SOME REMARKS ON LOWER SEMICONTINUITY

Irene Fonseca † & Giovanni Leoni ‡

Dedicated to James Serrin

Abstract. Classical lower semicontinuity results obtained by Serrin in the scalar case are extended to the vectorial setting for convex and polyconvex integrands.

§1. Introduction.

The following theorems on lower semicontinuity were proved by Serrin in [S]:

Theorem A. (Serrin [S, Th. 12]) Let f : Ω × ℝ × ℝ^{N} → [0, ∞) be a continuous function, with f(x, u, ·) convex in ℝ^{N}, where Ω is an open subset of ℝ^{N}. Assume that f satisfies any one of the following conditions:

(i) f(x, u, ξ) → ∞ as |ξ| → ∞ for each (x, u) ∈ Ω × ℝ. (ii) f(x, u, ·) is strictly convex in ℝ^{N} for each (x, u) ∈ Ω × ℝ. (iii) The derivatives f_{x}, f_{ξ} and f_{ξ x} exist and are continuous.

Then the functional

F (u, Ω) := ∫_{Ω} f(x, u(x), ∇u(x)) dx

is lower semicontinuous in W_{loc}^{1,1}(Ω; ℝ) with respect to local convergence in L^{1}.

Theorem B. (Serrin [S, Th. 13]) Let f : Ω × ℝ × ℝ^{N} → [0, ∞) be a continuous function, with f(x, u, ·) convex in ℝ^{N}, where Ω is an open subset of ℝ^{N}. Then the functional F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω; ℝ) with respect to weak convergence in BV_{loc}(Ω; ℝ).

We recall that a sequence of functions {u_{n}} ⊂ W_{loc}^{1,1}(Ω; ℝ) (or BV_{loc}(Ω; ℝ) ) converges weakly in BV_{loc}(Ω; ℝ) to some function u if u_{n} converges to u in L_{loc}^{1}(Ω; ℝ) and

(1.1)  sup_{n} ∫_{Ω} |∇u_{n}(x)| dx < ∞.

Theorem B was later improved by Ambrosio [A] who established the following result:

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 49J45, 49Q20.

Key words and phrases. Functions of bounded variation, lower semicontinuity, convexity, polyconvexity.

† Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

‡ Dipartimento di Scienze e Tecnologie Avanzate, Universitá del Piemonte Orientale, Alessandria, Italy 15100.

Theorem C. (Ambrosio [A, Th. 3.2]) Let f : Ω×ℝ×ℝ^{N} → [0, ∞] be a lower semicontinuous function, with f(x, u, ·) convex in ℝ^{N}, where Ω is an open bounded subset of ℝ^{N}. Assume that there exists a continuous function g : Ω × ℝ → ℝ^{N} such that the function

(1.2)  (x, u) → f(x, u, g(x, u))  is continuous and real valued.

Then the functional F (u, Ω) is lower semicontinuous in W^{1,1}(Ω; ℝ) with respect to weak convergence in BV (Ω; ℝ).

Theorems A and B were also extended to the vectorial case by Morrey in his book on Calculus of Variations [M, Thms. 4.1.1, 4.1.2]. However, several years later Eisen [E] found a gap in Morrey’s proof, thus placing in doubt the validity of Theorem A and B when d > 1, and constructed counterexamples for Theorem A(iii) when d > 1 (see also [St] for more details and an extensive bibliography). To our knowledge the validity of Theorems A(i)–(ii) and B when d > 1 remains open.

Note also that the techniques used by Ambrosio in Theorem C are rather different from those of Serrin’s paper, and are based on a paper of De Giorgi, Buttazzo and Dal Maso [DBDM], where the authors proved lower semicontinuity with respect to local convergence in L^{1} for integrands of the form f = f(u, ξ), which are only measurable in the u variable and not necessarily coercive. One of the main ideas in [DBDM] was to approximate from below f = f(u, ξ) by functions of the form (a(u) + b(u) · ξ)^{+} and then to study lower semicontinuity for these special integrands. A crucial observation is that b(u(x)) · ∇u(x) is the gradient of B(u(x)) := ∫_{0}^{u(x)} b(s) ds, and thus integration by parts allows one to eliminate the troublesome term b(u(x)) · ∇u(x). Clearly this approach cannot be extended to the vectorial case.

In this paper we show that Theorems A(i) and B continue to hold in the vectorial case. More precisely, we can prove the following:

Theorem 1.1. Let f : Ω × ℝ^{d} × 𝕄^{d×N} → [0, ∞] be a lower semicontinuous function, with f(x, u, ·) convex in 𝕄^{d×N}. Suppose that for all (x_{0}, u_{0}) ∈ Ω × ℝ^{d} either f(x_{0}, u_{0}, ξ) ≡ 0 for all ξ ∈ 𝕄^{d×N}, or there exist C, δ_{0} > 0, and a continuous function g : B(x_{0}, δ_{0}) × B(u_{0}, δ_{0}) → 𝕄^{d×N} such that

f(x, u, g(x, u)) ∈ L^{∞} (B(x_{0}, δ_{0}) × B(u_{0}, δ_{0}); ℝ) , (1.3)

f(x, u, ξ) ≥ C|ξ| − \frac{1}{C} (1.4)

for all (x, u) ∈ Ω × ℝ^{d} with |x − x_{0}| + |u − u_{0}| ≤ δ_{0} and for all ξ ∈ 𝕄^{d×N}. Let u ∈ BV_{loc}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W_{loc}^{1,1}(Ω; ℝ^{d}) converging to u in L_{loc}^{1}(Ω; ℝ^{d}). Then

∫_{Ω} f(x, u, ∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx.

Here ∇u is the Radon–Nikodym derivative of the distributional derivative Du of u, with respect to the N–dimensional Lebesgue measure L^{N}.

The argument of the proof of in Theorem 1.1 is based on the blow–up and truncation methods introduced by Fonseca and Müller [FM], [FM1], together with an approximation result for convex functions due to Ambrosio [A].

If in Theorem 1.1 we require only measurability in x, rather than lower semicontinuity, then the situation becomes significantly more delicate. This problem has been taken up by Acerbi, Bouchitté and Fonseca [ABF] for integrands of the form f = f(x, ξ), convex in ξ, which satisfy the growth condition

C|ξ|^{p} − \frac{1}{C} ≤ f(x, ξ) ≤ C_{2}(|ξ|^{q} + 1),

where p > max{1, q(N − 1)N} and C > 0.

Corollary 1.2. Let f : Ω × ℝ^{d} × 𝕄^{d×N} → [0, ∞] be a lower semicontinuous function, with f(x, u, ·) convex in 𝕄^{d×N}. Suppose that

f(x, u, ξ) → ∞ as |ξ| → ∞

and that f(x, u, 0) ∈ L_{loc}^{∞} (Ω × ℝ^{d}; ℝ). Let u ∈ BV_{loc}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W_{loc}^{1,1}(Ω; ℝ^{d}) converging to u in L_{loc}^{1}(Ω; ℝ^{d}). Then

∫_{Ω} f(x, u, ∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx.

Corollary 1.2 extends Theorem A(i) of Serrin to the vectorial case.

Corollary 1.3. Assume that f satisfies all the hypotheses of Theorem 1.1, maybe with exception of (1.4). Then the functional F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω; ℝ^{d}) with respect to weak convergence in BV_{loc}(Ω; ℝ^{d}).

The method used in Theorem 1.1 can also be applied to polyconvex integrands. For each matrix ξ ∈ 𝕄^{d×N} let M(ξ) ∈ ℝ^{τ} be the vector whose components are all the minors of ξ, where

τ = τ(d, N) := ∑_{k=1}^{min\{d,N\}} \binom{d}{k} \binom{N}{k} .

Theorem 1.4. Let h : Ω × ℝ^{d} × ℝ^{τ} → [0, ∞] be a lower semicontinuous function, with h(x, u, ·) convex in ℝ^{τ}. Suppose that for all (x_{0}, u_{0}) ∈ Ω × ℝ^{d} either h(x_{0}, u_{0}, v) ≡ 0 for all v ∈ ℝ^{τ}, or there exist C, δ_{0} > 0, and a continuous function g : B(x_{0}, δ_{0}) × B(u_{0}, δ_{0}) → ℝ^{τ} such that

h(x, u, g(x, u)) ∈ L^{∞} (B(x_{0}, δ_{0}) × B(u_{0}, δ_{0}); ℝ) , (1.5)

h(x, u, v) ≥ C|v| − \frac{1}{C} (1.6)

for all (x, u) ∈ Ω × ℝ^{d} with |x − x_{0}| + |u − u_{0}| ≤ δ_{0} and for all v ∈ ℝ^{τ}. Let u ∈ BV_{loc}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W_{loc}^{1,p}(Ω; ℝ^{d}) which converges to u in L_{loc}^{1}(Ω; ℝ^{d}), where p = min{d, N}. Then

∫_{Ω} h(x, u, M(∇u)) dx ≤ lim inf_{n→∞} ∫_{Ω} h(x, u_{n}, M(∇u_{n})) dx.

The conclusion of Theorem 1.4 continues to hold for integrands h : Ω × ℝ^{d} × ℝ^{τ} → [0, ∞] which can approximated from below by a monotone sequence of functions h_{j} : Ω × ℝ^{d} × ℝ^{τ} → [0, ∞] which satisfy the hypotheses of Theorem 1.4. As

(1.7)  h(x, u, v) = sup_{j} h_{j}(x, u, v)  for all (x, u, v) ∈ Ω × ℝ^{d} × ℝ^{τ},

h is clearly lower semicontinuous, convex, and coercive in v (in the weak sense (1.6) above), but it may not verify condition (1.5). As an example, consider

h(u, ξ) := A(u) h_{0}(v),  where  A(u) := \{ \begin{matrix}1 + 1/|u| & if u ≠ 0 \\ 1 & if u = 0, \end{matrix}

h_{0} ≥ 0 is convex and h_{0}(v) → ∞ as |v| → ∞. Clearly (1.5) is not satisfied for u_{0} = 0, but it is easy to approximate h, by taking A_{j}(u) = min{A(u), j} and h_{j} := A_{j}(u) h_{0}(v).

More generally (1.7) always holds if for every (x_{0}, u_{0}) ∈ Ω × ℝ^{d} and ε > 0 there exists δ > 0 such that

(1.8)  h(x, u, v) ≥ (1 − ε)h(x_{0}, u_{0}, v)

for all (x, u) ∈ Ω × ℝ with |x − x_{0}| ≤ δ, |u − u_{0}| ≤ δ and for all v ∈ ℝ^{τ} (see [DMS] for a proof). Thus we recover the lower semicontinuity results of Dal Maso and Sbordone [DMS], and of Fusco and Hutchinson [FH], where condition (1.5) is replaced by (1.8) (see also [FL] for related results).

It would be interesting to understand whether for any lower semicontinuous function h, convex and coercive in v, it is possible to find an approximating sequence h_{j} as above. This would entail that Theorem 1.4 (resp. 1.1 ) continues to hold without condition (1.5) (resp. (1.3)). Although this seems likely, we have not been able to prove it.^{1}

Corollary 1.5. Let φ : Ω×ℝ^{N} ×ℝ → [0, ∞) be a continuous function, with φ(x, u, ·) convex in ℝ. Suppose that for all (x, u) ∈ Ω × ℝ^{N}

(1.9)  φ(x, u, s) → ∞  as |s| → ∞.

Let u ∈ W^{1,N}(Ω; ℝ^{N}), and let {u_{n}} be a sequence of functions in W^{1,N}(Ω; ℝ^{N}) bounded in W^{1,N−1}(Ω; ℝ^{N}) and converging to u in L^{1}(Ω; ℝ^{N}). Then

∫_{Ω} φ(x, u, det ∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} φ(x, u_{n}, det ∇u_{n}) dx.

To the authors’ knowledge Corollary 1.5 is new in this generality. Lower semicontinuity for polyconvex and quasiconvex integrands of this type has been studied by several authors in the past years, see in particular the papers of [ADM], [B], [BM], [CDA], [CDM], [DMar], [DMS], [FH], [FMý1], [Fmar], [G], [Mý1], [Mý2], [Mý3] for the polyconvex case and of [BFMý], [FMý], [Mar1], [Mar2] for the quasiconvex case. It is rather interesting to observe that when φ depends only on the gradient variable s, rather than on the full set of variables (x, u, s), then condition (1.9) can be dropped. This was shown by Celada and Dal Maso in [CDM]. No analogous results without a coercivity condition of the type (1.6) are known for the case when d ≠ N (see Proposition 3 in the Appendix for the coercive case).

§2. Proof of Theorem 1.1.

Proof of Theorem 1.1. Without loss of generality we may assume that

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx = lim_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx < ∞.

Passing to a subsequence, if necessary, there exists a nonnegative Radon measure μ such that

f(x, u_{n}(x), ∇u_{n}(x)) L^{N}⌊ Ω ⇀^{⋆} μ

as n → ∞, weakly ⋆ in the sense of measures. We claim that

(2.1)  \frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} ≥ f(x_{0}, u(x_{0}), ∇u(x_{0}))  for L^{N} a.e. x_{0} ∈ Ω,

1Note added to proof: in a recent paper [CMý] Cerný and Malý have constructed an integrand f which satisfies all the hypotheses of Theorem 1.3 except (1.5) and such that the corresponding functional F fails to be lower semicontinuous with respect to L^{1} convergence.

where Q_{ν}(x_{0}, ε) := x_{0} +ε Q_{ν}. If (2.1) holds, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{c}(Ω; ℝ), 0 ≤ φ ≤ 1. We have

lim_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx ≥ lim inf_{n→∞} ∫_{Ω} φ f(x, u_{n}, ∇u_{n}) dx = ∫_{Ω} φ dμ

≥ ∫_{Ω} φ \frac{dμ}{dL^{N}} dx ≥ ∫_{Ω} φ f(x, u, ∇u) dx.

By letting φ → 1, and using Lebesgue Dominated Convergence Theorem, we obtain the desired result. Thus, to conclude the proof of the theorem, it suffices to show (2.1).

Take x_{0} ∈ Ω such that

(2.2) \frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} < ∞,  lim_{ε→0^{+}} \frac{1}{ε^{N+1}} ∫_{Q(x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})|dx = 0.

Choosing ε_{k} ↘ 0 such that μ(∂Q(x_{0}, ε_{k})) = 0, then

lim_{k→∞} \frac{μ(Q(x_{0}, ε_{k}))}{ε_{k}^{N}} = lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x, u_{n}, ∇u_{n}) dx

= lim_{k→∞} lim_{n→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{n,k}(y), ∇w_{n,k}(y)) dy,

where

w_{n,k}(y) := \frac{u_{n}(x_{0} + ε_{k}y) − u(x_{0})}{ε_{k}}.

Clearly w_{n,k} ∈ W^{1,1}(Q; ℝ^{d}), and by (2.2), lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{L^{1}(Q;ℝ^{d})} = 0, where w_{0}(y) := ∇u(x_{0})y. By

a standard diagonalization argument, we may extract a subsequence w_{k} := w_{n_{k},k} which converges to w_{0} in L^{1}(Q; ℝ^{d}) and such that

(2.3)  \frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy.

If f(x_{0}, u(x_{0}), ξ) ≡ 0 for all ξ then there is nothing to prove. Otherwise, let δ_{0} > 0 be given by (1.3), (1.4). From (2.3) and (1.4), and for k large

\begin{aligned}\frac{dμ}{dL^{N}} (x_{0}) + 1 & ≥ ∫_{\{|w_{k}|≤δ_{0}/ε_{k}\}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy \\ & ≥ C ∫_{\{|w_{k}|≤δ_{0}/ε_{k}\}} |∇w_{k}(y)| dy − 1/C, \end{aligned}

and so there exists a constant K > 0 such that

(2.4)  ∫_{{|w_{k}|≤δ_{0}/ε_{k}}} |∇w_{k}(y)| dy ≤ K.

By Proposition 1 and Remark 2(ii) in the Appendix, with M = (x_{0} +ε_{1}\overline{Q})×\overline{B(u(x_{0}), δ_{0}/2)} and V = 𝕄^{d×N}, there exist two sequences of continuous functions

a_{j} : M → ℝ, b_{j} : M → 𝕄^{d×N},

6 I. Fonseca and G. Leoni

such that

(2.5)  f(x, u, ξ) = sup_{j} (a_{j}(x, u) + b_{j}(x, u) · ξ)^{+}

for all (x, u) ∈ M and ξ ∈ 𝕄^{d×N}. Define

f_{j}(x, u, ξ) := (a_{j}(x, u) + b_{j}(x, u) · ξ)^{+} .

Then f_{j} is continuous, convex in ξ and

(2.6)  0 ≤ f_{j}(x, u, ξ) ≤ C_{j}(|ξ| + 1)

for all (x, u) ∈ M and ξ ∈ 𝕄^{d×N}, where

C_{j} := max{|a_{j}(x, u)| + |b_{j}(x, u)| : (x, u) ∈ M}.

For any fixed ε > 0 we may find 0 < δ_{j} ≤ δ_{0}/2 such that

|a_{j}(x, u) − a_{j}(x_{0}, u(x_{0}))| + |b_{j}(x, u) − b_{j}(x_{0}, u(x_{0}))| ≤ ε

for all (x, u) ∈ (x_{0}+δ_{j}\overline{Q})×\overline{B(u(x_{0}), δ_{j})}. Since the function g(s) := s^{+} is Lipschitz continuous with Lipschitz constant 1, we have

|f_{j}(x, u, ξ) − f_{j}(x_{0}, u(x_{0}), ξ)| ≤ |a_{j}(x, u) − a_{j}(x_{0}, u(x_{0}))| + |b_{j}(x, u) − b_{j}(x_{0}, u(x_{0}))| |ξ| ≤ ε(1 + |ξ|) for all (x, u) ∈ (x_{0} + δ_{j}\overline{Q}) × \overline{B(u(x_{0}), δ_{j})} and all ξ ∈ 𝕄^{d×N}. Hence, by (2.3), (2.4) and (2.5), for any fixed j

(2.7)

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{\{|w_{k}|≤δ_{j}/ε_{k}\}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy

≥ lim inf_{k→∞} ∫_{{|w_{k}|≤δ_{j}/ε_{k}}} f_{j}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy

≥ lim inf_{k→∞} (∫_{{|w_{k}|≤δ_{j}/ε_{k}}} f_{j}(x_{0}, u(x_{0}), ∇w_{k}(y)) dy − ε − ε ∫_{{|w_{k}|≤δ_{j}/ε_{k}}} |∇w_{k}(y)| dy)

≥ lim inf_{k→∞} ∫_{{|w_{k}|≤δ_{j}/ε_{k}}} f_{j}(x_{0}, u(x_{0}), ∇w_{k}(y)) dy − ε − ε K.

In order to truncate w_{k}, let s := ||w_{0}||_{L^{∞}(Q;ℝ^{d})} + 1 and let {m_{k}} ⊂ ℕ be an increasing sequence of positive integers such that 2^{m_{k}} ≤ δ_{j}/ε_{k}, with m_{k} → ∞. Choose an interval

(s_{k}, L_{k}) ∈ {(s, 2s), …, (2^{m_{k}−1}s, 2^{m_{k}}s)}

such that

(2.8)  ∫_{\{s_{k}≤|w_{k}|≤L_{k}\}} |∇w_{k}(y)| dy ≤ \frac{K}{m_{k}} .

Construct a smooth cut–off function g_{k} : ℝ^{d} → ℝ^{d} such that

(2.9)  g_{k}(u) = \{ \begin{matrix}u & if |u| ≤ s_{k}, \\ 0 & if |u| ≥ L_{k}, \end{matrix}

with |g_{k}(u)| ≤ |u| and |∇g_{k}(u)| ≤ C L_{k}/(L_{k}−s_{k}) for all u ∈ ℝ^{d}. Note that by construction L_{k}/(L_{k}−s_{k}) = 2. Define v_{k}(y) := g_{k}(w_{k}(y)), and

E_{k} := {y ∈ Q : |w_{k}(y)| < s_{k}}, E_{k}^{+} := {y ∈ Q : |w_{k}(y)| > L_{k}}, E_{k}^{−} := {y ∈ Q : s_{k} ≤ |w_{k}(y)| ≤ L_{k}}.

Then

∫_{Q} |v_{k}(y) − w_{0}(y)|dy = ∫_{E_{k}} |w_{k}(y) − w_{0}(y)|dy + ∫_{E_{k}^{+}} |w_{0}(y)|dy + ∫_{E_{k}^{−}} |g_{k}(w_{k}(y)) − w_{0}(y)|dy
≤ ||w_{k} − w_{0}||_{L^{1}(Q;ℝ^{d})} + ||w_{0}||_{L^{∞}(Q;ℝ^{d})}L^{N}(E_{k}^{+} ∪ E_{k}^{−}) + ∫_{E_{k}^{−}} |w_{k}(y)|dy
≤ 2||w_{k} − w_{0}||_{L^{1}(Q;ℝ^{d})} + 2||w_{0}||_{L^{∞}(Q;ℝ^{d})}L^{N}(E_{k}^{+} ∪ E_{k}^{−}) → 0 as k → ∞,

because

(2.10)  \begin{aligned}0 ≤ L^{N}(E_{k}^{+} ∪ E_{k}^{−}) & = L^{N}(\{y ∈ Q : |w_{k}(y)| ≥ s_{k}\}) \\ & ≤ L^{N}(\{y ∈ Q : |w_{k}(y) − w_{0}(y)| ≥ 1\}) ≤ ||w_{k} − w_{0}||_{L^{1}(Q;ℝ^{d})}. \end{aligned}

Moreover

(2.11)  \begin{matrix}∫_{Q} f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy = & ∫_{E_{k}} f_{j}(x_{0}, u(x_{0}), ∇w_{k}(y)) dy \\ + & ∫_{E_{k}^{+}} f_{j}(x_{0}, u(x_{0}), 0) dy + ∫_{E_{k}^{−}} f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy. \end{matrix}

We claim that the last two integral are infinitesimal as k → ∞. Indeed, by (2.6) and (2.10)

(2.12)  0 ≤ ∫_{E_{k}^{+}} f_{j}(x_{0}, u(x_{0}), 0) dy ≤ C_{j}L^{N}(E_{k}^{+}) → 0,

while, from (2.6) and (2.8)

(2.13)  _{∫E_{k}^{−}}f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy ≤ C_{j} ∫_{E_{k}^{−}}(1 + |∇g_{k}(w_{k})∇w_{k}| dy

≤ C_{j} (L^{N}(E_{k}^{−}) + ||∇g_{k}||_{∞} ∫_{E_{k}^{−}} |∇w_{k}| dy) ≤ C_{j} (L^{N}(E_{k}^{−}) + 2C \frac{K}{m_{k}} ) → 0 (2.14)

as k → ∞. Therefore from (2.7), (2.11), (2.12), (2.13) and (2.14)

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{Q} f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy − ε − ε K,

where we used the fact that E_{k} ⊂ {y ∈ Q : |w_{k}| ≤ δ_{j}/ε_{k}} by choice of m_{k}. By a result of Serrin [S,Theorem 2] we obtain

\frac{dμ}{dL^{N}} (x_{0}) ≥ f_{j}(x_{0}, u(x_{0}), ∇u(x_{0})) − ε − ε K.

By taking the sup in j and using (2.5) we have

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ∇u(x_{0})) − ε − ε K.

Now we let ε → 0^{+}.

□

Proof of Corollary 1.2. We only need to show that condition (1.4) in Theorem 1.1 is satisfied. The argument follows closely that of Morrey [M] in Lemma 4.1.3.

Suppose that (1.4) is false. Then we may find (x_{0}, u_{0}) ∈ Ω × ℝ^{d} and a sequence {(x_{n}, u_{n}, ξ_{n})} ⊂ Ω × ℝ^{d} × 𝕄^{d×N}, with (x_{n}, u_{n}) → (x_{0}, u_{0}) and |ξ_{n}| → ∞ as n → ∞, such that

(2.15)  f(x_{n}, u_{n}, ξ_{n}) ≤ \frac{1}{n} |ξ_{n}| − 1.

By hypothesis f(x, u, 0) ∈ L_{loc}^{∞} (Ω × ℝ^{d}; ℝ), hence there exists M > 0 such that

(2.16)  f(x_{n}, u_{n}, 0) ≤ M for all n.

Define v_{n} := ξ_{n}/|ξ_{n}|. By extracting a subsequence, if necessary, we may assume that v_{n} → v_{0} ∈ 𝕄^{d×N}\{0} as n → ∞. Since f(x, u, ·) is convex, for any R > 0 we have

\begin{aligned}f (x_{n}, u_{n}, R v_{n}) & = f (x_{n}, u_{n}, \frac{R}{|ξ_{n}|} ξ_{n}) \\ & ≤ (1 − \frac{R}{|ξ_{n}|}) f(x_{n}, u_{n}, 0) + \frac{R}{|ξ_{n}|}f(x_{n}, u_{n}, ξ_{n}) ≤ (1 − \frac{R}{|ξ_{n}|}) M + \frac{R}{|ξ_{n}|} (\frac{1}{n} |ξ_{n}| − 1) , \end{aligned}

where we have used (2.15) and (2.16). If we now let n → ∞ in the previous inequality, and use the facts that |ξ_{n}| → ∞, (x_{n}, u_{n}, R v_{n}) → (x_{0}, u_{0}, Rv_{0}), together with the lower semicontinuity of the function f, we get

f (x_{0}, u_{0}, R v_{0}) ≤ M,

which is a contradiction since f (x_{0}, u_{0}, R v_{0}) → ∞ as R → ∞.

□

Proof of Corollary 1.3. Let {u_{n}} ⊂ W_{loc}^{1,1}(Ω; ℝ) be a sequence of functions converging weakly in BV_{loc}(Ω; ℝ) to u. Then by (1.1)

sup_{n} ∫_{Ω} |∇u_{n}(x)| dx ≤ K < ∞,

for some constant K > 0 independent of n. Fix ε > 0. The integrand

f_{ε}(x, u, ξ) := f(x, u, ξ) + ε |ξ|

satisfies all the hypotheses of Theorem 1. Therefore

∫_{Ω} f(x, u, ∇u) dx ≤ ∫_{Ω} f_{ε}(x, u, ∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} f_{ε}(x, u_{n}, ∇u_{n}) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx + ε K.

It suffices to let ε → 0^{+}. □

§3. Proof of Theorem 1.4.

Proof of Theorem 1.4. The proof of Theorem 1.4 is similar to the previous one, and we will only indicate the main differences. Let f(x, u, ξ) := h(x, u, M(ξ)). We proceed as in the proof of Theorem 1.1 up to (2.3), where now w_{k} ∈ W^{1,p}(Q; ℝ^{d}). Inequality (2.4) should be replaced by

(3.1)  ∫_{{|w_{k}|≤δ_{0}/ε_{k}}} |M(∇w_{k}(y))| dy ≤ K.

By Proposition 1 and Remark 2(ii) in the Appendix, with M = (x_{0} + ε_{1}\overline{Q}) × \overline{B(u(x_{0}), δ_{0}/2)} and V = ℝ^{τ}, there exist two sequences of continuous functions

a_{j} : M → ℝ, b_{j} : M → ℝ^{τ} ,

such that

h(x, u, v) = sup_{j} (a_{j}(x, u) + b_{j}(x, u) · v)^{+}

for all (x, u) ∈ M and v ∈ ℝ^{τ}. Define

h_{j}(x, u, v) := (a_{j}(x, u) + b_{j}(x, u) · v)^{+} , and f_{j}(x, u, ξ) := h_{j}(x, u, M(ξ)).

Then h_{j} is continuous, convex in v and

(3.2)  0 ≤ h_{j}(x, u, v) ≤ C_{j}(|v| + 1),

for all (x, u) ∈ M and v ∈ ℝ^{τ}, where

C_{j} := max{|a_{j}(x, u)| + |b_{j}(x, u)| : (x, u) ∈ M}.

We may now proceed as in Theorem 1.1, replacing (2.8) with

(3.3)  ∫_{\{s_{k}≤|w_{k}|≤L_{k}\}} |M(∇w_{k}(y))| dy ≤ \frac{K}{m_{k}} ,

up to (2.13) which becomes

(3.4)  ∫_{E_{k}^{−}} f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy ≤ C_{j} ∫_{E_{k}^{−}} (1 + |M(∇g_{k}(w_{k})∇w_{k})|) dy.

Since ||∇g_{k}||_{∞} ≤ C for some constant C, and in view of the fact that for x ∈ E_{k}^{−}

|M(∇g_{k}(w_{k})∇w_{k})| ≤ |M(∇g_{k}(w_{k})| |M(∇w_{k})| ,

it follows that for x ∈ E_{k}^{−}

(3.5)  |M(∇g_{k}(w_{k})∇w_{k})| ≤ C|M(∇w_{k})|.

Then, by (3.3) and (3.5),

(3.6)  C_{j} ∫_{E_{k}^{−}} (1 + |M(∇g_{k}(w_{k})∇w_{k})|) dy ≤ C_{j} (L^{N}(E_{k}^{−}) + \frac{CK}{m_{k}} ) → 0.

Consequently, in view of (2.7), (2.11), (2.12), (3.4) and (3.6)

(3.7)  \frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{Q} f_{j}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy − ε − ε K.

At this stage we are not yet in position to apply Proposition 2 in the Appendix due to the lack of coercivity of h_{j}. By (3.1), (3.6) and since v_{k} = 0 on E_{k}^{+}, we have

(3.8)  sup_{k} ∫_{Q} |M(∇v_{k}(y))| dy ≤ K_{1} < ∞,

where K_{1} is independent of δ and j. Define

g_{j,ε}(v) := h_{j}(x_{0}, u(x_{0}), v) + ε |v|.

By (3.7) and (3.8)

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{Q} g_{j,ε}(M(∇v_{k}(y))) dy − ε − ε(K + K_{1}).

By Proposition 2 in the Appendix we obtain

\frac{dμ}{dL^{N}} (x_{0}) ≥ g_{j,ε}(M(∇u(x_{0}))) − ε − ε(K + K_{1}) = f_{j}(x_{0}, u(x_{0}), ∇u(x_{0})) + ε |M(∇u(x_{0}))| − ε − ε(K + K_{1}).

By taking the sup in j we have

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ∇u(x_{0})) + ε |M(∇u(x_{0}))| − ε − ε(K + K_{1}).

Now we let ε → 0^{+}. This concludes the proof of the theorem □

Remark 3.1. The truncation arguments in the proofs of Theorem 1.1 and 1.4 simplify respectively the ones of Fonseca and Müller [FM] and of Fusco and Hutchinson [FH]. Moreover they can be used when the target function w_{0} is any L^{∞} function, not necessarily smooth. See [FL] for more details.

Proof of Corollary 1.5. The proof is similar to that of Corollary 1.3. In this case we consider the integrand

f_{ε}(x, u, ξ) := h(x, u, M(ξ)) + ε |ξ|^{N−1}, ε > 0.

We omit the details. □

§4. Appendix.

Proposition 1. Let M be a closed set of ℝ^{p}, let V be a reflexive and separable Banach space. Let f : M × V → (−∞, +∞] be a M × w-V sequentially lower semicontinuous function, convex in the last variable and such that there exists a continuous function v_{0} : M → V with

(4.1)  (f(·, v_{0}(·)))^{+} ∈ L^{∞}(M; ℝ).

Then there exist two sequences of continuous functions

a_{j} : M → ℝ, b_{j} : M → V^{∗},

where V^{∗} is the dual space of V, such that

f(t, v) = sup_{j} (a_{j}(t)+ < b_{j}(t), v >)

for all t ∈ M and v ∈ V.

Moreover if f is bounded from below, then (4.1) can be weakened to

(4.2)  (f(·, v_{0}(·)))^{+} ∈ L_{loc}^{∞}(M; ℝ).

Remark 2. (i) Proposition 1 was proved by Ambrosio in [A, Lemma 2.5], in the case where (4.1) is replaced by the assumption that f(·, v_{0}(·)) is continuous.

(ii) If f is non negative then, without loss of generality, we may assume that

f(t, v) = sup_{j} (a_{j}(t)+ < b_{j}(t), v >)^{+} .

Proof. Define

(4.3)  g(t, v) := f(t, v + v_{0}(t)) − C_{0},  where C_{0} := ||(f(·, v_{0}(·)))^{+}||_{L^{∞}(M;ℝ)}.

Clearly g satisfies the same hypotheses of f and

(4.4)  −∞ < g(t, 0) = f(t, v_{0}(t)) − C_{0} ≤ 0.

Following Ambrosio [A], we consider the multifunction Γ : M → {C ⊂ ℝ × V^{∗} : C ≠ ∅, convex, closed} defined by

Γ(t) := {(α, β) ∈ ℝ × V^{∗} : g(t, v) ≥ α+ < β, v > for all v ∈ V } .

It is easy to see by the hypotheses on g that Γ(t) is well defined. We only note that (4.4)_{1} and the convexity of g(t, ·) imply that Γ(t) ≠ ∅. We claim that Γ is lower semicontinuous. Fix an open set W ⊂ ℝ × V^{∗}. We need to show that

Γ^{−}(W) := {t ∈ M : Γ(t) ∩ W ≠ ∅}

is open in M. Assume that Γ^{−}(W) ≠ ∅ and fix t_{0} ∈ Γ^{−}(W), (α_{0}, β_{0}) ∈ W and ε > 0 such that

(4.5)  \begin{matrix}g(t_{0}, v) ≥ α_{0}+ < β_{0}, v > for all v ∈ V, \\ \{(α, β) ∈ ℝ × V^{∗} : |α − α_{0}| + ||β − β_{0}|| < 4ε\} ⊂ W. \end{matrix}

We claim that there exists an open ball B in M, containing t_{0}, such that

(4.6)  g(t, v) ≥ α_{0}+ < β_{0}, v > −ε(1 + ||v||) for all t ∈ B and v ∈ V .

Assume that (4.6) is false. Then there exist two sequences {t_{n}} ⊂ ℝ converging to t_{0} in M and {v_{n}} ⊂ V such that

(4.7)  g(t_{n}, v_{n}) < α_{0}+ < β_{0}, v_{n} > −ε(1 + ||v_{n}||) for all n ∈ ℕ.

If sup_{n} ||v_{n}|| < ∞ then we can assume that v_{n} converges weakly to some v_{0} ∈ V and, due to the lower semicontinuity of g, letting n → ∞ in (4.7) we get

g(t_{0}, v_{0}) ≤ α_{0}+ < β_{0}, v_{0} > −ε(1 + ||v_{0}||)

which contradicts (4.5). Therefore sup_{n} ||v_{n}|| = ∞, and without loss of generality we can assume that ||v_{n}|| → ∞, with p_{n} := v_{n}/||v_{n}|| converging weakly to some p_{0} ∈ V as n → ∞. Fix γ > 0 and let n be so large that ||v_{n}|| γ > 1; by the convexity of g(t, ·) and (4.7)

\begin{aligned}g(t_{n}, p_{n}/γ) − α_{0}− < β_{0}, p_{n}/γ > & ≤ \frac{1}{γ ||v_{n}||} [g(t_{n}, v_{n}) − α_{0}− < β_{0}, v_{n} >] + (1 − \frac{1}{γ ||v_{n}||}) (g(t_{n}, 0) − α_{0}) \\ & ≤ −\frac{ε(1 + ||v_{n}||)}{γ ||v_{n}||} − α_{0} (1 − \frac{1}{γ ||v_{n}||}) , \end{aligned}

where we have used the fact that g(t_{n}, 0) ≤ 0 by (4.4)_{2}. Letting n → ∞ and using the lower semicontinuity of g and (4.5) gives

0 ≤ g(t_{0}, p_{0}/γ) − α_{0}− < β_{0}, p_{0}/γ >≤ −\frac{ε}{γ} − α_{0} < 0,

provided γ is taken sufficiently small. We arrive at a contradiction, and this proves (4.6). We can now proceed as in [A].

We claim that B ⊂ Γ^{−}(W). Fix b ∈ B. By the Hahn–Banach Separation Theorem there exist α ∈ ℝ and β ∈ V^{∗} such that

g(b, v) − α_{0}− < β_{0}, v >≥ α+ < β, v >≥ −2ε − ε||v|| for all v ∈ V .

Consequently α ≥ −2ε and ||β|| ≤ ε. Let ω := inf{α, 2ε}. Then

g(b, v) ≥ α_{0} + ω+ < β_{0} + β, v > for all v ∈ V

and thus, by (4.5) it follows that b ∈ Γ^{−}(W) and the claim is proved. Since Γ is lower semicontinuous by the Continuous Selection Theorem (see [A, Lemma 1.1]) for every (t, v) ∈ M × V we can write

g(t, v) = sup {σ_{1}(t)+ < σ_{2}(t), v >: (σ_{1}, σ_{2}) is a continuous selection for Γ}

and by Lindelöf Theorem (see e.g. Lemma 9.2 in [FL]) we get the desired result. This completes the proof of the proposition in the case where (4.1) holds.

When f is bounded from below and (4.2) is satisfied, without loss of generality we can assume that f is nonnegative. Let ψ_{n} ∈ C_{0}^{∞}(ℝ^{p}; ℝ), 0 ≤ ψ_{n}(t) ≤ 1 be a cut–off function such that ψ_{n}(t) ≡ 1 for |t| ≤ n and ψ_{n}(t) ≡ 0 for |t| ≥ n + 1. Define

f_{n}(t, v) := ψ_{n}(t) f(t, v).

Then f_{n} satisfies the same hypotheses of f and also (4.1). Thus we can apply the first part of the Proposition to f_{n}. To conclude the proof it suffices to observe that, since f ≥ 0,

f(t, v) = sup_{n} f_{n}(t, v).

□

Proposition 2. Let h : ℝ^{τ} → [0, ∞) be a convex function such that

h(v) → ∞ as |v| → ∞.

Let u ∈ W^{1,p}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W^{1,p}(Ω; ℝ^{d}) which converges to u in L^{1}(Ω; ℝ^{d}), where p = min{d, N}. Then

∫_{Ω} h(M(∇u)) dx ≤ lim inf_{n→∞} ∫_{Ω} h(M(∇u_{n})) dx.

Proposition 2 has been proved by Dal Maso and Sbordone (cf. Theorem 2.2 in [DMS]) using cartesian currents and by Fusco and Hutchinson (cf. Theorem 2.6 in [FH]).

Acknowledgments. The research of I. Fonseca was partially supported by the National Science Foundation through the Center for Nonlinear Analysis under Grants No. DMS–9500531 and DMS–9731957, and that of G. Leoni by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through GNAFA and the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”. G. Leoni would like to thank the hospitality of the Center for Nonlinear Analysis, where part of this work was carried out during his visit in the spring of 1999.

The authors would like to thank the referee for some observations which led to significant simplifications of the proofs.

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