac{c_{h}}{2} = \frac{λ_{h}}{b_{h}} = h^{\frac{1−q(1−α)}{α} −q} = h^{\frac{1−q(1−α)−qα}{α}} = h^{\frac{1−q}{α}}

which converges to zero, since q > 1.

3 Preliminaries

In this section we give some preliminary results (standard in the context of lower semicontinuity problems) that will be used in the proof of the Theorem 2.

Definition 4 Let Ω be a bounded open set in ℝ^{n}. We say that a function v ∈ W^{1,∞}(Ω) is piecewise affine in Ω if there exists a finite partition of Ω into open sets {Ω_{j}}_{j=1}^{N}, N ∈ ℕ, and a set of measure zero, i.e.,

Ω_{j} ∩ Ω_{h} = ∅, ∀j, h ∈ {1, ..., N}, j ≠ h, |Ω\ ⋃_{j=1}^{N} Ω_{j}| = 0,

such that v is affine on each Ω_{j}, i.e., there exists a ξ_{j} ∈ ℝ^{n} and a q_{j} ∈ ℝ such that

v(x) = (ξ_{j}, x) + q_{j}, ∀x ∈ Ω_{j}, j ∈ {1, ..., N}.

We shall denote the space of such functions with Aff(Ω).

The following theorem holds (see Theorem 1.8, Chapter 2 in [4]).

Theorem 5 Let Ω be a bounded open set with Lipschitz boundary and let u ∈ W_{0}^{1,p}(Ω), 1 ≤ p < ∞. Then for every ε > 0 there exists v_{ε} ∈ Aff_{0}(Ω) such that

||u − v_{ε}||_{W^{1,p}(Ω)} < ε.

Here it is the classical generalization of Lusin’s Theorem by Scorza Dragoni, that has revealed to be very useful when dealing with Carathéodory integrands (see [9]).

Theorem 6 (Scorza Dragoni) Let Ω be an open set of ℝ^{n} of finite measure, let g : Ω × ℝ → ℝ^{n} be a Carathéodory function. Then, for every τ > 0, there exists an open set Ω_{τ} with |Ω_{τ} | < τ such that g is continuous on (Ω\Ω_{τ}) × ℝ.

The following approximation result has been given by De Giorgi (see [6]).

Lemma 7 Assume that f = f(x, s, ξ) satisfies (7) and has compact support in Ω × ℝ. Then there exists an increasing sequence of functions {f_{j}(x, s, ξ)}_{j∈ℕ} such that:

(i) f_{j}(x, s, ξ) converges to f(x, s, ξ) on Ω × ℝ × ℝ^{n};

(ii) for every j ∈ ℕ, f_{j}(x, s, ξ) satisfies (7), and has compact support in Ω×ℝ;

(iii) for every j ∈ ℕ there exist constants M_{j} with the following properties:

\{ \begin{matrix}|f_{j}(x, s, ξ)| ≤ M_{j} (1 + |ξ|) , ∀ (x, s, ξ) ∈ Ω × ℝ × ℝ^{n}, \\ |f_{j}(x, s, ξ_{1}) − f_{j}(x, s, ξ_{2})| ≤ M_{j} |ξ_{1} − ξ_{2}| , ∀ (x, s) ∈ Ω × ℝ, ξ_{1}, ξ_{2} ∈ ℝ^{n}. \end{matrix}

(30)

Proof. Here follows a description of the f_{j}. The complete proof can be found in [6]. For a suitable sequence of mollifiers {α^{(t)}}_{t=1}^{∞}, i.e., α^{(t)} ∈ C_{c}^{∞}(ℝ^{n}), with α^{(t)} ≥ 0 and ∫_{ℝ^{n}} α^{(t)}(η) dη = 1, the functions

a_{h}^{(t)}(x, s) = − ∫_{ℝ^{n}} f(x, s, η)α_{ξ_{h}}^{(t)}(η) dη , ∀ h = 1, 2, …, n,  (31)

a_{0}^{(t)}(x, s) = ∫_{ℝ^{n}} f(x, s, η) {(n + 1)α^{(t)}(η) + ∑_{h=1}^{n} η_{h}α_{ξ_{h}}^{(t)}(η)} dη ,

are considered, and then put together in

f_{j}(x, s, ξ) = max_{1≤j≤t} {0, a_{0}^{(t)}(x, s) + ∑_{h=1}^{n} a_{h}^{(t)}(x, s)ξ_{h}} .

4 Proof of the lower semicontinuity theorem

The proof is structured in the following way. We first prove a lower semicontinuity result under technical hypotheses (Lemma 8); this is the core of the proof. Then we show how to apply Lemma 7 in order to get Theorem 2 from the previous results.

Lemma 8 Let us assume that f satisfies (7) and that:

(i) the derivative f_{ξ}(x, s, ξ) exists and is a Carathéodory function in Ω×ℝ×ℝ^{n} such that f_{ξ}(·, s, ξ) ∈ W_{loc}^{1,1} (Ω, ℝ^{n}), for every (s, ξ) ∈ ℝ × ℝ^{n}; moreover, for each Ω′ × H × K ⊂⊂ Ω × ℝ × ℝ^{n}, there exists a constant L_{Ω′×H×K} such that

∫_{Ω′} |\frac{∂f_{ξ}}{∂x} (x, s, ξ)| dx ≤ L_{Ω′×H×K}, ∀ (s, ξ) ∈ H × K;

(ii) there exists a constant M such that,

\{ \begin{matrix}|f_{ξ}(x, s, ξ)| ≤ M, ∀ (x, s, ξ) ∈ Ω × ℝ × ℝ^{n}, \\ |f_{ξ}(x, s, ξ_{1}) − f_{ξ}(x, s, ξ_{2})| ≤ M |ξ_{1} − ξ_{2}| , ∀ (x, s) ∈ Ω × ℝ, ξ_{1}, ξ_{2} ∈ ℝ^{n}. \end{matrix}

(32)

Then (2) holds.

Proof. Let {γ_{i}}_{i∈ℕ} be an increasing sequence of smooth functions with compact support in Ω × ℝ, converging pointwise to 1 in Ω × ℝ. Since γ_{i}(x, s)f(x, s, ξ) is an increasing sequence of functions which pointwise converges to f(x, s, ξ), by a standard argument, it is sufficient to prove the stated lower semicontinuity assuming directly that there exists Ω′ × H ⊂⊂ Ω × ℝ such that f(x, s, ξ) = 0, for every (x, s, ξ) ∈ (Ω\Ω′) × (ℝ\H) × ℝ^{n}, (which implies F (u, Ω) = F (u, Ω′), ∀u ∈ W_{loc}^{1,1}(Ω)). In this position hypotheses (i) of Lemma 8 become f_{ξ}(·, s, ξ) ∈ W^{1,1} (Ω), for every (s, ξ) ∈ ℝ × ℝ^{n}, and

∫_{Ω} |\frac{∂f_{ξ}}{∂x} (x, s, ξ)| dx ≤ L_{K} , ∀ (s, ξ) ∈ ℝ×K ,  (33)

for a suitable constant L_{K} depending on K ⊂⊂ ℝ^{n}.

Let u_{h}, u ∈ W_{loc}^{1,1} (Ω) such that u_{h} → u in L_{loc}^{1} (Ω). We will prove that

lim inf_{h→+∞} F (u_{h}, Ω) ≥ F (u, Ω) .  (34)

Without loss of generality, we can assume that

lim inf_{h→+∞} F (u_{h}, Ω) = lim_{h→+∞} F (u_{h}, Ω) < ∞,

and that u_{h} converges almost everywhere to u in Ω. By the local boundedness of f and by (32), it results that there exists a constant M′ such that

|f(x, s, ξ)| ≤ M′ (1 + |ξ|) , ∀(x, s, ξ) ∈ Ω × ℝ × ℝ^{n}.

In particular we have F (u, Ω) < ∞.

Since u ∈ W^{1,1}(Ω′), and since ∂Ω′ can be supposed Lipschitz, we can extend u outside Ω′ to a function of W_{0}^{1,1}(Ω′′), where Ω′′ ⊂⊂ Ω has Lipschitz boundary and Ω′ ⊂⊂ Ω′′.

Let us fix ε > 0. By Theorem 5 there exists v_{ε} ∈ Aff_{0}(Ω′′) such that, in particular,

∫_{Ω′} |Du − Dv_{ε}| dx ≤ ε;  (35)

by Fatou Lemma and the finiteness of F (u, Ω), we can also choose v_{ε} such that

∫_{Ω′} f(x, u, Dv_{ε}) dx ≥ ∫_{Ω′} f(x, u, Du) dx − ε .  (36)

Since v_{ε} ∈ Aff(Ω′), referring to definition 4 we shall consider {Ωj^{′}}_{j=1}^{N} such that Dv_{ε} = ξ_{j} ∈ ℝ^{n} in Ω′_{j}.

Now let us take a sequence of functions {β_{k}^{ε}}_{k=1}^{∞} ⊂ C^{∞}(Ω′) such that,

\{ \begin{matrix}0 ≤ β_{k}^{ε}(x) ≤ β_{k+1}^{ε}(x) ≤ 1, ∀x ∈ Ω′, \\ β_{k}^{ε}|_{Ω′_{j}} ∈ C_{c}^{∞}(Ω′_{j}), lim_{k→∞} β_{k}^{ε}(x) = 1, a.e. x ∈ Ω′. \end{matrix}

By Beppo Levi’s Theorem, there exists an index k_{ε} such that, for every k ≥ k_{ε}, we have

∫_{Ω′} β_{k}^{ε}(x)f(x, u, Dv_{ε}) dx ≥ ∫_{Ω′} β_{k}^{ε}(x)f(x, u, Du) dx − 2ε .  (37)

We estimate the difference of the integrands in (34)

f(x, u_{h}, Du_{h}) − f(x, u, Du) = f(x, u_{h}, Du_{h}) − f(x, u_{h}, Dv_{ε})  (38)
+f(x, u_{h}, Dv_{ε}) − f(x, u, Dv_{ε}) + f(x, u, Dv_{ε}) − f(x, u, Du) .

By the convexity of f(x, s, ξ) with respect to ξ, for the first difference in the right side of (38) we have:

f(x, u_{h}, Du_{h}) − f(x, u_{h}, Dv_{ε}) ≥ (f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h} − Dv_{ε})
= (f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)
+ (f_{ξ}(x, u, Dv_{ε}), Du − Dv_{ε}) + (f_{ξ}(x, u, Dv_{ε}) − f_{ξ}(x, u_{h}, Dv_{ε}), Dv_{ε}) .

Now by multiplying for β_{k}^{ε} and integrating over Ω′, we obtain

∫_{Ω′} β_{k}^{ε}(x) {f(x, u_{h}, Du_{h}) − f(x, u, Du)} dx

≥ ∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx
+ ∫_{Ω′} β_{k}^{ε}(x) (f_{ξ}(x, u, Dv_{ε}), Du − Dv_{ε}) dx
+ ∫_{Ω′} β_{k}^{ε}(x) (f_{ξ}(x, u, Dv_{ε}) − f_{ξ}(x, u_{h}, Dv_{ε}), Dv_{ε}) dx
+ ∫_{Ω′} β_{k}^{ε}(x) {f(x, u_{h}, Dv_{ε}) − f(x, u, Dv_{ε})} dx
+ ∫_{Ω′} β_{k}^{ε}(x) {f(x, u, Dv_{ε}) − f(x, u, Du) } dx.

We remember that, by (32), |f_{ξ}(x, s, Dv_{ε}(x))| ≤ M for every (x, s) ∈ Ω × ℝ and for every v_{ε}; so, by (35), we have

∫_{Ω′} β_{k}^{ε}(x) (f_{ξ}(x, u, Dv_{ε}), Du − Dv_{ε}) dx ≥ −M ∫_{Ω′} |Du − Dv_{ε}| dx ≥ −Mε;

moreover, being (x, s) → f(x, s, Dv_{ε}(x)) and (x, s) → f_{ξ}(x, s, Dv_{ε}(x)) bounded functions, by the Lebesgue’s dominated convergence theorem we have,

lim_{h→∞} ∫_{Ω′} β_{k}^{ε}(x) {f(x, u_{h}, Dv_{ε}) − f(x, u, Dv_{ε})} dx = 0,

lim_{h→∞} ∫_{Ω′} β_{k}^{ε}(x) (f_{ξ}(x, u, Dv_{ε}) − f_{ξ}(x, u_{h}, Dv_{ε}), Dv_{ε}) dx = 0;

eventually we remember of (37), to get that, for every ε > 0 and for every v_{ε}, k ≥ k_{ε}, we have:

lim inf_{h→∞} ∫_{Ω′} β_{k}^{ε}(x) {f(x, u_{h}, Du_{h}) − f(x, u, Du)} dx

≥lim inf_{h→∞} ∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx − Mε − 2ε.

Then, to complete the proof, it remains to show that for every fixed v_{ε} and k ≥ k_{ε}, we have

lim_{h→+∞} ∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx = 0 . (39)

Indeed, since 0 ≤ β_{k}^{ε}(x) ≤ 1, we shall have that

lim inf_{h→∞} ∫_{Ω′} f(x, u_{h}, Du_{h})dx ≥ ∫_{Ω′} β_{k}^{ε}(x)f(x, u, Du)dx − Mε − 2ε,

for every k ≥ k_{ε}; letting k → ∞, by Beppo Levi’s theorem, we obtain

lim inf_{h→∞} ∫_{Ω′} f(x, u_{h}, Du_{h})dx ≥ ∫_{Ω′} f(x, u, Du)dx − Mε − 2ε.

Now the dependence from v_{ε} is vanished and so we can let ε → 0 to gain the conclusion (34).

So it remains to prove (39); we stress that it suffices to prove this relation for every v_{ε} and k fixed. In order to achieve this we prove that

|∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx |

≤ ∫_{Ω′} n ; ∑_{i=1} |∫_{u(x)}^{u_{h}(x)} \frac{∂}{∂x_{i}} (β_{k}^{ε}(x)f_{ξ_{i}}(x, s, ξ_{j})) ds| dx.  (40)

Indeed let us consider the function g(x, s) = β_{k}^{ε}(x)f_{ξ}(x, s, Dv_{ε}(x)). For σ small enough

g_{σ}(x, s) = ∫_{B_{σ}(x)} k_{σ}(x − y)g(y, s)dy,

is defined on Ω′ × ℝ, with g_{σ}(·, s) ∈ C^{∞}(Ω′; ℝ^{n}) for every s fixed (here k_{σ} is the usual mollifier). In particular for every small σ, and u, v ∈ W^{1,1}(Ω),

∫_{Ω′} (g_{σ}(x, v(x)), Dv(x)) − (g_{σ}(x, u(x)), Du(x)) dx

= − ∫_{Ω′} \{ n ; ∑_{i=1} ∫_{u(x)}^{v(x)} \frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s)ds\} dx;

now we prove that we can pass to the limit as σ → 0 in this formula to gain (40). If we look, for example, at

lim_{σ→0} ∫_{Ω′} (g_{σ}(x, v(x)), Dv(x)) dx = ∫_{Ω′} (g(x, v(x)), Dv(x)) dx ,

since the boundedness of g and the summability of Dv, we only need to prove that

lim_{σ→0} g_{σ}(x, v(x)) = g(x, v(x)),  (41)

for a.e. x ∈ Ω′. The following proof of (41) has been suggested to us by the referee, shortening the original argument. Let us note that, from standard properties of convolutions, we only know that for every fixed s ∈ ℝ there exists a set K_{s} of full measure in Ω′ such that lim_{σ→0} g_{σ}(x, s) = g(x, s) for every x ∈ K_{s}. Yet to prove (41) we need to show that a set of full measure K_{s} with this property can be chosen independently from s. To show this, we apply Scorza-Dragoni’s Theorem in order to find a numerable disjoint family of compact sets (K_{i})_{i=1}^{∞} such that L^{n} (Ω′\ ⋃_{i=1}^{∞} K_{i}) = 0 and g is (uniformly) continuous on K_{i} × ℝ for every i ∈ ℝ. Then we consider the set C = ⋃_{i=1}^{∞} C_{i} where we put

C_{i} = {x ∈ K_{i} : θ^{n}(K_{i}, x) = 1},

(here θ^{n}(K_{i}, x) = lim_{ρ→0} (L^{n}(B_{ρ}(x) ∩ K_{i})/ω_{n}ρ^{n}) is as usual the n-dimensional density of the set K_{i} at the point x). By Lebesgue Density Theorem it is L^{n}(K_{i}\C_{i}) = 0, and hence C is of full measure in Ω′. By proving that

lim_{σ→0} g_{σ}(x, s) = g(x, s), ∀(x, s) ∈ C × ℝ,  (42)

we conclude the proof of (41). To see the validity of (42) it is sufficient to show that every x ∈ C is a Lebesgue point of g(·, s) for every s ∈ ℝ. Let us fix x ∈ C_{i} for some i and ε > 0. By the choice of K_{i} there exists ρ_{ε} > 0 such that

|g(x, s) − g(y, s)| ≤ ε, ∀(y, s) ∈ (B_{ρ_{ε}}(x) ∩ K_{i}) × ℝ,

and hence, for every ρ < ρ_{ε}, we have

\frac{1}{ω_{n}ρ^{n}} ∫_{B_{ρ}(x)} |g(x, s) − g(y, s)|dy ≤ ε\frac{L^{n}(B_{ρ}(x) ∩ K_{i})}{ω_{n}ρ^{n}} + 2M \frac{L^{n}(B_{ρ}(x)\K_{i})}{ω_{n}ρ^{n}},

where we have used the boundedness of g. Letting ρ → 0, by the definition of n-dimensional density and by x ∈ C_{i}, we find

lim_{ρ→0} \frac{1}{ω_{n}ρ^{n}} ∫_{B_{ρ}(x)} |g(x, s) − g(y, s)|dy ≤ εθ^{n}(K_{i}, x) + 2Mθ^{n}(ℝ^{n}\K_{i}, x) = ε,

for every ε > 0. This proves (42), and then, as explained, (41).

To have the thesis of the lemma it remains to prove that, for each i = 1, ..., n,

lim_{σ→0} ∫_{Ω′} \{∫_{u(x)}^{v(x)} \frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s)ds\} dx = ∫_{Ω′} \{∫_{u(x)}^{v(x)} \frac{∂g^{(i)}}{∂x_{i}} (x, s)ds\} dx.

Indeed

∫_{Ω′} |∫_{u(x)}^{v(x)} \frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s)ds − ∫_{u(x)}^{v(x)} \frac{∂g^{(i)}}{∂x_{i}} (x, s)ds| dx

≤ ∫_{Ω′} \{∫_{H} |\frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s) − \frac{∂g^{(i)}}{∂x_{i}} (x, s)| ds\} dx

= ∫_{H} \{∫_{Ω′} |\frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s) − \frac{∂g^{(i)}}{∂x_{i}} (x, s)| dx\} ds.

Once again, by standard properties of convolution, we have that, for each s ∈ ℝ,

t_{σ}(s) = ∫_{Ω′} |\frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s) − \frac{∂g^{(i)}}{∂x_{i}} (x, s)| dx → 0

when σ → 0; moreover it results

|t_{σ}(s)| ≤ ∫_{Ω′} |\frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s)| dx + ∫_{Ω′} |\frac{∂g^{(i)}}{∂x_{i}} (x, s)| dx ≤ 2L .

Since t_{σ}(s) is bounded on H uniformly on σ, by Lebesgue’s dominated convergence Theorem we conclude. Hence (40) is established.

By (40) we have

|∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx |

≤ n ; ∑_{i=1} ∑_{j=1}^{N} ∫_{Ω′_{j}} |∫_{u(x)}^{u_{h}(x)} |\frac{∂β_{k}^{ε}}{∂x_{i}} (x)| |f_{ξ_{i}}(x, s, ξ_{j})| ds| dx

+ n ; ∑_{i=1} ∑_{j=1}^{N} ∫_{Ω′_{j}} |∫_{u(x)}^{u_{h}(x)} |\frac{∂f_{ξ_{i}}}{∂x_{i}} (x, s, ξ_{j})| ds| dx.  (43)

Using the first inequality in (32), we have

n ; ∑_{i=1} ∑_{j=1}^{N} ∫_{Ω′_{j}} |∫_{u(x)}^{u_{h}(x)} |\frac{∂β_{k}^{ε}}{∂x_{i}} (x)| |f_{ξ_{i}}(x, s, ξ_{j})| ds| dx

≤ nM ||Dβ_{k}^{ε}||_{∞} ∫_{Ω′} |u_{h} − u| dx,

which goes to zero for h → +∞. On the other hand,

n ; ∑_{i=1} ∑_{j=1}^{N} ∫_{Ω′_{j}} |∫_{u(x)}^{u_{h}(x)} |\frac{∂f_{ξ_{i}}}{∂x_{i}} (x, s, ξ_{j})| ds| dx ≤ n ∑_{j=1}^{N} ∫_{D_{j,h}} |\frac{∂f_{ξ}}{∂x} (x, s, ξ_{j})| dxds,

for

D_{j,h} = {(x, s) ∈ Ω′_{j} × H : min{u_{h}(x), u(x)} ≤ s ≤ max{u_{h}(x), u(x)}}.

Now

|D_{j,h}| ≤ ∫_{Ω′} |u_{h} − u| dx → 0, as h → ∞,

while, if we define K = {ξ_{j}}_{j=1}^{N} and consider L_{K} as in hypothesis (33) we have that, for every j,

∫_{Ω×ℝ} |\frac{∂f_{ξ}}{∂x} (x, s, ξ_{j})| dxds = ∫_{H} ds ∫_{Ω′} |\frac{∂f_{ξ}}{∂x} (x, s, ξ_{j})| dx ≤ |H| L_{K}.

Hence

lim_{h→∞} ∫_{D_{j,h}} |\frac{∂f_{ξ}}{∂x} (x, s, ξ_{j})| dxds = 0, ∀j = 1, ..., N,

from which we conclude that, for ε fixed,

lim_{h→∞} |∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, u_{h}, Dv_{ε}), Du_{h}) − (f_{ξ}(x, u, Dv_{ε}), Du)} dx |

≤ lim_{h→∞} nM ||Dβ_{k}^{ε}||_{∞} ∫_{Ω′} |u_{h} − u| dx

+ lim_{h→∞} nN max_{1≤j≤N} ∫_{D_{j,h}} |\frac{∂f_{ξ}}{∂x} (x, s, ξ_{j})| dxds = 0.

This ends the proof.

We are ready for the proof of Theorem 2. We start preparing the integrand f to satisfy the hypotheses of Lemma 7, then we manipulate De Giorgi’s approximating functions in order to apply on them Lemma 8. As usual, the increasing convergence of the approximating integrands give the result for the initial functional.

Proof of theorem 2. Arguing as in the beginning of Lemma 8 we can assume that there exists Ω′ × H ⊂⊂ Ω × ℝ such that

f(x, s, ξ) = 0, ∀(x, s, ξ) ∈ (Ω\Ω′) × (ℝ\H) × ℝ^{n}.

In this position the hypotheses of Theorem 2 become

f(·, s, ξ) ∈ W^{1,1} (Ω) , ∀ (s, ξ) ∈ ℝ × ℝ^{n},  (44)

and

∫_{Ω} |\frac{∂f}{∂x} (x, s, ξ)| dx ≤ L_{K} , ∀ (s, ξ) ∈ ℝ×K ,  (45)

for a suitable constant L_{K} depending on K ⊂⊂ ℝ^{n}.

Since f has compact support in (x, s), we can approximate it with an increasing sequence {f_{j}(x, s, ξ)}_{j∈ℕ} as in Lemma 7. We would like to apply Lemma 8 to such functions, but this is not possible. So we denote by φ_{ρ} a mollifier in ℝ^{n} (φ_{ρ} ∈ C_{c}^{∞}(ℝ^{n}), φ_{ρ} ≥ 0, φ_{ρ}(η) = 0 if |η| ≥ ρ, ∫_{ℝ^{n}} φ_{ρ}(η) dη = 1) and we consider the functions

g_{j}(x, s, ξ) = ∫_{ℝ^{n}} f_{j}(x, s, ξ − η)φ_{ρ_{j}}(η) dη,

where we define, for M_{j} as in the statement of Lemma 7, ρ_{j} = (jM_{j})^{−1}. By the Lipschitz continuity (30) of f_{j} with respect to ξ ∈ ℝ^{n}, we have

|g_{j}(x, s, ξ) − f_{j}(x, s, ξ)| ≤ ∫_{ℝ^{n}} |f_{j}(x, s, ξ − η) − f_{j}(x, s, ξ)| φ_{ρ_{j}}(η) dη

≤ M_{j} ∫_{B_{ρ_{j}} (0)} |η| φ_{ρ_{j}}(η) dη ≤ \frac{1}{j} ,

so that, for every (x, s, ξ) ∈ Ω × ℝ × ℝ^{n},

f_{j}(x, s, ξ) − \frac{2}{j} ≤ g_{j}(x, s, ξ) − \frac{1}{j} ≤ f_{j}(x, s, ξ) ≤ f(x, s, ξ) .  (46)

By the monotone convergence theorem we have

lim_{j→+∞} ∫_{Ω′} f_{j} (x, u(x), Du(x)) dx = ∫_{Ω′} f (x, u(x), Du(x)) dx .

Thus, if we consider the sequence of integrals

F_{j}(u, Ω′) = ∫_{Ω′} \{g_{j} (x, u(x), Du(x)) − \frac{1}{j}\} dx ,  (47)

by (46) we obtain that F_{j}(u, Ω′) converges, as j → +∞, to the main integral F (u, Ω′) = F (u, Ω) , and that, at the same time, F_{j}(u, Ω′) ≤ F (u, Ω′) for every j ∈ ℕ. Therefore it suffices to show that every F_{j}(u, Ω′) is lower semicontinuous. To achieve this we shall invoke Lemma 8, and so all that we have to do is to verify that the g_{j} are in the hypotheses of such Lemma.

Clearly every g_{j} is Carathéodory, non negative, convex in the ξ variable, locally bounded, and has compact support in (x, s), being a convolution in ξ of f_{j} ≤ f. Furthermore (g_{j})_{ξ} exists and is a Carathéodory function.

Next we verify hypothesis (32). By the Lipschitz continuity (6) of f_{j}, we have

|g_{j}(x, s, ξ_{1}) − g_{j}(x, s, ξ_{2})| ≤ ∫_{ℝ^{n}} |f_{j}(x, s, ξ_{1} − η) − f_{j}(x, s, ξ_{2} − η)| φ_{ρ_{j}}(η) dη

≤ M_{j} |ξ_{1} − ξ_{2}| ,

and so |(g_{j})_{ξ}(x, s, ξ)| ≤ M_{j}. By the definition of convolution

|(g_{j})_{ξ}(x, s, ξ_{1}) − (g_{j})_{ξ}(x, s, ξ_{2})|

≤ ∫_{ℝ^{n}} |f_{j}(x, s, ξ_{1} − η) − f_{j}(x, s, ξ_{2} − η)| |(φ_{ρ_{j}})_{ξ} (η)| dη ≤ M_{j}T_{j} |ξ_{1} − ξ_{2}| , where

T_{j} = ∫_{ℝ^{n}} |(φ_{ρ_{j}})_{ξ} (η)| dη.

Therefore the assumption (32) is satisfied with M = max{M_{j}, M_{j}T_{j}}.

It remains to study the properties of weak derivability and summability of (g_{j})_{ξ} in the x variable. We shall start examining such properties for the coefficients a_{h}^{(t)} in Lemma 7. From (31) we have that a_{h}^{(t)} is Carathéodory. For every ψ ∈ C_{c}^{∞}(Ω), by (44) we have that

∫_{Ω} a_{h}^{(t)}(x, s)\frac{∂ψ}{∂x_{i}} (x)dx = ∫_{Ω} (− ∫_{ℝ^{n}} f(x, s, η)α_{ξ_{h}}^{(t)}(η) dη) \frac{∂ψ}{∂x_{i}} (x)dx

= − ∫_{ℝ^{n}} (∫_{Ω} f(x, s, η)\frac{∂ψ}{∂x_{i}} (x)dx) α_{ξ_{h}}^{(t)}(η) dη

= ∫_{ℝ^{n}} (∫_{Ω} \frac{∂f}{∂x_{i}} (x, s, η)ψ(x)dx) α_{ξ_{h}}^{(t)}(η) dη

= ∫_{Ω} (∫_{ℝ^{n}} \frac{∂f}{∂x_{i}} (x, s, η)α_{ξ_{h}}^{(t)}(η)dη) ψ(x)dx;

furthermore, thanks to the main assumption (45), we obtain

∫_{Ω} |\frac{∂a_{h}^{(t)}}{∂x_{i}} (x, s)| dx = ∫_{Ω} |∫_{ℝ^{n}} \frac{∂f}{∂x_{i}} (x, s, η)α_{ξ_{h}}^{(t)}(η)dη| dx

≤ ||α_{ξ_{h}}^{(t)}||_{∞} ∫_{spt(α)} (∫_{Ω} |\frac{∂f}{∂x_{i}} (x, s, η)| dx) dη

≤ ||α_{ξ_{h}}^{(t)}||_{∞} |spt(α^{(t)})| L_{spt(α^{(t)})} .

Hence a_{h}^{(t)}(·, s) ∈ W^{1,1}(Ω), and there exists a constant R_{h}^{(t)} such that

∫_{Ω} |\frac{∂a_{h}^{(t)}}{∂x_{i}} (x, s)| dx ≤ R_{h}^{(t)}, ∀s ∈ ℝ.

The same analysis carried over for a_{h}^{(t)} applies unchanged to a_{0}^{(t)}, so that a_{0}^{(t)}(x, s) is Carathéodory, with a_{0}^{(t)}(·, s) ∈ W^{1,1}(Ω), and there exists a constant R_{0}^{(t)} such that

∫_{Ω} |\frac{∂a_{0}^{(t)}}{∂x_{i}} (x, s)| dx ≤ R_{0}^{(t)}, ∀s ∈ ℝ.

In particular a_{0}^{(t)}(·, s) + ∑_{i=1}^{n} a_{h}^{(t)}(·, s)ξ_{h} ∈ W^{1,1}(Ω), from which,

f_{j}(·, s, ξ) = max_{t=1,...,j} {0, a_{0}^{(t)}(·, s) + ∑_{h=1}^{n} a_{h}^{(t)}(·, s)ξ_{i}} ∈ W^{1,1}(Ω),

and finally we have (g_{j})_{ξ} (·, s, ξ) ∈ W^{1,1}(Ω), with

\frac{∂(g_{j})_{ξ}}{∂x_{i}} (x, s, ξ) = ∫_{ℝ^{n}} \frac{∂f_{j}}{∂x_{i}} (x, s, ξ − η) (φ_{ρ_{j}})_{ξ} (η) dη.

Since,

∫_{Ω} |\frac{∂f_{j}}{∂x_{i}} (x, s, ξ)| dx ≤ ∑_{t=1}^{j} \{∫_{Ω} |\frac{∂}{∂x_{i}} (a_{0}^{(t)}(x, s) + ∑_{h=1}^{n} a_{h}^{(t)}(x, s)ξ_{h})| dx\}

≤ ∑_{t=1}^{j} ∫_{Ω} |\frac{∂a_{0}^{(t)}}{∂x_{i}} (x, s)| dx + |ξ| ∑_{t=1}^{j} \{ max_{h=1....,n} ∫_{Ω} |\frac{∂a_{h}^{(t)}}{∂x_{i}} (x, s)| dx\} ≤ S_{j} (1 + |ξ|) ,

for a suitable constant S_{j} (depending on R_{0}^{(t)}, R_{h}^{(t)}, j), we deduce that

∫_{Ω} |\frac{∂(g_{j})_{ξ}}{∂x_{i}} (x, s, ξ)| dx ≤ ∫_{Ω} dx ∫_{ℝ^{n}} |\frac{∂f_{j}}{∂x_{i}} (x, s, ξ − η)| |(φ_{ρ_{j}})_{ξ} (η)| dη

≤ ∫_{B_{ρ_{j}} (0)} |(φ_{ρ_{j}})_{ξ} (η)| dη ∫_{Ω} |\frac{∂f_{j}}{∂x_{i}} (x, s, ξ − η)| dx

≤ ∫_{B_{ρ_{j}} (0)} S_{j} (1 + |ξ − η|) |(φ_{ρ_{j}})_{ξ} (η)| dη ≤ S_{j}T_{j}(1 + |ξ| + ρ_{j}).

Hence if K ⊂⊂ ℝ^{n} for (L_{j})_{K} = S_{j}T_{j}(1+T +ρ_{j}), (where T is equal to the radius of a ball centered at the origin and containing K), we have

∫_{Ω} |\frac{∂(g_{j})_{ξ_{i}}}{∂x_{i}} (x, s, ξ)| dx ≤ (L_{j})_{K}, ∀(s, ξ) ∈ ℝ × K.

So Lemma 8 applies to every g_{j}, providing the lower semicontinuity of each F_{j}. By the preceding arguments this end the proof of Theorem 2.

5 Some results in the vector-valued case

In this section we deal with a function f : Ω × ℝ^{m} × ℝ^{m×n} → [0, ∞), where Ω is an open set of ℝ^{n}, n, m ≥ 1, and for every (x, s) ∈ Ω × ℝ^{m} the function f(x, s, ·) is convex in ℝ^{m×n}. We seek conditions on f sufficient to prove

\{ \begin{matrix}u_{h}, u ∈ W_{loc}^{1,1}(Ω; ℝ^{m}) \\ u_{h} → u in L_{loc}^{1}(Ω; ℝ^{m}) \end{matrix} ⟹ F (u, Ω) ≤ lim inf_{h→∞} F (u_{h}, Ω) .  (48)

The lower semicontinuity results by Serrin [14] were proved for the scalar case m = 1. Some effort has been spent to understand the question of their validity in the vector-valued case m > 1. See in particular Eisen’s [8] and Černý-Malý [2], [3] counterexamples; see also the results by Fonseca and Leoni [10].

The main difference with respect to the scalar case is the need of an assumption of coercivity of the integrand, at least when we assume the dependence on the s variable, as Eisen’s counterexample shows [8]. Under the investigation of the scalar case made in the previous sections, we became aware of the fact that, in particular, two results, without coercivity assumptions, hold in the vectorvalued case too, provided the dependence on the s variable is dropped. The first one is a vector-valued version of Theorem 1.

Proposition 9 Let us assume that f(x, ξ) satisfies (3) and, for every ξ ∈ ℝ^{m×n}, f(·, ξ) ∈ W_{loc}^{1,∞} (Ω). Let us also assume that, for every Ω′ ⊂⊂ Ω, K ⊂⊂ ℝ^{m×n}, there exists a constant L = L_{Ω′×K} such that

ess sup_{x∈Ω′} |\frac{∂}{∂x}f(x, ξ)| ≤ L,

for every ξ ∈ K. Then the lower semicontinuity (48) holds.

The second result we give is an extension to the vector-valued case of the Serrin’s result related to strictly convex integrands. Since the counterexample by Černý and Malý [3], is natural here to drop the dependence on s.

Proposition 10 Let us assume that f(x, ξ) satisfies (3) and that f(x, ·) is strictly convex for every x ∈ Ω.Then (48) holds.

For comparison with Proposition 12 below, we quote a result proved in [10].

Theorem 11 (Fonseca - Leoni) Let us assume that f is lower semicontinuous on Ω × ℝ^{m} × ℝ^{m×n} and it is convex with respect to ξ ∈ ℝ^{m×n}. Moreover, for every (x_{0}, s_{0}) ∈ Ω × ℝ^{m} there exists δ_{0} > 0 and a continuous function g : B_{δ_{0}}(x_{0}, s_{0}) → ℝ^{m×n} such that

(x, s) ⟼ f(x, s, g(x, s)) ∈ L^{∞}(B_{δ_{0}}(x_{0}, s_{0})).  (49)

Finally, let us assume that either f(x_{0}, s_{0}, ξ) = 0 for every ξ ∈ ℝ^{m×n}, or there exists δ_{0} > 0 and c_{0} > 0 such that

f(x, s, ξ) ≥ c_{0} |ξ| − \frac{1}{c_{0}} , ∀(x, s) ∈ B_{δ_{0}}(x_{0}, s_{0}).

Then the lower semicontinuity (48) holds.

A question posed in [10] is about the necessity of assumption (49). This hypothesis comes out by the use of the approximation theorem by Ambrosio (Lemma 1.5, Statement A, in [1]). A counterexample by Černý and Malý [2] shows that such assumption in general cannot be dropped. However it may be interesting to note that, if we assume a local coerciveness with superlinear growth, then Theorem 11 holds without (49), as stated in the next proposition.

Proposition 12 Let us assume that f is lower semicontinuous on Ω × ℝ^{m} × ℝ^{m×n} and convex in the ξ variable. Then (48) holds if we also assume that, for every (x_{0}, s_{0}) ∈ Ω × ℝ^{m}, either f(x_{0}, s_{0}, ξ) = 0 for every ξ ∈ ℝ^{m×n}, or there exists δ_{0} > 0, c_{0} > 0 and p_{0} > 1 such that

f(x, s, ξ) ≥ c_{0} |ξ|^{p_{0}} − \frac{1}{c_{0}} , ∀(x, s) ∈ B_{δ_{0}}(x_{0}, s_{0}).  (50)

We notice that, under assumption (50), if the coerciveness constant doesn’t depend on s as in Proposition 12, we can consider more generally that f is only measurable with respect to x (see Proposition 5.6 in [12]).

Below we give some details of the proofs. We start with a lemma useful in the proof of Proposition 9.

Lemma 13 Let us assume that f satisfies (3) and that: (i) there exists an open set Ω′ ⊂⊂ Ω, such that f(x, ξ) = 0, for every x ∈ Ω\Ω′ and for every ξ ∈ ℝ^{m×n}; (ii) the derivative f_{ξ}(x, ξ) exists and is a continuous function in Ω×ℝ^{m×n} such that f_{ξ}(·, ξ) ∈ W^{1,∞} (Ω; ℝ^{m×n}), for every ξ ∈ ℝ^{m×n}; (iii) there exists a constant M such that

\{ \begin{matrix}|f_{ξ}(x, ξ)| ≤ M, ∀ (x, ξ) ∈ Ω × ℝ^{m×n}, \\ |f_{ξ}(x, ξ_{1}) − f_{ξ}(x, ξ_{2})| ≤ M |ξ_{1} − ξ_{2}| , ∀ x ∈ Ω, ξ_{1}, ξ_{2} ∈ ℝ^{m×n}. \end{matrix}  (51)

Then the lower semicontinuity inequality (48) holds.

Proof. Let u_{h}, u ∈ W_{loc}^{1,1} (Ω; ℝ^{m}) such that u_{h} → u in L_{loc}^{1}(Ω; ℝ^{m}). Without loss of generality, we can assume that

lim inf_{h→+∞} F (u_{h}, Ω) = lim_{h→+∞} F (u_{h}, Ω) < ∞,

and that u_{h} converges almost everywhere to u in Ω. We also remark that, from the first assumption in (51), we have F (u, Ω) < ∞, so that F (u_{h}, Ω) = F (u_{h}, Ω′) and F (u, Ω) = F (u, Ω′).

As in the proof of Lemma 8, for every ε > 0, there exists v_{ε} ∈ Aff(Ω′′; ℝ^{m}) such that,

∫_{Ω′} |Du − Dv_{ε}| dx ≤ ε , ∫_{Ω′} f(x, Dv_{ε}) dx ≥ ∫_{Ω′} f(x, Du) dx − ε . (52)

Since v_{ε} ∈ Aff(Ω′; ℝ^{m}), referring to definition 4, we consider {Ωj^{′}}_{j=1}^{N} such that Dv_{ε} = ξ_{j} ∈ ℝ^{m×n} in Ω′_{j}. Now let us consider the sequence of functions {β_{k}^{ε}}_{k=1}^{∞} ⊂ C^{∞}(Ω′) as in Lemma 8 and using the same argument we have semicontinuity provided that, for every v_{ε} and k fixed,

lim_{h→+∞} ∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, Dv_{ε}), Du_{h}) − (f_{ξ}(x, Dv_{ε}), Du)} dx = 0 .  (53)

Decomposing Ω′ on the sets where Dv_{ε} is constant and observing that β_{k}^{ε}(x)f(x, ξ_{j}) ∈ W_{0}^{1,∞} (Ω′_{j}; ℝ^{m×n}), we have that

|∫_{Ω′} β_{k}^{ε}(x) {(f_{ξ}(x, Dv_{ε}), Du_{h}) − (f_{ξ}(x, Dv_{ε}), Du)} dx |

≤ ∑_{j=1}^{N} |∫_{Ω′_{j}} (β_{k}^{ε}(x)f_{ξ}(x, ξ_{j}), Du_{h} − Du)| dx

= ∑_{j=1}^{N} |∫_{Ω′_{j}} ∑_{i=1}^{n} ∑_{t=1}^{m} β_{k}^{ε}(x)f_{ξ_{t,i}}(x, ξ_{j}) · \frac{∂}{∂x_{i}} (u_{h}^{(t)} − u^{(t)})| dx

= ∑_{j=1}^{N} |∫_{Ω′_{j}} ∑_{i=1}^{n} ∑_{t=1}^{m} \frac{∂}{∂x_{i}} (β_{k}^{ε}(x)f_{ξ_{t,i}}(x, ξ_{j})) · (u_{h}^{(t)} − u^{(t)})| dx

≤ ∑_{j=1}^{N} ∑_{t=1}^{m} ∫_{Ω′_{j}} C_{j} |u_{h}^{(t)} − u^{(t)}| dx → 0,

for h → +∞, where C_{j} = sup \{|\frac{∂}{∂x_{i}} (β_{k}^{ε}(x)f_{ξ}(x, ξ_{j}))| : x ∈ Ω′_{j}\}.

Proof of proposition 9. We only have to use the same (in fact simpler) argument used by the authors in [12] for the proof of Theorem 1.6.

Proof of proposition 10. We refer here to theorems and lemmas in the paper [14] by Serrin: the proof of Theorem 12, valid in the scalar case, is based on Lemmas 4, 5, 7, 8. However, Lemmas 4, 5, 8 are also true in the vectorial setting; moreover the presence of the variable s is not determinant for their validity. So Lemma 7 is the only one which makes Serrin’s Theorem 12 work only in the scalar setting; on the contrary, if f is independent of s, we can easily extend Lemma 7 to the vectorial setting and complete the proof.

Proof of proposition 12. The usual blow up method reduce the problem of lower semicontinuity to that of proving the inequality

lim inf_{k→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), Dw_{k}(y))dy ≥ f(x_{0}, u(x_{0}), Du(x_{0})),

where x_{0} ∈ Ω, Q is the unit cube of ℝ^{n}, ε_{k} tends decreasing to zero, and w_{k} ∈ W^{1,1}(Q; ℝ^{m}) converges strongly L^{1}(Q; ℝ^{m}) to the affine limit w_{0}(y) := (Du(x_{0}), y) , y ∈ Q. This is trivial if, for the fixed value (x_{0}, u(x_{0})) it results f(x_{0}, u(x_{0}), ·) = 0. Otherwise there exist δ_{0}, c_{0} positive and p_{0} > 1 such that

f(x, s, ξ) ≥ c_{0} |ξ|^{p_{0}} − \frac{1}{c_{0}} ,

holds for every (x, s, ξ) ∈ \overline{B_{δ_{0}}(x_{0}, u(x_{0}))} × ℝ^{m×n}. If we set M = \overline{B_{δ_{0}}(x_{0}, u(x_{0}))} we are in the hypotheses of Lemma 1.5, Statement B by Ambrosio [1]; thus there exist continuous functions a_{h} : M → ℝ and b_{h} : M → ℝ^{m×n} such that

f(x, s, ξ) = sup_{h∈ℕ} {a_{h}(x, s) + (b_{h}(x, s), ξ)} ,

for every (x, s, ξ) ∈ \overline{B_{δ_{0}}(x_{0}, u(x_{0}))} × ℝ^{m×n}. Now we can conclude the proof as in Fonseca-Leoni [10], Theorem 1.1.

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