ON LOWER SEMICONTINUITY

AND RELAXATION

Irene Fonseca † & Giovanni Leoni ‡

Abstract. Lower semicontinuity and relaxation results in BV are obtained for multiple integrals

F (u, Ω) := ∫_{Ω} f(x, u(x), ∇u(x)) dx, u ∈ W^{1,1}(Ω; ℝ^{d}),

where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the Calculus of Variations.

§1. Introduction.

In this paper we address lower semicontinuity and relaxation properties for multiple functionals of the form

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

where Ω is an open subset of ℝ^{N}, and u(x) is a ℝ^{d}–valued function defined on Ω.

In [30] Serrin considered the scalar–valued case where

f ∈ C(Ω × ℝ × ℝ^{N}; [0, ∞)) and f(x, u, ·) is convex in ℝ^{N}.  (1.1)

Among his results we select Theorems A and B below.

Theorem A. (cf. [30, Theorem 11]) Assume that f satisfies (1.1). Let u ∈ BV_{loc}(Ω; ℝ), and let {u_{n}} be a sequence of functions in W_{loc}^{1,1}(Ω; ℝ) converging to u in L_{loc}^{1}(Ω; ℝ). Let λ, ρ be moduli of continuity such that

(i) ρ(s) ≤ C s for C > 0 and all s > 0 large, and

|f(x, u, ξ) − f(x_{0}, u_{0}, ξ)| ≤ λ(|x − x_{0}|)(1 + f(x, u, ξ)) + ρ(|u − u_{0}|)

for all (x, u), (x_{0}, u_{0}) ∈ Ω × ℝ, and for all ξ ∈ ℝ^{N}; or (ii) |f(x, u, ξ) − f(x_{0}, u_{0}, ξ)| ≤ λ(|x − x_{0}| + |u − u_{0}|)(1 + f(x, u, ξ)) for all (x, u), (x_{0}, u_{0}) ∈ Ω × ℝ, and for

all ξ ∈ ℝ^{N}. Suppose, in addition, that u(x) is continuous. Then

∫_{Ω} f(x, u(x), ∇u(x)) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) 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}. Also, we intend by modulus of continuity a nonnegative, increasing, continuous function ρ such that ρ(0) = 0.

1991 Mathematics Subject Classification. 49J45, 49Q20.

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

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

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

Theorem B. (cf. [30, Theorem 12]) Assume that f satisfies (1.1) and 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 F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω; ℝ) with respect to local convergence in L^{1}.

The prototype of integrands that we want to study is represented by f = h(x, u) |ξ|, where h ≥ 0, for which conditions (i)–(iii) of Theorem B may be violated; hence, in this paper we will focus our attention mainly on Theorem A. Note also that while conditions (i) and (ii) of Theorem A are trivially satisfied when f = f(ξ), so that L_{loc}^{1} lower semicontinuity holds in this case only under assumption (1.1), Theorem B is more stringent as it imposes extra conditions on the dependence of f on the gradient variable ξ.

It is worth noticing that Theorem A requires no coercivity hypothesis, i.e., a condition of the type

f(x, u, ξ) ≥ C|ξ| − \frac{1}{C}

for some constant C > 0. One of the main purposes of this paper is to try to understand the deep relation between coercivity (or the lack of it) and lower semicontinuity. A drawback in Theorem A(ii) is that, in practice, one seldom knows whether the target function u(x) is continuous or not. Important examples of integrands which satisfy (i) and (ii) of Theorem A are given by

f = f(x, ξ) = h(x)g(ξ), f = f(x, u, ξ) = h(x, u)g(ξ),

where h(x) and h(x, u) are uniformly continuous functions bounded away from zero and g is a nonnegative convex function. Conditions (i) and (ii) of Theorem A appear often in the study of lower semicontinuity and relaxation and were exploited by several authors. Dal Maso [9] obtained an integral representation formula for the relaxation F(u, Ω) with respect to the L_{loc}^{1} topology of the functional F, namely

F(u, Ω) := inf_{{u_{n}}} {lim inf_{n→∞} F (u_{n}, Ω) : u_{n} ∈ W_{loc}^{1,1}(Ω; ℝ), u_{n} → u in L_{loc}^{1}(Ω; ℝ)} ,

under a weak form of condition (ii) in Theorem A and assuming coercivity. Let

H(u, Ω) := ∫_{Ω} f(x, u, ∇u) dx + ∫_{Ω} f^{∞}(x, u, dC(u)) + ∫_{S(u)∩Ω} (∫_{u^{−}(x)}^{u^{+}(x)} f^{∞}(x, s, ν_{u}) ds) dH^{N−1}(x), (1.2)

where f^{∞} is the recession function of f, that is f^{∞}(x, u, ξ) := lim sup_{t→∞} \frac{f(x, u, t ξ)}{t}, C(u) is the Cantor part

of Du, and (u^{+} − u^{−}) is the jump of u across the interface S(u).

As a corollary of Dal Maso’s general results^{1}, we have the following theorem

Theorem C. (cf. [9, Theorem 3.2]) Assume that Ω is bounded and that f is a Borel function which satisfies (1.1) for H^{N} a.e (x, u) ∈ Ω × ℝ. Suppose also that for every r > 0 there exist C > 0 and three functions c, a and A ∈ C(Ω; [0, ∞)) ∩ L^{1}(Ω; [0, ∞)), with c(x) > 0 in Ω, such that

c(x)|ξ| − a(x) ≤ f(x, u, ξ) ≤ C|ξ| + A(x)  (1.3)

for all (x, u, ξ) ∈ Ω × ℝ × ℝ^{N} with |u| ≤ r. Finally, assume that for H^{N} a.e (x_{0}, u_{0}) ∈ Ω × ℝ and for every ε > 0 there exists δ > 0 such that

|f(x, u, ξ) − f(x_{0}, u_{0}, ξ)| ≤ ε(1 + |ξ|)  (1.4)

1Dal Maso’s results are given in terms of Γ–convergence of a family of functionals F_{n}(u, Ω) := ∫_{Ω} f_{n}(x, u(x), ∇u(x)) dx.

We take here f_{h} ≡ f and refer to [9] for the more general statement of Theorems 3.2 and 3.5.

for all (x, u) ∈ Ω × ℝ with |x − x_{0}| + |u − u_{0}| ≤ δ and for all ξ ∈ ℝ^{N}. Then for all u ∈ BV (Ω; ℝ) ∩ L^{∞}(Ω; ℝ) we have

F(u, Ω) = H(u, Ω).  (1.5)

Furthermore, if there exist b ∈ L^{1}(Ω; [0, ∞)), C ≥ 0, α ≥ 1, such that f(x, u, 0) ≤ C|u|^{α} + b(x) for all (x, u) ∈ Ω × ℝ, then (1.5) holds for all u ∈ BV (Ω; ℝ) ∩ L^{α}(Ω; ℝ).

Note that (1.5) implies, in particular, that F is L^{1}-lower semicontinuous in W^{1,1}. Indeed, as f^{∞} ≥ 0 it follows that ∫_{Ω} f(x, u(x), ∇u(x)) dx ≤ F(u, Ω) for u ∈ BV (Ω; ℝ). Therefore, Dal Maso’s result extends Theorem A(ii) of Serrin to target functions u of bounded variation which are not necessarily continuous. However, the “price” to pay for this extension is the coercivity and growth assumption (1.3). Dal Maso established also an integral representation result for functionals which are not necessarily coercive.

Theorem D. (cf. [9, Theorem 3.5]) Assume that Ω is bounded, that f is a Borel function, and that f(x, u, ·) is positively homogeneous of degree one and convex. Suppose also that there exist λ : Ω × ℝ → [0, ∞), with λ(·, u) continuous and λ(0, u) = 0 for L^{1} a.e. u ∈ ℝ, and a function P ∈ C(Ω; [0, ∞)) such that

|f(x, u, ξ) − f(x_{0}, u, ξ)| ≤ λ(|x − x_{0}|, u)(1 + f(x, u, ξ)),  0 ≤ f(x, u, ξ) ≤ P (u)|ξ|,  (1.6)

for all x, x_{0} ∈ Ω, u ∈ ℝ and ξ ∈ ℝ^{N}. Then (1.5) holds for all u ∈ BV (Ω; ℝ).

The main difference between hypotheses (1.4) and (1.6)_{1} is that (1.4) is a local hypothesis in (x, u), while (1.6)_{1} may be interpreted as a global restriction in u. The conditions of Theorem D are satisfied by

f(x, u, ξ) = h(x)B(u)|ξ|

where h is a positive, bounded, uniformly continuous function and B is a nonnegative continuous function. Note also that condition (1.6)_{1} is trivially satisfied when f does not depend on x, that is f = f(u, ξ). Lower semicontinuity for these integrands of the form f = f(u, ξ) was later studied by De Giorgi, Buttazzo and Dal Maso [13], who proved the following result:

Theorem E. (cf. [13, Theorem 1]) Assume that f = f(u, ξ) is nonnegative, measurable in the variable u, and convex in ξ. Suppose also that f(u, 0) is lower semicontinuous and that

lim sup_{ξ→0} \frac{(f(u, 0) − f(u, ξ))^{+}}{|ξ|} ∈ L_{loc}^{1}(ℝ; ℝ).

Then for every u ∈ W_{loc}^{1,1}(Ω; ℝ) the function f(u(x), ∇u(x)) is measurable and the functional F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω; ℝ) with respect to local convergence in L^{1}.

Theorem E was extended by De Cicco in [12] to functions of bounded variation. More precisely, De Cicco showed that when f = f(u, ξ) satisfies the hypotheses of Theorem E then the functional H(u) defined in (1.2) is lower semicontinuous in BV_{loc}(Ω; ℝ) with respect to local convergence in L^{1}. The hypotheses of Theorem E were significantly weakened by Ambrosio in [3] (see also [11]), where the sequence {u_{n}} is assumed to be bounded in W^{1,1}(Ω; ℝ). This condition is somewhat related to coercivity, and we will not dwell more on it here.

Unlike the case where f = f(u, ξ), without some kind of coercivity one cannot expect in general lower semicontinuity in the L^{1} topology for functionals of the form

∫_{Ω} f(x, ∇u(x)) dx.

Indeed, in [9] Dal Maso, following a counterexample of Aronszajn, constructed a continuous function ω : Ω → ℝ, where Ω = (0, 1)×(0, 1) and x = (x_{1}, x_{2}), and a sequence of functions {u_{n}} converging to u(x) = x_{2} in L^{∞}(Ω; ℝ), such that

∫_{Ω} |(sin ω(x), cos ω(x)) · ∇u(x))| dx > lim inf_{n→∞} ∫_{Ω} |(sin ω(x), cos ω(x)) · ∇u_{n}(x))| dx.

Since the target function u(x_{1}, x_{2}) = x_{2} is continuous, f(x, ξ) = |(sin ω(x), cos ω(x))·ξ| cannot satisfy either condition (i) or (ii) of Theorem A, for in this case we would obtain a contradiction. This example suggests that, when there is no coercivity, lower semicontinuity in the L^{1} topology may fail unless we strengthen (1.1) with an uniform continuity condition.

We are now ready to present the main result of the paper.

Theorem 1.1. Assume that f : Ω × ℝ × ℝ^{N} → [0, ∞) is a Borel integrand, f(x, u, ·) is convex in ℝ^{N}, and for all (x_{0}, u_{0}) ∈ Ω × ℝ and ε > 0 there exists δ > 0 such that

f(x_{0}, u_{0}, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))  (1.7)

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

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

The main tool in the proof of Theorem 1.1 is the blow–up method introduced by Fonseca and Müller [17], [18], where we reduce the domain Ω to a ball and the target function u becomes a piecewise affine function. Since affine functions are locally bounded, in the scalar case we may replace the truncation used in [17], [18], in a vectorial setting, and which required a degenerate coercivity condition, by a considerably simpler argument.

Theorem 1.1 improves Serrin’s Theorem A, not only because continuity of the target function u is assumed in Theorem A(ii), and is not needed here, but also because condition (1.7) is significantly weaker than (ii), as the following result illustrates:

Corollary 1.2. Let g : ℝ^{N} → [0, ∞) be a convex function, and let h : Ω × ℝ → [0, ∞) be a lower semicontinuous function. If u ∈ BV_{loc}(Ω; ℝ) and {u_{n}} ⊂ W_{loc}^{1,1}(Ω; ℝ) converges to u in L_{loc}^{1}(Ω; ℝ), then

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

This result seems to be new in this generality. Note that conditions (1.3) and (1.4) in Theorem C of Dal Maso imply the validity of (1.7), while f(x, u, ξ) = h(x, u)g(ξ) as in Corollary 1.2 satisfies (1.7), but not, in general, (1.4), (1.6)_{1}, and (i), (ii) of Theorem A.

Conditions of the type (1.7) appeared already in the papers of Fonseca and Müller [17], [18], Dal Maso and Sbordone [10], Fusco and Hutchinson [20]. All these results deal with the vectorial case and require some type of coercivity conditions.

In the special case where

h = h(x) := χ_{A}(x) for some measurable set A ⊂ Ω,

then h(x)g(ξ) satisfies (1.7) if and only if L^{N}(∂A) = 0 (i.e. if χ_{A}(x) has a lower semicontinuous representative), and thus we recover the condition obtained by Gangbo [21]. Corollary 1.2 attests to the sharpness of condition (1.7). Indeed, when N = 1 and Ω is bounded, Fusco [19] proved that the functional

F (u) := ∫_{Ω} h(x)|u′(x)| dx, u ∈ W^{1,1}(Ω; ℝ),

where h(x) is a bounded, nonnegative measurable function, is lower semicontinuous in L^{1}(Ω; ℝ) if and only if h(x) is lower semicontinuous.

Theorem 1.3. Assume that Ω is bounded, f : Ω×ℝ×ℝ^{N} → [0, ∞) is a Borel integrand, f(x, u, ·) is convex in ℝ^{N}, and there exists a constant C > 0 such that

0 ≤ f(x, u, ξ) ≤ C(1 + |ξ|)  for all (x, u, ξ) ∈ Ω × ℝ × ℝ^{N}.  (1.8)

Then F(u, ·) is the trace of a finite, Radon measure on the open subsets of Ω, and

(i) if f is Carathéodory or f(·, ·, ξ) is upper semicontinuous then F(u, Ω\(S(u)∪M(u))) ≤ ∫_{Ω} f(x, u, ∇u) dx;

(ii) if f^{∞}(·, ·, ξ) is upper semicontinuous then F(u, M(u)) ≤ ∫_{Ω} f^{∞}(x, u, dC(u));

(iii) if f^{∞}(·, u, ξ) is upper semicontinuous then F(u, S(u)) ≤ ∫_{S(u)∩Ω} (∫_{u^{−}(x)}^{u^{+}(x)} f^{∞}(x, s, ν_{u}) ds) dH^{N−1}(x).

Here, and in what follows, M(u) is the support of the Cantor part of Du. Theorem 1.3 is based on a recent work by Bouchitté, Fonseca and Mascarenhas [7]. We have thus obtained the following relaxation result:

Corollary 1.4. Under the hypotheses of Theorems 1.1 and 1.3(i), (ii), (iii), we have F(u, Ω) = H(u, Ω) for all u ∈ BV (Ω; ℝ).

If we require (1.8) to be satisfied locally u in compact sets of ℝ, i.e., for every r > 0 there exist C > 0 and A ∈ L^{1}(Ω; [0, ∞)), such that

0 ≤ f(x, u, ξ) ≤ C|ξ| + A(x)

for all (x, u, ξ) ∈ Ω × ℝ × ℝ^{N} with |u| ≤ r, then it can be shown that Corollary 1.4 continues to hold for all u ∈ BV (Ω; ℝ) ∩ L^{∞}(Ω; ℝ). Thus, when (1.3) and (1.4) are satisfied for all (x_{0}, u_{0}) ∈ Ω × ℝ (see also Section 8), Corollary 1.4 improves Theorem C since conditions (1.3) and (1.4) imply condition (1.7), and Corollary 1.4 does not require any coercivity properties.

Next we extend Theorem E to integrands f = f(x, u, ξ) which depend on x. As we pointed out before, there are already several results in this direction, e.g. due to Ambrosio [3] and later extended by De Cicco [11] to BV functions, and where local convergence in L^{1} is replaced by weak convergence in BV.

Theorem 1.5. Assume that f : Ω × ℝ × ℝ^{N} → [0, ∞) is a Borel integrand, f(x, u, ·) is convex in ℝ^{N}, and for all x_{0} ∈ Ω and ε > 0 there exists δ > 0 such that

|f(x_{0}, u, ξ) − f(x, u, ξ)| ≤ ε(1 + f(x, u, ξ))  (1.9)

for all x ∈ Ω with |x−x_{0}| ≤ δ and for all (u, ξ) ∈ ℝ×ℝ^{N}. Suppose also that f(x_{0}, ·, 0) is lower semicontinuous and

lim sup_{|ξ|→0} \frac{(f(x_{0}, u, 0) − f(x_{0}, u, ξ))^{+}}{|ξ|} ∈ L_{loc}^{1}(ℝ; ℝ).

Then H(u, Ω) ≤ F(u, Ω) for all u ∈ BV (Ω; ℝ).

Note that in Theorem 1.5 we do not require f(x, u, ·) to be positively homogeneous of degree one as in Theorem D of Dal Maso. The proof of Theorem 1.5 relies on the blow-up method of Fonseca and Müller [17], and on the original proof of De Giorgi, Buttazzo and Dal Maso [13].

Theorem 1.6. Assume that Ω is bounded and that f : Ω × ℝ × ℝ^{N} → [0, ∞) is a Borel integrand which satisfies (1.8), with f(x, u, ·) convex in ℝ^{N}, and f(·, u, ξ) continuous in Ω. Then F(u, Ω) ≤ H(u, Ω).

We now turn our attention to the vectorial case, and consider nonnegative integrands

f : Ω × ℝ^{d} × ℝ^{dN} → [0, ∞), where d > 1.

The situation is considerably more complicated, even when f(x, u, ·) is assumed to be convex, rather than quasiconvex, which is the natural assumption when d > 1 (see [6], [8], [26]). In his book on Calculus of Variations [26, Theorems 4.1.1, 4.1.2], Morrey extended Serrin’s Theorems A and B to the vectorial case. Several years later, Eisen [14] studied the case where d > 1 and proved that Lemma 4.14 in [26], which is the core of Theorem B, ceases to be true when d > 1, thus placing in doubt the validity of theorem itself. In addition, he constructed counterexamples for Theorems A(ii) and B(iii). Theorem A(ii) seems to fail in the vectorial case due mainly to the truncation techniques of the type used in Lemma 3 of [30] (see also [29, pp. 30–31]) and in our Theorem 1.1, suitable only for the scalar case. On the other hand, Serrin’s Theorem A(i) continues to hold in the vectorial case, while the validity of Theorem B(i)–(ii) when d > 1 remains open. Note that Eisen’s counterexamples were both of the form

f = f(u, ξ) = h(u)g(ξ).

Thus we cannot hope to fully extend either Theorem E of De Giorgi, Buttazzo and Dal Maso or our Theorems 1.1 and 1.5 to the vectorial case. However, we can prove the following:

Theorem 1.7. Let f be a nonnegative Borel integrand. Suppose that for all (x_{0}, u_{0}) ∈ Ω × ℝ^{d} and ε > 0 there exist δ > 0 and a modulus of continuity ρ, with ρ(s) ≤ C(1 + s) for s > 0 and for some C > 0, such that

f(x_{0}, u_{0}, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ)) + ρ(|u − u_{0}|)  (1.10)

for all x ∈ Ω with |x − x_{0}| ≤ δ, and for all (u, ξ) ∈ ℝ^{d} × ℝ^{dN}. Assume also that either

(a) f(x_{0}, u_{0}, ·) is convex in ℝ^{dN} or

(b) f(x_{0}, u_{0}, ·) is quasiconvex in ℝ^{dN} and

0 ≤ f(x_{0}, u_{0}, ξ) ≤ C(|ξ|^{q} + 1)  for all ξ ∈ ℝ^{dN},  (1.11)

where C > 0 and the exponent q ≥ 1 may depend on (x_{0}, u_{0}). In addition, if q > 1 then assume that

f(x_{0}, u_{0}, ξ) ≥ \frac{1}{C} |ξ|^{q} − C  for all ξ ∈ ℝ^{dN}.  (1.12)

Let u ∈ BV_{loc}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W_{loc}^{1,1}(Ω; ℝ^{d}) which converges to u in L_{loc}^{1}(Ω; ℝ^{d}). Then:

(i) ∫_{Ω} f(x, u, ∇u) dx + ∫_{Ω} f^{∞}(x, u, dC(u)) ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx;

(ii) if f = f(x, ξ) then ∫_{S(u)∩Ω} f^{∞}(x, (u^{+}(x) − u^{−}(x)) ⊗ ν_{u}) dH^{N−1} ≤ lim inf_{n→∞} ∫_{Ω} f(x, ∇u_{n}) dx.

Theorem 1.7 improves Theorem A(i) of Serrin, since condition (1.10) is significantly weaker than the corresponding (i). Moreover, Theorem 1.7 is also closely related to a recent result of Acerbi, Bouchitté and Fonseca [1] for integrands of the form f = f(x, ξ), convex in ξ, where condition (1.10) is replaced by the growth condition

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

with

p > q \frac{N − 1}{N}.

In the quasiconvex case we use a result of Ambrosio and Dal Maso [4] for functions g = g(ξ) such that

0 ≤ g(ξ) ≤ C(1 + |ξ|).

This growth condition is of vital importance for their argument to work.

When (1.11) and (1.12) hold then, by a recent result of Kristensen [22], we can approximate f(x_{0}, u_{0}, ξ) by an increasing sequence of quasiconvex functions g_{j}(ξ) which grow at most linearly, and thus we can still use [4] for each g_{j}. Note that without (1.12) L_{loc}^{1} lower semicontinuity may fail even for the simplest case when f = f(ξ). This has been shown by Malý [23] for

f = f(ξ) = |det ξ|, d = N,

who constructed a sequence in W^{1,N} which converges to u(x) = x weakly in W^{1,p}, where p < N − 1, and for which lower semicontinuity fail (see also Fonseca and Malý [16]). Thus, by Theorem 1.7 it follows that f(ξ) = |det ξ| cannot be approximated from below by an increasing sequence of functions g_{j}(ξ) which grow at most linearly (see also example 7.9 in [22] for a different proof). This is in sharp contrast with the convex case, where it is well known that this approximation can always be done (see e.g. Proposition 9.1).

In Theorem 1.7(ii) we have chosen to restrict ourselves to integrands f of the form f = f(x, ξ) because in this case there is a simple integral representation formula for the relaxation of F on the jump set S(u), while, when f depends on the full set of variables and d > 1, then the formula is rather complicated (see Theorem 1.10 below).

Theorem 1.8. Theorem 1.7(i) still holds if we replace condition (1.10) with the following: for all (x_{0}, u_{0}) ∈ Ω × ℝ^{d} either f(x_{0}, u_{0}, ξ) ≡ 0 for all ξ ∈ ℝ^{dN}, or for every ε > 0 there exist C_{1}, C_{2}, δ > 0 such that

f(x_{0}, u_{0}, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))  (1.13)
f(x, u, ξ) ≥ C_{1}|ξ| − C_{2}  (1.14)

for all (x, u) ∈ Ω × ℝ^{d} with |x − x_{0}| + |u − u_{0}| ≤ δ and for all ξ ∈ ℝ^{d N}.

Theorem 1.8 was proven by Fonseca and Müller [18], under somewhat stronger hypotheses, and in the case where assumption (b) of Theorem 1.7 holds with q = 1. The convex case can be thought of as a natural extension of Theorem A(ii) of Serrin to the vectorial case.

Theorems 1.7 and 1.8 are complemented by the following result:

Theorem 1.9. Assume that the hypotheses of Theorem 1.3 are verified in the vectorial case, with f(x, u, ·) quasiconvex in ℝ^{dN}. Then Theorem 1.3(i)–(ii) continues to hold. Furthermore, if in Theorem 1.3(iii) we assume that f^{∞} = f^{∞}(x, ξ), then

F(u, S(u)) ≤ ∫_{S(u)∩Ω} f^{∞}(x, (u^{+}(x) − u^{−}(x)) ⊗ ν_{u}) dH^{N−1}(x).

A similar extension holds for Theorem 1.6.

To obtain an integral representation formula for the relaxation F over the jump set S(u) in the vectorial case we need a different set of hypotheses.

Theorem 1.10. Assume that f is a nonnegative Borel integrand which satisfies (1.8). Suppose also that for all x_{0} ∈ Ω and ε > 0 there exist two constants δ, L > 0 such that

f^{∞}(x_{0}, u, ξ) − f^{∞}(x, u, ξ) ≤ ε(1 + f^{∞}(x, u, ξ))  (1.15)

for all x ∈ Ω with |x − x_{0}| ≤ δ, and for all (u, ξ) ∈ ℝ^{d} × ℝ^{dN}; and

f^{∞}(x, u, ξ) − \frac{f(x, u, tξ)}{t} ≤ ε (1 + \frac{f(x, u, tξ)}{t})  (1.16)

for all x ∈ Ω, with |x − x_{0}| ≤ δ, and for all (u, ξ) ∈ ℝ^{d} × ℝ^{dN} and t > L. Then

F(u, S(u)) ≥ ∫_{S(u)∩Ω} h(x, u^{+}(x), u^{−}(x), ν_{u}) dH^{N−1}(x),  (1.17)

where

h(x_{0}, λ, θ, ν) := inf {∫_{Q_{ν}} f^{∞}(x_{0}, w(y), ∇w(y)) dy : w ∈ W^{1,1}(Q_{ν}), w|_{∂Q_{ν}(x_{0},ε)} = u_{λ,θ,ν}}  (1.18)

and

u_{λ,θ,ν}(y) := \{ \begin{matrix}λ & if y · ν > 0, \\ θ & if y · ν ≤ 0. \end{matrix}

Furthermore if (1.16) is replaced by

|f^{∞}(x, u, ξ) − \frac{f(x, u, tξ)}{t}| ≤ ε (1 + \frac{f(x, u, tξ)}{t})  (1.16)′

and f^{∞}(·, u, ξ) is upper semicontinuous, then (1.17) is an equality.

Here, and in what follows, Q_{ν} := R_{ν}(−\frac{1}{2}, \frac{1}{2})^{N}, where R_{ν} denotes a rotation such that R_{ν}e_{N} = ν, and Q = (−\frac{1}{2}, \frac{1}{2})^{N}. Also, C will denote a generic constant which may vary from line to line.

It is not difficult to see that conditions (H2) and (H4) in Thm. 4.1.4 of [7] imply (1.16).

In addition to the novelty of the results in this paper, which significantly improve upon classical theorems in the literature, we would like to close this section pointing out some aspects of our approach. One of the main tools exploited in the paper is the blow–up method introduced by Fonseca and Müller [17], [18]. This method was first used to deal with quasiconvex integrands, since many of the techniques in convex analysis available for the scalar case could not be easily extended to the vectorial case. It turns out that blow–up arguments in the scalar case, combined with some classical methods for convex integrands, may improve and simplify some important results in the literature. Also, we use the very recent global method of relaxation introduced by Bouchitté, Fonseca and Mascarenhas [7], to show that the relaxed energy density may be written in terms of a Dirichlet problem. Most of the proofs are carried out firstly for f which grow at most linearly in the gradient variable ξ. While this approach is standard in the convex setting, and in the literature there are several results which allow to approximate from below convex functions by an increasing sequence of convex functions which grow at most linearly, it was only very recently that Kristensen brought this idea to the vectorial setting, exploiting his approximation result for quasiconvex functions (see [22]; see also [24]).

In the presentation of the paper, and whenever it was possible, we have tried to treat separately the energies corresponding to the Lebesgue, Cantor, and Jump part of Du, in order to better understand the corresponding scaling and the necessity and sufficiency of our hypotheses. It is interesting to observe that the Lebesgue and Cantor measures may be treated in a similar fashion and, more importantly, under the same hypotheses on the integrand f. On the other hand lower semicontinuity for the jump part requires hypotheses and methods which depart from the above mentioned.

Although the hypotheses on the integrand f are rather mild, they are not by any means minimal. Indeed, it was not our purpose to obtain necessary conditions for lower semicontinuity, but, rather, to find simple sufficient assumptions, which would be easy to verify in the applications. It seems, however, that lower semicontinuity of f in the x variable is almost necessary, but it is not clear if it should always be uniform in ξ (at least for functionals which are allowed to vanish). Dal Maso’s example (see [9])

f(x, ξ) = |(sin ω(x), cos ω(x)) · ξ|

certainly seems to imply that it should. The lower semicontinuity of f in the u variable is not necessary, as proved by Theorem D, but in order to drop it, stronger assumptions on the dependence on x seem to be needed.

§2. Proof of Theorem 1.1.

Throughout this work we will use often truncation arguments, and the result below will be instrumental.

Proposition 2.1. (Truncation) Let f : Ω × ℝ × ℝ^{N} → [0, ∞) be a Borel integrand satisfying (1.8). Suppose that there exists a sequence (ε_{k}, λ_{k}, t_{k}, u_{0k}) ∈ ℝ^{4} such that

ε_{k} → 0^{+},  λ_{k} → λ ∈ [0, ∞),  t_{k} → T ∈ (0, ∞],  u_{0k} → u_{0} ∈ ℝ,

and

lim_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0} + ε_{k}y, u_{0k} + λ_{k}w_{k}(y), t_{k}∇w_{k}(y)) dy =: ℓ < ∞,

where x_{0} ∈ Ω, {w_{k}} ⊂ W^{1,1}(Q_{ν}; ℝ) converges in L^{1}(Q_{ν}; ℝ) to a function w_{0} ∈ L^{∞}(Q_{ν}; ℝ).

Let [λ essinf_{Q_{ν}} w_{0}, λ esssup_{Q_{ν}} w_{0}] ⊂ (α_{1}, α_{2}), for some α_{1}, α_{2} ∈ ℝ. Then there exists a new sequence {v_{k}} ⊂ W^{1,1}(Q_{ν}; ℝ), converging to w_{0} in L^{1}(Q_{ν}; ℝ), such that

lim_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0} + ε_{k}y, u_{0k} + λ_{k}v_{k}(y), t_{k}∇v_{k}(y)) dy ≤ ℓ

and

u_{0k} + λ_{k}v_{k}(y) ∈ [u_{0} + α_{1}, u_{0} + α_{2}] for L^{N} a.e. y ∈ Q_{ν}.

Remark 2.2. (i) It is easy to check that the conclusion of Proposition 2.1 still holds if we replace Q_{ν} by any bounded, open, convex subset of ℝ^{N} containing the origin.

(ii) Condition (1.8) can be significantly weakened if we specialize the sequences t_{k}, ε_{k} and x_{0} (see Lemma 8.4).

Proof. Take 0 < 2ε < min{λ essinf_{Q_{ν}} w_{0}−α_{1}, α_{2}−λ esssup_{Q_{ν}} w_{0}}, and let k be so large that |u_{0}−u_{0k}| < ε/2. Define

\begin{matrix}E_{k} := \{y ∈ Q_{ν} : u_{0k} + λ_{k} w_{k}(y) ∈ [u_{0} + α_{1}, u_{0} + α_{2}]\}, \\ E_{k}^{+} := \{y ∈ Q_{ν} : u_{0k} + λ_{k} w_{k}(y) > u_{0} + α_{2}\}, \\ E_{k}^{−} := \{y ∈ Q_{ν} : u_{0k} + λ_{k} w_{k}(y) < u_{0} + α_{1}\}, \end{matrix} v_{k}(y) := \{ \begin{matrix}w_{k}(y) & in E_{k}, \\ (u_{0} − u_{0k} + α_{2})/λ_{k} & in E_{k}^{+}, \\ (u_{0} − u_{0k} + α_{1})/λ_{k} & in E_{k}^{−}. \end{matrix}

Then v_{k} ∈ W^{1,1}(Q_{ν}; ℝ), and for k large enough

α_{2} > ε + |u_{0} − u_{0k}| + λ_{k} esssup_{Q_{ν}} w_{0},  α_{1} + ε < |u_{0} − u_{0k}| + λ_{k} essinf_{Q_{ν}} w_{0},  (2.1)

with

\begin{aligned}∫_{Q_{ν}} |v_{k}(y) − w_{0}(y)|dy & = ∫_{E_{k}} |w_{k}(y) − w_{0}(y)|dy + ∫_{E_{k}^{+}} (\frac{u_{0} − u_{0k} + α_{2}}{λ_{k}} − w_{0}(y)) dy \\ & + ∫_{E_{k}^{−}} (w_{0}(y) − \frac{u_{0} − u_{0k} + α_{1}}{λ_{k}} ) dy ≤ ||w_{k} − w_{0}||_{L^{1}(Q_{ν};ℝ)} → 0 \end{aligned}

as k → ∞. Moreover

\begin{matrix}\frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0} + ε_{k}y, u_{0k} + λ_{k}v_{k}(y), t_{k}∇v_{k}(y)) dy = \frac{1}{t_{k}} ∫_{E_{k}} f(x_{0} + ε_{k}y, u_{0k} + λ_{k}w_{k}(y), t_{k}∇w_{k}(y)) dy \\ + \frac{1}{t_{k}} ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u_{0} + α_{2}, 0) dy + \frac{1}{t_{k}} ∫_{E_{k}^{−}} f(x_{0} + ε_{k}y, u_{0} − α_{1}, 0) dy. \end{matrix}

By (1.8) and (2.1)

\begin{aligned}0 & ≤ \frac{1}{t_{k}} ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u_{0} + α_{2}, 0) dy ≤ \frac{C}{t_{k}} L^{N}(E_{k}^{+}) \\ & ≤ \frac{C}{t_{k}} L^{N}(\{y ∈ Q_{ν} : |w_{k}(y) − w_{0}(y)| > ε/λ_{k}\}) ≤ \frac{Cλ_{k}}{t_{k}ε} ||w_{k} − w_{0}||_{L^{1}(Q_{ν};ℝ)} → 0. \end{aligned}  (2.2)

Similarly, we can show that the integral over E_{k}^{−} approaches zero as k → ∞. □

Proof of Theorem 1.1. Without loss of generality we may assume that f is continuous, f(x, u, ·) is convex, f satisfies (1.7), (1.8), and for all (x_{0}, u_{0}) ∈ Ω × ℝ and ε > 0 there exists δ > 0 such that

f(x_{0}, u, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))  (2.3)

for all (x, u) ∈ Ω × ℝ with |x − x_{0}| + |u − u_{0}| ≤ δ and for all ξ ∈ ℝ^{N}.

Indeed, suppose that the conclusion of the theorem is true under these additional hypotheses. By applying Proposition 9.3 to the function f, and noting that (1.7) and (9.2) are equivalent if f ≥ 1, we may find an increasing sequence of nonnegative continuous functions f_{j} convex in ξ, which satisfy (1.7), (1.8), and (2.3), and such that f(x, u, ξ) + 1= sup_{j} f_{j}(x, u, ξ). Let {u_{n}} ⊂ W_{loc}^{1,1}(Ω; ℝ) converge to u ∈ BV_{loc}(Ω; ℝ) in L_{loc}^{1}(Ω; ℝ), and let A ⋐ Ω. For any fixed j

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx ≥ lim inf_{n→∞} ∫_{A} f_{j}(x, u_{n}(x), ∇u_{n}(x)) dx − L^{N}(A) ≥ H_{j}(u, A) − L^{N}(A),

where H_{j} is the functional given in (1.2) and corresponding to f_{j}. If now we let j → ∞, and use Lebesgue Monotone Convergence Theorem and Proposition 9.3, we conclude that

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx ≥ ∫_{A} f(x, u, ∇u) dx + ∫_{A} f^{∞}(x, u, dC(u))

+ ∫_{S(u)∩A} (∫_{u^{−}(x)}^{u^{+}(x)} f^{∞}(x, s, ν_{u}) ds) dH^{N−1}
= H(u, A),

where we have used the fact that (f + 1)^{∞} = f^{∞}. The result now follows by letting A ↗ Ω, and using Lebesgue Monotone Convergence Theorem once again.

Thus, in what follows f is continuous, verifies (1.7), (1.8) and (2.3), {u_{n}} ⊂ W_{loc}^{1,1}(Ω; ℝ) converges to u ∈ BV_{loc}(Ω; ℝ) in L_{loc}^{1}(Ω; ℝ), and, 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 < ∞.  (2.4)

Passing to a subsequence, if necessary, we may find 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.

(i) (Lebesgue part) We claim that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} ≥ f(x_{0}, u_{n}(x_{0}), ∇u_{n}(x_{0}))  for L^{N} a.e. x_{0} ∈ Ω,  (2.5)

where Q(x_{0}, ε) := x_{0} + ε Q. If (2.5) holds, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{0}(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 in what concerns the absolutely continuous part, it suffices to show (2.5).

Fix x_{0} ∈ Ω such that

\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. (2.6)

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^{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), and by (2.6), lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{L^{1}(Q;ℝ)} = 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) and such that

\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.  (2.7)

Fix ε > 0 and let δ be provided by (1.7). By Proposition 2.1, with

λ_{k} := ε_{k},  t_{k} :≡ 1,  u_{0k} :≡ u(x_{0}),  −α_{1} = α_{2} := δ,

we may find a new sequence {v_{k}} ⊂ W^{1,1}(Q; ℝ), still convergent to w_{0} in L^{1}(Q; ℝ), such that

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy,

and |ε_{k}v_{k}(y)| ≤ δ for L^{N} a.e. y ∈ Q. Since ε_{k} → 0, by (1.7) we obtain

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇v_{k}(y)) dy.

We can now apply Serrin’s Theorem A(i) to the integrand g(ξ) := f(x_{0}, u(x_{0}), ξ) to conclude that

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

The result follows by letting ε → 0^{+}. □

(ii) (Cantor part) The proof for the Cantor part is somewhat similar to the previous one, and we will only indicate the main differences. The inequality (2.5) is now replaced by

\frac{dμ}{d|C(u)|}(x_{0}) ≥ f^{∞} (x_{0}, u(x_{0}), \frac{dC(u)}{d|C(u)|}(x_{0})) for C(u) a.e. x_{0} ∈ Ω,

where (see [2])

\frac{dC(u)}{d|C(u)|}(x_{0}) = a_{u}(x_{0})ν_{u}(x_{0}),  (2.8)

with a_{u}(x_{0}) ∈ ℝ and ν_{u}(x_{0}) ∈ S^{N−1} := {x ∈ ℝ^{N} : |x| = 1}. For simplicity of the notation, from now on we will write a and ν to designate a_{u}(x_{0}) and ν_{u}(x_{0}), respectively. It is known (see [4], [7], [18]) that for C(u) a.e. x_{0} ∈ Ω the following hold

\frac{dμ}{d|C(u)|}(x_{0}) = lim_{ε→0^{+}} \frac{μ(Q_{ν}(x_{0}, ε))}{|Du|(Q_{ν}(x_{0}, ε))} < ∞,  lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν}(x_{0},ε)} |u(x) − u(x_{0})|dx = 0,
lim_{ε→0^{+}} \frac{|Du|(Q_{ν}(x_{0}, ε))}{ε^{N−1}} = 0,  lim_{ε→0^{+}} \frac{|Du|(Q_{ν}(x_{0}, ε))}{ε^{N}} = ∞.

(2.9)

Fix x_{0} ∈ Ω so that (2.9) holds. By Lemma 3.9 in [7] (see also Theorem 2.3 in [4]) there exist ε_{k} → 0^{+} such that μ(∂Q_{ν}(x_{0}, ε_{k})) = 0, and a non decreasing function Ψ : (−\frac{1}{2}, \frac{1}{2}) → ℝ such that the followings hold:

Ψ(\frac{1}{2} − 0) − Ψ(−\frac{1}{2} + 0) = 1,  ∫− \frac{1}{2}^{\frac{1}{2}} Ψ(s) ds = 0,

z_{k}(y) : = \frac{u(x_{0} + ε_{k}y) − \frac{1}{ε_{k}^{N}} \frac{1}{ε_{k}^{N}} ∫_{Q_{ν}} u(x_{0} + ε_{k}z)dz}{λ_{k}} → w_{0}(y) := Ψ(y · ν) a in L^{1}(Q_{ν}; ℝ),

lim_{k→∞} |Dz_{k}|(Q_{ν}) = |Dw_{0}|(Q_{ν}),

(2.10)

where, by (2.9), λ_{k} := |Du|(Q_{ν}(x_{0}, ε_{k}))/ε_{k}^{N−1} → 0 and t_{k} := λ_{k}/ε_{k} → ∞ as k → ∞. Then

\frac{dμ}{d|C(u)|}(x_{0}) = lim_{k→∞} \frac{μ(Q_{ν}(x_{0}, ε_{k}))}{|Du|(Q_{ν}(x_{0}, ε_{k}))} = lim_{k→∞} lim_{n→∞} \frac{1}{|Du|(Q_{ν}(x_{0}, ε_{k}))} ∫_{Q_{ν}(x_{0},ε_{k})} f(x, u_{n}, ∇u_{n}) dx

= lim_{k→∞} lim_{n→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0} + ε_{k}y, u_{0n,k} + λ_{k}w_{n,k}(y), t_{k}∇w_{n,k}(y)) dy,

where

w_{n,k}(y) := \frac{u_{n}(x_{0} + ε_{k}y) − u_{0n,k}}{λ_{k}} , u_{0n,k} := \frac{1}{ε_{k}^{N}} ∫_{Q_{ν}} u_{n}(x_{0} + ε_{k}z)dz.

Clearly w_{n,k} ∈ W^{1,1}(Q_{ν}; ℝ), and by (2.10), (2.9)_{2}, and the fact that {u_{n}} converges to u in L^{1},

lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{L^{1}(Q_{ν};ℝ)} = 0,  lim_{k→∞} lim_{n→∞} u_{0n,k} = u(x_{0}).

By a standard diagonalization argument, we may extract two subsequences {w_{k} := w_{n_{k},k}}, {u_{0k} := u_{0n_{k},k}}, which converge to w_{0} in L^{1}(Q_{ν}; ℝ) and u(x_{0}), respectively, such that

\frac{dμ}{d|C(u)|}(x_{0}) = lim_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0} + ε_{k}y, u_{0k} + λ_{k}w_{k}(y), t_{k}∇w_{k}(y)) dy.  (2.11)

We can now continue as in part (i), using Proposition 2.1 and then (1.7) to conclude that

(1 + ε)\frac{dμ}{d|C(u)|}(x_{0}) + ε ≥ lim inf_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0}, u(x_{0}), t_{k}∇v_{k}(y)) dy.  (2.12)

Due to the presence of the sequence t_{k}, we cannot apply directly Serrin’s Theorem A(i) to the integrand g(ξ) := f(x_{0}, u(x_{0}), ξ) as we did in part(i). Although the adaptations to the present setting are quite straightforward, here we present an alternative proof which can be extended to the vectorial case and to quasiconvex functions. Assume, for simplicity, that ν = e_{N} and construct a sequence of smooth functions v_{h}(y) = v̄_{h}(y_{N}) such that

||v_{h} − w_{0}||_{L^{1}(Q;ℝ)} ≤ \frac{1}{h}  and  |∇v_{h}|(Q) − |Dw_{0}|(Q) → 0.

as h → ∞. Since v_{h} depends only on y_{N}, its trace on ∂Q agrees with the trace of a function

A_{h} y + p(y), A_{h} := (v̄_{h}(1/2) − v̄_{h}(−1/2)) ⊗ e_{N} = ∇v_{h}(Q),

where p is Q–periodic. Choose open sets Ω_{i}, i = 1, 2, 3, such that Ω_{1} ⋐ Ω_{2} ⋐ Ω_{3} ⋐ Q. Since g satisfies (1.8), by Lemma 2.5 of Ambrosio and Dal Maso [4] we may find a new sequence

v_{h,k}(y) = φ(y)w_{k}(y) + (1 − φ(y))v_{h}(y)

such that

\frac{1}{t_{k}} ∫_{Ω_{3}} g(t_{k}∇w_{k}) dy + \frac{1}{t_{k}} ∫_{Q\Ω_{1}} g(t_{k}∇v_{h}) dy + \frac{4C}{δ} ∫_{Q} |w_{k} − v_{h}| dy + \frac{1}{k} ≥ \frac{1}{t_{k}} ∫_{Q} g(t_{k}∇v_{h,k}) dy

where φ is a cut–off function such that φ ≡ 1 in a neighborhood of Ω_{2}, φ ≡ 0 in a neighborhood of ℝ^{N}\Ω_{3}, and δ < dist (Ω_{2}, ∂Ω_{3}). By virtue of the quasiconvexity of g, together with the growth (1.8) (recall that in the scalar case quasiconvexity is equivalent to convexity), we obtain

\frac{1}{t_{k}} ∫_{Q} g(t_{k}∇w_{k}) dy + C (\frac{L^{N}(Q\Ω_{1})}{t_{k}} + |∇v_{h}|(Q\Ω_{1})) + \frac{4C}{δ} ∫_{Q} |w_{k} − v_{h}| dy + \frac{1}{k} ≥ \frac{1}{t_{k}} g(t_{k}∇v_{h}(Q)).

Letting k → ∞ gives

lim_{k→∞} \frac{1}{t_{k}} ∫_{Q} g(t_{k}∇w_{k}) dy + C|∇v_{h}|(Q\Ω_{1}) + \frac{4C}{δ} ∫_{Q} |w_{0} − v_{h}| dy ≥ g^{∞}(∇v_{h}(Q));

hence, taking the limit as h → ∞, we conclude that

lim_{k→∞} \frac{1}{t_{k}} ∫_{Q} g(t_{k}∇w_{k}) dy + C|Dw_{0}|(Q\Ω_{1}) ≥ g^{∞}(∇w_{0}(Q)),

where we used the continuity of g^{∞} (see Proposition 9.1), and the fact that |∇v_{n}|(Q) → |Dw_{0}|(Q). If we now let Ω_{1} ↗ Q we get

lim_{k→∞} \frac{1}{t_{k}} ∫_{Q} g(t_{k}∇w_{k}) dy ≥ g^{∞}(a ν);

thus,

(1 + ε)\frac{dμ}{d|C(u)|}(x_{0}) + ε ≥ f^{∞} (x_{0}, u(x_{0}), \frac{dC(u)}{d|C(u)|}(x_{0})) .

It suffices now to let ε → 0^{+}. □

(iii)(Jump part) To complete the proof of Theorem 1.1 it remains to show that

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) ≥ ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} f^{∞}(x_{0}, s, ν) ds for H^{N−1} a.e. x_{0} ∈ S(u),

where ν = ν_{u}(x_{0}) is the normal to S(u). It is known that (see [7]) for H^{N−1} a.e. x_{0} ∈ S(u)

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) = lim_{ε→0^{+}} \frac{μ(Q_{ν}(x_{0}, ε))}{ε^{N−1}} < ∞,
lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν}^{+}(x_{0},ε)} |u(x) − u^{+}(x_{0})|dx = 0,  lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν}^{−}(x_{0},ε)} |u(x) − u^{−}(x_{0})|dx = 0,

(2.13)

where Q_{ν}^{+}(x_{0}, ε) := {y ∈ Q_{ν}(x_{0}, ε) : y · ν > 0} and Q_{ν}^{−}(x_{0}, ε) := {y ∈ Q_{ν}(x_{0}, ε) : y · ν < 0}. Fix x_{0} ∈ S(u) such that (2.13) holds, and choose a sequence ε_{k} → 0^{+} with μ(∂Q_{ν}(x_{0}, ε_{k})) = 0. Then

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) = lim_{k→∞} \frac{μ(Q_{ν}(x_{0}, ε_{k}))}{ε_{k}^{N−1}} = lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{N−1}} ∫_{Q_{ν}(x_{0},ε_{k})} f(x, u_{n}, ∇u_{n}) dx

= lim_{k→∞} lim_{n→∞} ∫_{Q_{ν}} ε_{k}f (x_{0} + ε_{k}y, w_{n,k}(y), \frac{1}{ε_{k}} ∇w_{n,k}(y)) dy,

where w_{n,k}(y) := u_{n}(x_{0} + ε_{k}y). Clearly w_{n,k} ∈ W^{1,1}(Q_{ν}; ℝ), and by (2.13) together with the fact that u_{n} converges to u in L^{1},

lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{L^{1}(Q_{ν};ℝ)} = 0,  where  w_{0}(y) := \{ \begin{matrix}u^{+}(x_{0}) & if y · ν > 0, \\ u^{−}(x_{0}) & if y · ν ≤ 0. \end{matrix}

As before, by a standard diagonalization argument we may extract a subsequence {w_{k} := w_{n_{k},k}} converging to w_{0} in L^{1}(Q_{ν}; ℝ), with

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) = lim_{k→∞} ∫_{Q_{ν}} ε_{k}f (x_{0} + ε_{k}y, w_{k}(y), \frac{1}{ε_{k}} ∇w_{k}(y)) dy.  (2.14)

Fix ε > 0. By (2.3) for each u_{1} ∈ [u^{−}(x_{0}), u^{+}(x_{0})] there is δ_{u_{1}} > 0 such that

f(x_{0}, u, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))

for all |x − x_{0}| ≤ δ_{u_{1}}, |u − u_{1}| ≤ δ_{u_{1}}, and for all ξ ∈ ℝ^{N}. Since

⋃_{u_{1}∈[u^{−}(x_{0}),u^{+}(x_{0})]} B(u_{1}, δ_{u_{1}}) ⊃ [u^{−}(x_{0}), u^{+}(x_{0})],

we may find a finite subcovering

⋃_{i=1}^{M} B(u_{i}, δ_{i}) ⊃ [u^{−}(x_{0}), u^{+}(x_{0})].

Set δ := min{δ_{1}, …, δ_{M}, δ^{+}, δ^{−}}, where δ^{±} are provided by (2.3) corresponding to the points (x_{0}, u^{±}(x_{0})), respectively. Then

f(x_{0}, u, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))  (2.15)

for all |x − x_{0}| ≤ δ, u ∈ [u^{−}(x_{0}) − δ, u^{+}(x_{0}) + δ], and for all ξ ∈ ℝ^{N}.

By Proposition 2.1, with

λ_{k} :≡ 1,  t_{k} := 1/ε_{k},  u_{0k} :≡ 0,  α_{1} = u^{−}(x_{0}) − δ,  α_{2} := u^{+}(x_{0}) + δ,

there exists a new sequence v_{k} ∈ W^{1,1}(Q_{ν}; ℝ), convergent to w_{0} in L^{1}(Q_{ν}; ℝ), such that v_{k}(y) ∈ [u^{−}(x_{0}) − α_{1}, u^{+}(x_{0}) + α_{2}], and by (2.14) and (2.15) we have

(1 + ε)\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) ≥ lim inf_{k→∞} ∫_{Q_{ν}} ε_{k}f (x_{0}, v_{k}(y), \frac{1}{ε_{k}} ∇v_{k}(y)) dy.

Since h(u, ξ) := f(x_{0}, u, ξ) is continuous, by an approximation result due to Ambrosio [3] we can write

h(u, ξ) = sup_{i∈ℕ}[a_{i}(u) + b_{i}(u) · ξ]^{+}

where the functions a_{i} : ℝ → ℝ and b_{i} : ℝ → ℝ^{N} are bounded and continuous. It is not difficult to see that

h^{∞}(u, ξ) = sup_{i}[b_{i}(u) · ξ]^{+}.

Therefore (see [12, Lemma 6])

∫_{u^{−}(x_{0})}^{u^{+}(x_{0})}h^{∞}(s, ν) ds = sup_{j∈ℕ} sup{∑_{i=1}^{j} ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})}ψ_{i}(s)[b_{i}(s)·ν]^{+} ds : ψ_{i} ∈ C_{0}^{∞}((u^{−}(x_{0})^{,}u^{+}(x_{0})); [0, 1]), ∑_{i=1}^{j} ψ ≤ 1}.

(2.16)

Fix j ∈ ℕ, ψ_{1}, …, ψ_{j}, as in (2.16), and let φ ∈ C_{0}^{∞}(Q_{ν}; [0, 1]). We have

(1 + ε)\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) ≥ lim inf_{k→∞} ∑_{i=1}^{j} ∫_{Q_{ν}} φ(y)ε_{k}ψ_{i}(v_{k}(y)) [a_{i}(v_{k}(y)) + \frac{1}{ε_{k}} b_{i}(v_{k}(y)) · ∇v_{k}(y)]^{+} dy

≥ lim inf_{k→∞} ∑_{i=1}^{j} ∫_{Q_{ν}} φ(y)ψ_{i}(v_{k}(y)) [b_{i}(v_{k}(y)) · ∇v_{k}(y)]^{+} dy,

where we have used the inequality (α + β)^{+} ≥ (β)^{+} − |α| for α, β ∈ ℝ, Lebesgue Dominated Convergence Theorem, and the fact that

ε_{k}|a_{i}(v_{k}(y))| ≤ ||a_{i}||_{L^{∞}(ℝ)} ε_{k} → 0.

By a result of De Cicco [12, Theorem 1], we have

(1 + ε)\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) ≥ ∑_{i=1}^{j} ∫_{S(w_{0})} φ(y) ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} ψ_{i}(s) [b_{i}(s) · ν]^{+} ds dH^{N−1}(y),

and taking φ ↗ 1, we obtain

\begin{aligned}(1 + ε)\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) & ≥ ∑_{i=1}^{j} ∫_{S(w_{0})} ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} ψ_{i}(s) [b_{i}(s) · ν]^{+} ds dH^{N−1}(y) \\ & = ∑_{i=1}^{j} ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} ψ_{i}(s) [b_{i}(s) · ν]^{+} ds. \end{aligned}

In view of (2.16), the proof is concluded by taking the supremum over all j ∈ ℕ, ψ_{1}, …, ψ_{j}, and letting ε → 0^{+}. □

§3. Proof of Theorem 1.3.

(i)(Lebesgue part) Assume first that f is Carathéodory. Consider F : BV (Ω; ℝ) × A(Ω) → [0, ∞], where A(Ω) stands for the family of open subsets of Ω. It can be proved that F(u; ·) is the restriction to A(Ω) of a Radon measure, F(·; A) is L^{1}(A) lower semicontinuous, and

0 ≤ F(u, A) ≤ C (L^{N}(A) + |Du|(A)) .

For a proof we refer to Lemma 4.1.2 of [7] (see also [5], [18]). Let F_{1}(u, A) := F(u, A) + |Du|(A). By Theorem 3.7 of [7] we have

\frac{dF_{1}(u, ·)}{dL^{N}}(x_{0}) = \frac{dF(u, ·)}{dL^{N}}(x_{0}) + |∇u(x_{0})| = f_{1}(x_{0}, u(x_{0}), ∇u(x_{0})) for L^{N} a.e. x_{0} ∈ Ω,

where

f_{1}(x_{0}, u_{0}, ξ) := lim sup_{ε→0^{+}} \frac{1}{ε^{N}} inf \{F_{1}(v, Q(x_{0}, ε)) : v ∈ BV (Q(x_{0}, ε)), v|_{∂Q(x_{0},ε)} = u_{0} + ξ · (x − x_{0})\} .

Thus, the proof of part (i) is completed provided we show that

f_{1}(x_{0}, u_{0}, ξ) ≤ f(x_{0}, u_{0}, ξ) + |ξ|

for L^{N} a.e. x_{0} ∈ Ω and for all (u_{0}, ξ) ∈ ℝ × ℝ^{N}. Clearly

f_{1}(x_{0}, u_{0}, ξ) ≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} F(u_{0}+ξ·(x−x_{0}), Q(x_{0}, ε))+|ξ| ≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f(x, u_{0}+ξ·(x−x_{0}), ξ) dx+|ξ|.

Since f is Carathéodory, by the Scorza-Dragoni Theorem for each i ∈ ℕ there exists a compact set K_{i} ⊂ Ω, with L^{N}(Ω\K_{i}) ≤ 1/i, such that f : K_{i} × ℝ × ℝ^{N} → [0, ∞] is continuous. Let K_{i}^{∗} be the set of Lebesgue points of χ_{K_{i}}, and set ω := ∪_{i}^{∞}(K_{i} ∩ K_{i}^{∗}). Then

L^{N}(Ω\ω) ≤ L^{N}(Ω\K_{i}) ≤ \frac{1}{i} → 0 as i → ∞.

If x_{0} ∈ ω then x_{0} ∈ K_{i} ∩ K_{i}^{∗} for some index i. Since g(x) := f(x, u_{0} + ξ(x − x_{0}), ξ) is continuous over K_{i}, given δ > 0 there exists η > 0 such that g(x) ≤ g(x_{0}) + δ for all x ∈ K_{i} with |x − x_{0}| ≤ η. Therefore, by (1.8) we have

f_{1}(x_{0}, u_{0}, ξ) − |ξ| ≤ (f(x_{0}, u_{0}, ξ) + δ) lim sup_{ε→0^{+}} \frac{L^{N}(Q(x_{0}, ε) ∩ K_{i})}{ε^{N}} + C(1 + |ξ|) lim sup_{ε→0^{+}} \frac{L^{N}(Q(x_{0}, ε)\K_{i})}{ε^{N}}
= f(x_{0}, u_{0}, ξ) + δ,

where we have used the fact that x_{0} is a Lebesgue point of χ_{K_{i}}. By letting δ → 0^{+} we obtain the desired inequality. The argument for the case where f(·, ·, ξ) is upper semicontinuous is very similar to the one used in Theorem 1.3(ii) below, and therefore we omit the details. □

(ii)(Cantor part) By Lemma 3.9 of [7], for C(u) a.e. x_{0} ∈ Ω there exist a double indexed sequence {t_{ε}^{(k)}, u_{ε}^{(k)}} such that for every k

t_{ε}^{(k)} → ∞,  ε t_{ε}^{(k)} → 0^{+},  u_{ε}^{(k)} → u(x_{0}) as ε → 0^{+},  (3.1)

and

\frac{dF_{1}(u, ·)}{d|C(u)|}(x_{0}) = \frac{dF(u, ·)}{d|C(u)|}(x_{0}) + |a|

= lim_{k→∞} lim sup_{ε→0^{+}} \frac{inf \{F_{1}(v, Q_{ν}^{(k)}(x_{0}, ε)) : v ∈ BV (Q_{ν}^{(k)}(x_{0}, ε)), v|_{∂Q_{ν}^{(k)}(x_{0},ε)} = u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0})\}}{k^{N−1}ε^{N}t_{ε}^{(k)}},

where \frac{dC(u)}{d|C(u)|}(x_{0}) = aν, a = a(u, x_{0}) ∈ ℝ, ν = ν(u, x_{0}) ∈ S^{N−1}, Q_{ν}^{(k)}(x_{0}, ε) := x_{0} + ε Q_{ν}^{(k)} with

Q_{ν}^{(k)} := R_{ν} ((−\frac{k}{2} , \frac{k}{2} )^{N−1} × (−\frac{1}{2}, \frac{1}{2})) ,

where R_{ν} denotes a rotation such that R_{ν}e_{N} = ν. Take x_{0} ∈ Ω, so that all the limits above exist and are finite. Then

\frac{dF(u, ·)}{d|C(u)|}(x_{0}) + |a| ≤ lim_{k→∞} lim sup_{ε→0^{+}} \frac{1}{k^{N−1}ε^{N}t_{ε}^{(k)}} F_{1}(u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), Q_{ν}^{(k)}(x_{0}, ε))

≤ lim_{k→∞} lim sup_{ε→0^{+}} \frac{1}{k^{N−1}ε^{N}t_{ε}^{(k)}} ∫_{Q_{ν}^{(k)}(x_{0},ε)} f(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), t_{ε}^{(k)}aν) dx + |a|.

(3.2)

By Proposition 9.1, (9.1), and (1.8),

\begin{aligned}\frac{f(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν(x − x_{0}), t_{ε}^{(k)}aν)}{t_{ε}^{(k)}} & ≤ f^{∞}(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), aν) + \frac{f(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), 0)}{t_{ε}^{(k)}} \\ & ≤ f^{∞}(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), aν) + \frac{C}{t_{ε}^{(k)}} . \end{aligned}

Therefore, by (3.1) and (3.2),

\frac{dF(u, ·)}{d|C(u)|}(x_{0}) ≤ lim_{k→∞} lim sup_{ε→0^{+}} \frac{1}{k^{N−1}ε^{N}} ∫_{Q_{ν}^{(k)}(x_{0},ε)} f^{∞}(x, u_{ε}^{(k)} + t_{ε}^{(k)}aν · (x − x_{0}), aν)dx

= lim_{k→∞} lim sup_{ε→0^{+}} \frac{1}{k^{N−1}} ∫_{Q_{ν}^{(k)}} f^{∞}(x_{0} + εy, u_{ε}^{(k)} + ε t_{ε}^{(k)}aν · y, aν) dy.

Since the function f^{∞}(·, ·, aν) is upper semicontinuous, given δ > 0 there exists η > 0 such that f^{∞}(x, u, aν) ≤ f^{∞}(x_{0}, u(x_{0}), aν) + δ for all |x − x_{0}| ≤ η and |u − u(x_{0})| ≤ η. By (3.1), for each fixed k if ε is small enough then x_{0} + εy ∈ B(x_{0}, η) and u_{ε}^{(k)} + ε t_{ε}^{(k)}aν · y ∈ [u(x_{0}) − η, u(x_{0}) + η] for all y ∈ Q_{ν}^{(k)}. Hence

∫_{Q_{ν}^{(k)}} f^{∞}(x_{0} + εy, u_{ε}^{(k)} + ε t_{ε}^{(k)}aν · y, aν) dy ≤ (f^{∞}(x_{0}, u(x_{0}), aν) + δ) k^{N−1}

and, in turn,

\frac{dF(u, ·)}{d|C(u)|}(x_{0}) ≤ f^{∞}(x_{0}, u(x_{0}), aν) + δ.

We now let δ → 0^{+}. □

(iii) (Jump part) By Theorem 3.7 of [7], for H^{N−1} a.e. x_{0} ∈ S(u)

\frac{dF_{1}(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) = \frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) + |u^{+}(x_{0}) − u^{−}(x_{0})|

= lim sup_{ε→0^{+}} \frac{inf \{F_{1}(v, Q_{ν}(x_{0}, ε)) : v ∈ BV (Q_{ν}(x_{0}, ε)), v|_{∂Q_{ν}(x_{0},ε)} = w_{0}\}}{ε^{N−1}},

where ν = ν_{u}(x_{0}) is the normal to S(u) and

w_{0}(y) := \{ \begin{matrix}u^{+}(x_{0}) & if y · ν > 0, \\ u^{−}(x_{0}) & if y · ν ≤ 0. \end{matrix}

18 I. Fonseca and G. Leoni

Take x_{0} ∈ S(u) so that all the limit above exists and is finite. Then

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) + |u^{+}(x_{0}) − u^{−}(x_{0})| ≤ lim sup_{ε→0^{+}} \frac{F(w_{0}, Q_{ν}(x_{0}, ε))}{ε^{N−1}} + |u^{+}(x_{0}) − u^{−}(x_{0})|.

In what follows we assume for simplicity that x_{0} = 0 and ν = e_{N}, and we set

u_{n,ε}(x_{N}) := \{ \begin{matrix}u^{+}(x_{0}) & if x_{N} ≥ ε/2n \\ (u^{+}(x_{0}) − u^{−}(x_{0}))\frac{n}{ε}x_{N} + \frac{u^{+}(x_{0}) + u^{−}(x_{0})}{2} & if −ε/2n ≤ x_{N} ≤ ε/2n \\ u^{−}(x_{0}) & if x_{N} ≤ −ε/2n. \end{matrix}

Clearly ||u_{n,ε} − w_{0}||_{L^{1}(Q_{ν}(x_{0},ε))} → 0 as n → ∞; thus, by a standard diagonalization argument

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ lim_{ε→0^{+}} lim inf_{n→∞} \frac{1}{ε^{N−1}} ∫_{Q_{ν}(x_{0},ε)} f(x, u_{n,ε}(x_{N}), 0, …, 0, u′_{n,ε}(x_{N})) dx

≤ lim inf_{k→∞} ∫_{Q_{ν}} ε_{k}f (x_{0} + ε_{k}y, v_{k}(y_{N}), 0, …, 0, \frac{1}{ε_{k}} v′_{k}(y_{N})) dy,

where

v_{k}(y_{N}) = \{ \begin{matrix}u^{+}(x_{0}) & if y_{N} ≥ 1/2n_{k} \\ (u^{+}(x_{0}) − u^{−}(x_{0}))n_{k} y_{N} + \frac{u^{+}(x_{0}) + u^{−}(x_{0})}{2} & if −1/2n_{k} ≤ y_{N} ≤ 1/2n_{k} \\ u^{−}(x_{0}) & if y_{N} ≤ −1/2n_{k} \end{matrix}

and n_{k} → ∞ as k → ∞. By Proposition 9.1, (9.1), and (1.8), we have

\begin{aligned}ε_{k}f(x_{0} + ε_{k} & y, v_{k}(y_{N}), 0, …, 0, \frac{1}{ε_{k}} v′_{k}(y_{N})) ≤ f^{∞}(x_{0} + ε_{k}y, v_{k}(y_{N}), 0, …, 0, v′_{k}(y_{N})) \\ & + ε_{k}f(x_{0} + ε_{k}y, v_{k}(y_{N}), 0) ≤ f^{∞}(x_{0} + ε_{k}y, v_{k}(y_{N}), 0, …, 0, v′_{k}(y_{N})) + C ε_{k}. \end{aligned}

Therefore, Fubini’s Theorem yields

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ lim inf_{k→∞} ∫_{Q} f^{∞}(x_{0} + ε_{k}y, v_{k}(y_{N}), 0, …, 0, v′_{k}(y_{N})) dy

= lim inf_{k→∞} n_{k} ∫_{Q′} (∫_{−1/2n_{k}}^{1/2n_{k}} f^{∞}(x_{0} + ε_{k}y, v_{k}(y_{N}), (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) dy_{N}) dy′,

(3.3)

where Q′ is the unit cube in ℝ^{N−1}, and where we have used the fact that f^{∞} is positively homogeneous in ξ. Following [5], we introduce now the Yosida transforms

f_{λ}(x, u, ξ) := sup{f^{∞}(x′, u, ξ) − λ|x′ − x| : x′ ∈ Ω},  (3.4)

for λ > 0. For λ ≤ η and by (1.8) it follows that

0 ≤ f^{∞}(x, u, ξ) ≤ f_{η}(x, u, ξ) ≤ f_{λ}(x, u, ξ) ≤ C|ξ|.  (3.5)

We claim that

lim_{λ→∞} f_{λ}(x, u, ξ) = f^{∞}(x, u, ξ).  (3.6)

Indeed, let λ > 1 and choose x_{λ} such that

f_{λ}(x, u, ξ) ≤ f^{∞}(x_{λ}, u, ξ) − λ|x_{λ} − x| + \frac{1}{λ}.

By (3.5)

f^{∞}(x, u, ξ) ≤ f_{λ}(x, u, ξ) ≤ f_{λ}(x, u, ξ) + λ|x_{λ} − x| ≤ f^{∞}(x_{λ}, u, ξ) + \frac{1}{λ}.  (3.7)

Since the right hand side is bounded by C|ξ| + 1 and f_{λ} ≥ 0, it follows that x_{λ} → x as λ → ∞. If we now let λ → ∞ in (3.7), and use the fact that f^{∞}(·, u, ξ) is upper semicontinuous, we obtain (3.6).

Next we show that f_{λ} is Lipschitzian. Fix ε > 0, x, x_{1} ∈ Ω, and find x_{ε} such that

\begin{aligned}f_{λ}(x, u, ξ) & ≤ f^{∞}(x_{ε}, u, ξ) − λ|x_{ε} − x| + ε \\ & ≤ f^{∞}(x_{ε}, u, ξ) − λ|x_{ε} − x_{1}| + λ|x − x_{1}| + ε ≤ f_{λ}(x_{1}, u, ξ) + λ|x − x_{1}| + ε. \end{aligned}

If we now let ε → 0 we obtain

f_{λ}(x, u, ξ) − f_{λ}(x_{1}, u, ξ) ≤ λ|x − x_{1}|,

and, in a similar way,

f_{λ}(x_{1}, u, ξ) − f_{λ}(x, u, ξ) ≤ λ|x − x_{1}|.

We conclude that

|f_{λ}(x, u, ξ) − f_{λ}(x_{1}, u, ξ)| ≤ λ|x − x_{1}|  (3.8)

for all x, x_{1} ∈ Ω, u ∈ ℝ, ξ ∈ ℝ^{N}.

Fix λ > 0. By (3.3), (3.5), and the fact that f_{λ} is Lipschitzian, we have

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ lim inf_{k→∞} n_{k} ∫_{Q′} (∫_{−1/2n_{k}}^{1/2n_{k}} f^{∞}(x_{0} + ε_{k}y, v_{k}(y_{N}), (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) dy_{N}) dy′

≤ lim inf_{k→∞} n_{k} ∫_{Q′} (∫_{−1/2n_{k}}^{1/2n_{k}} f_{λ}(x_{0} + ε_{k}y, v_{k}(y_{N}), (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) dy_{N}) dy′

≤ lim inf_{k→∞} n_{k} ∫_{Q′} (∫_{−1/2n_{k}}^{1/2n_{k}} f_{λ}(x_{0}, v_{k}(y_{N}), (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) + λ ε_{k}|y| dy_{N}) dy′

= lim inf_{k→∞} n_{k} ∫_{−1/2n_{k}}^{1/2n_{k}} f_{λ}(x_{0}, v_{k}(y_{N}), (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) dy_{N}.

A simple change of variables now yields

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ \frac{1}{u^{+}(x_{0}) − u^{−}(x_{0})} ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} f_{λ}(x_{0}, s, (u^{+}(x_{0}) − u^{−}(x_{0})) e_{N}) ds.

Letting λ → ∞, by (3.5), (3.6), and Lebesgue Dominated Convergence Theorem, we obtain

\begin{aligned}\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) & ≤ lim_{λ→∞} \frac{1}{u^{+}(x_{0}) − u^{−}(x_{0})} ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} f_{λ}(x_{0}, s, (u^{+}(x_{0}) − u^{−}(x_{0}))e_{N}) ds \\ & = ∫_{u^{−}(x_{0})}^{u^{+}(x_{0})} f^{∞}(x_{0}, s, e_{N}) ds, \end{aligned}

where we have used again the fact that f^{∞} is positively homogeneous in ξ. □

§4. Proof of Theorem 1.5.

(i) (Lebesgue part) We claim that g(x) := f(x, u(x), ∇u(x)) ∈ L_{loc}^{1}(Ω; ℝ). To show this it is clearly enough to prove that

∫_{Q(x_{0},δ)} g(x) dx < ∞

for any x_{0} ∈ Ω and δ > 0 sufficiently small. Let δ > 0 correspond to ε = 1/2 in (1.9). Then

∞ > lim_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx ≥ \frac{2}{3} lim inf_{n→∞} ∫_{Q(x_{0},δ)} f(x_{0}, u_{n}(x), ∇u_{n}(x)) dx − \frac{1}{3} δ^{N}.

The functional

F_{0}(v, A) := ∫_{A} f(x_{0}, v, ∇v) dx + ∫_{A} f^{∞}(x_{0}, v, dC(v)) + ∫_{S(v)∩A} (∫_{v^{−}(x)}^{v^{+}(x)} f^{∞}(x_{0}, s, ν_{u}) ds) dH^{N−1}

satisfies all the conditions of Theorem 2 in [12], and F_{0}(u_{n}, A) = ∫_{A} f(x_{0}, u_{n}(x), ∇u_{n}(x)) dx since u_{n} ∈ W_{loc}^{1,1}(Ω; ℝ); thus, taking A := Q(x_{0}, δ),

∞ > lim_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx ≥ \frac{2}{3} F_{0}(u, Q(x_{0}, δ)) − \frac{1}{3}δ^{N} ≥ \frac{1}{3} ∫_{Q(x_{0},δ)} g(x) dx − \frac{2}{3}δ^{N},  (4.1)

where we have used (1.9) a second time and H is defined in (1.2). Therefore the claim is proved.

Let Ω(u) be the set of Lebesgue points of g. Since g ∈ L_{loc}^{1}(Ω; ℝ), we have that L^{N}(Ω\Ω(u)) = 0. We now proceed essentially as in the proof of Theorem 1.1, starting from (2.4) up to (2.6), where without loss of generality, we may also assume that x_{0} ∈ Ω(u). Fix ε > 0 and let δ > 0 be such that (1.9) holds. Choose ε_{k} → 0^{+} such that μ(∂Q(x_{0}, ε_{k})) = 0, ε_{k} ≤ δ, and

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

≥ \frac{1}{1 + ε} lim inf_{k→∞} lim inf_{n→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x_{0}, u_{n}, ∇u_{n}) dx − \frac{ε}{1 + ε}.

(4.2)

For fixed k, again by Theorem 2 in [12] applied this time to F_{0}(·, Q(x_{0}, ε_{k})), we obtain

lim inf_{n→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x_{0}, u_{n}(x), ∇u_{n}(x)) dx ≥ \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x_{0}, u(x), ∇u(x)) dx

(recall that f^{∞} ≥ 0) and, consequently,

\frac{dμ}{dL^{N}} (x_{0}) ≥ \frac{1}{1 + ε} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x_{0}, u, ∇u) dx − \frac{ε}{1 + ε}

≥ \frac{1 − ε}{1 + ε} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x, u, ∇u) dx − \frac{2ε}{1 + ε} = \frac{1 − ε}{1 + ε} f(x_{0}, u(x_{0}), ∇u(x_{0})) − \frac{2ε}{1 + ε}

where we have used (1.9) and the fact that x_{0} ∈ Ω(u). We now let ε → 0^{+}. □

(ii)(Cantor part) By (4.1) the function h(x) := f^{∞} (x, u(x), \frac{dC(u)}{d|C(u)|}(x)) ∈ L_{loc}^{1}(Ω; |C(u)|); thus, by Lebesgue–Besicovitch Differentiation Theorem,

lim_{ε→0^{+}} \frac{1}{|C(u)|(Q_{ν}(x_{0}, ε))} ∫_{Q_{ν}(x_{0},ε)} h d|C(u)| = h(x_{0})

for |C(u)| a.e. x_{0} ∈ Ω. Moreover, it is known that for C(u) a.e. x_{0} ∈ Ω

lim_{ε→0^{+}} \frac{|C(u)|(Q_{ν}(x_{0}, ε))}{|Du|(Q_{ν}(x_{0}, ε))} = 1;

hence,

lim_{ε→0^{+}} \frac{1}{|Du|(Q_{ν}(x_{0}, ε))} ∫_{Q_{ν}(x_{0},ε)} h(x) d|C(u)|(x) = h(x_{0})  (4.3)

for |C(u)| a.e. x_{0} ∈ Ω. Let M_{1}(u) be the set of all points of Ω which satisfy (4.3). Then |C(u)|(Ω\M_{1}(u)) = 0. We now proceed as in the proof of part (ii) in Theorem 1.1 up to (2.10), with the only difference that we impose the further restriction that x_{0} ∈ M_{1}(u), to obtain

\frac{dμ}{d|C(u)|}(x_{0}) = lim_{k→∞} lim_{n→∞} \frac{1}{|Du|(Q_{ν}(x_{0}, ε_{k}))} ∫_{Q_{ν}(x_{0},ε_{k})} f(x, u_{n}, ∇u_{n})dx.

The remaining of the proof follows an argument similar to that of part (ii) above after (4.2), except that the integral is now averaged over Q_{ν}(x_{0}, ε_{k}), and we use (4.3) at the end. We omit the details. □

(iii)(Jump part) It suffices to use arguments similar to those of parts (i) and (ii) above. □

§5. Proof of Theorem 1.6.

(i) (Lebesgue part) Fix n ∈ ℕ. Applying the Scorza–Dragoni Theorem to the function f : Ω × [−n, n] × ℝ^{N} → [0, ∞) for each i ∈ ℕ there exists a compact set K_{i,n} ⊂ ℝ, with L^{1}([−n, n]\K_{i,n}) ≤ 1/(i2^{n}), such that f : Ω × K_{i,n} × ℝ^{N} → [0, ∞) is continuous. Let K_{i,n}^{∗} be the set of Lebesgue points of χ_{K_{i,n}}, and set ω := ∪_{n}^{∞} ∪_{i}^{∞} (K_{i,n} ∩ K_{i,n}^{∗}). Then

L^{1}(ℝ\ω) ≤ ∑_{n=1}^{∞} L^{1}([−n, n]\K_{i,n}) ≤ \frac{1}{i} → 0 as i → ∞,

and so |Du|(A) = 0, where A := {x ∈ Ω\S(u) : u(x) ∈ ℝ\ω)}. Fix x_{0} ∈ Ω\S(u). If x_{0} ∈ A then, up to a set of N−dimensional Lebesgue measure zero, we may assume that ∇u(x_{0}) = 0, so that, as in the proof of Theorem 1.3(i), we have

f_{1}(x_{0}, u(x_{0}), 0) = lim_{ε→0^{+}} \frac{1}{ε^{N}} F(u(x_{0}), Q(x_{0}, ε)) ≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f(x, u(x_{0}), 0) dx = f(x_{0}, u(x_{0}), 0),

where we have used the fact that f(·, u(x_{0}), 0) is continuous. If x_{0} ∈ (Ω\S(u))\A and ∇u(x_{0}) = 0, then we proceed as above. If x_{0} ∈ (Ω\S(u))\A and ∇u(x_{0}) ≠ 0, then set

ν := \frac{∇u(x_{0})}{|∇u(x_{0})|},  ξ := ∇u(x_{0}),  w_{0}(x) := u(x_{0}) + ξ(x − x_{0}).

Find n ∈ ℕ such that ||w_{0}||_{L^{∞}(Q_{ν}(x_{0},1))} ≤ n and let i ∈ ℕ be such that u(x_{0}) ∈ K_{i,n} ∩ K_{i,n}^{∗}. Since g(x) := f(x, w_{0}(x), ξ) is continuous over w_{0}^{−1}(K_{i,n}), given δ > 0 there exists η > 0 such that g(x) ≤ g(x_{0})+δ for all x ∈ w_{0}^{−1}(K_{i,n}) with |x − x_{0}| ≤ η. Therefore, as in proof of Theorem 1.3(i), and by (1.8),

f_{1}(x_{0}, u(x_{0}), ξ) − |ξ| = lim_{ε→0^{+}} \frac{1}{ε^{N}} F(u(x_{0}), Q(x_{0}, ε))

≤ (f(x_{0}, u(x_{0}), ξ) + δ) lim sup_{ε→0^{+}} \frac{L^{N}(Q_{ν}(x_{0}, ε) ∩ w_{0}^{−1}(K_{i,n}))}{ε^{N}}

+ C(1 + |ξ|) lim sup_{ε→0^{+}} \frac{L^{N}(Q_{ν}(x_{0}, ε)\w_{0}^{−1}(K_{i,n}))}{ε^{N}}

≤ f(x_{0}, u(x_{0}), ξ) + δ, (5.1) since, by the coarea formula (see [31, Theorem 2.7.1]), and setting η_{ε}(x) := u(x_{0}) + εξ · x for x ∈ Q_{ν}, we have

lim sup_{ε→0^{+}} \frac{L^{N}(Q_{ν}(x_{0}, ε)\w_{0}^{−1}(K_{i,n}))}{ε^{N}} = 1 − lim inf_{ε→0^{+}} \frac{L^{N}(Q_{ν}(x_{0}, ε) ∩ w_{0}^{−1}(K_{i,n}))}{ε^{N}}

= 1 − lim inf_{ε→0^{+}} \frac{1}{|ξ| ε^{N}} ∫_{Q_{ν}(x_{0},ε)∩w_{0}^{−1}(K_{i,n})} |Dw_{0}(x)| dx

= 1 − lim inf_{ε→0^{+}} \frac{1}{|ξ| ε^{N}} ∫_{K_{i,n}} H^{N−1}(w_{0}^{−1}(s) ∩ Q_{ν}(x_{0}, ε)) ds.

For simplicity, we assume that x_{0} = 0, u(x_{0}) = 0, ν = e_{N}, so that

w_{0}^{−1}(s) ∩ Q_{ν}(x_{0}, ε) = \{ \begin{matrix}ε[−\frac{1}{2}, \frac{1}{2}]^{N−1} × \{s\} & if |s| ≤ \frac{ε}{2} |ξ|, \\ ∅ & if |s| > \frac{ε}{2} |ξ|; \end{matrix}

thus,

lim sup_{ε→0^{+}} \frac{L^{N}(Q_{ν}(x_{0}, ε)\w_{0}^{−1}(K_{i,n}))}{ε^{N}} = 1 − lim inf_{ε→0^{+}} \frac{L^{1}([−\frac{ε}{2} |ξ|, \frac{ε}{2} |ξ|] ∩ K_{i,n})}{|ξ|ε} = 0

where we have used the fact that u(x_{0}) is a Lebesgue point of χ_{K_{i,n}}. By letting δ → 0^{+} in (5.1) we obtain the desired inequality. □

(ii) (Cantor part)

The proof for the Cantor part is very similar to the previous one (see also the proof of Theorem 1.3(ii)), and therefore we omit the details. □

(iii) (Jump part) The proof follows the same arguments of the proof of Theorem 1.3(iii). □

§6. Proof of Theorem 1.7.

(i) (Lebesgue part) We proceed as in the proof of Theorem 1.1(i) up to (2.7), where instead of using the truncated sequence, we apply condition (1.10) to get

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇w_{k}(y)) dy − ∫_{Q} ρ(ε_{k} |w_{k}(y)|) dy.

By Fatou’s Lemma, and since ρ is continuous with ρ(0) = 0, we have

C − lim sup_{k→∞} ∫_{Q} ρ(ε_{k} |w_{k}(y)|) dy = lim inf_{k→∞} ∫_{Q}[C(1 + ε_{k} |w_{k}(y)|) − ρ(ε_{k} |w_{k}(y)|)] dy

≥ ∫_{Q} lim inf_{k→∞}[C(1 + ε_{k} |w_{k}(y)|) − ρ(ε_{k} |w_{k}(y)|)] dy = C,

and so

∫_{Q} ρ(ε_{k} |w_{k}(y)|) dy → 0 as k → ∞.

Thus

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇w_{k}(y)) dy.  (6.1)

If g(ξ) := f(x_{0}, u(x_{0}), ξ) is convex then we may apply Serrin’s Theorem A(i), which continues to hold in the vectorial case. If g is quasiconvex and q = 1 in (1.11), then we apply a result of Ambrosio and Dal Maso [4] (see also [18]) to conclude that

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

It is now sufficient to let ε → 0^{+} to obtain the desired result. When q > 1 and g is quasiconvex in (1.11), then we can apply an approximation result of Kristensen [22, Proposition 1.9] to write

f(x_{0}, u(x_{0}), ξ) = sup_{j} g_{j}(ξ),

where g_{j}(ξ) is quasiconvex, g_{j}(ξ) ≤ g_{j+1}(ξ), and g_{j}(ξ) = a_{j}|ξ| + b_{j} for |ξ| large, say |ξ| ≥ r_{j}. From (6.1) and for any fixed j, applying [4] we have

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} g_{j}(∇w_{k}(y)) dy ≥ g_{j}(∇u(x_{0})),

and then let j → ∞. □

(ii) (Cantor part) We proceed as in Theorem 1.1(ii) until (2.11), where (2.8) should now be written as

\frac{dC(u)}{d|C(u)|}(x_{0}) = a_{u}(x_{0}) ⊗ ν_{u}(x_{0}),  (6.2)

with a_{u}(x_{0}) ∈ ℝ^{d} and ν_{u}(x_{0}) ∈ S^{N−1}, and where we have used Alberti’s result [2]. By (2.11) and (1.10)

\begin{aligned}(1 + ε)\frac{dμ}{d|C(u)|}(x_{0}) + ε & ≥ lim_{k→∞} \{\frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0}, u(x_{0}), t_{k}∇w_{k}(y)) dy \\ & − \frac{1}{t_{k}} ∫_{Q_{ν}} ρ(|u(x_{0}) − u_{0k} + λ_{k}w_{k}(y)|) dy\} . \end{aligned}  (6.3)

As ρ(s) ≤ C(1 + s) for all s > 0 and for some C > 0, w_{k} converges to w_{0} in L^{1}(Q_{ν}; ℝ^{d}), u_{0k} converges to u(x_{0}), λ_{k} → 0, and t_{k} → ∞ as k → ∞, we have

\frac{1}{t_{k}} ∫_{Q_{ν}} ρ(|u(x_{0}) − u_{0k} + λ_{k} w_{k}(y)|) dy → 0

and thus

(1 + ε)\frac{dμ}{d|C(u)|}(x_{0}) + ε ≥ lim_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} f(x_{0}, u(x_{0}), t_{k}∇w_{k}(y)) dy.  (6.5)

If g is quasiconvex and q = 1 in (1.11) then we can proceed as in Theorem 1.1(ii) starting from (2.12). When g(ξ) := f(x_{0}, u(x_{0}), ξ) is convex, or g is quasiconvex q > 1 in (1.11), then we use Proposition 9.1 below or Proposition 1.9 of Kristensen [22], to deduce from (6.3) and (6.4):

(1 + ε)\frac{dμ}{d|C(u)|}(x_{0}) + ε ≥ lim_{k→∞} \frac{1}{t_{k}} ∫_{Q_{ν}} g_{j}(t_{k}∇w_{k}(y)) dy.

Proceeding as in the case q = 1, we obtain

\frac{dμ}{d|C(u)|}(x_{0}) ≥ g_{j}^{∞}(a ⊗ ν).

Since the function h_{j}(t) := g_{j}(ta ⊗ ν) is convex, by Proposition 9.1, (9.1), we have for t > 1

\frac{dμ}{d|C(u)|}(x_{0}) ≥ \frac{g_{j}(t a ⊗ ν)}{t} − \frac{g_{j}(0)}{t}, t > 1.

As f(x_{0}, u(x_{0}), 0) = sup_{j} g_{j}(0) ≤ C, letting j → ∞ yields

\frac{dμ}{d|C(u)|}(x_{0}) ≥ \frac{f(x_{0}, u(x_{0}), t a ⊗ ν)}{t} − \frac{C}{t} .

We now let t → ∞. □

Remark. Note that when (1.12) holds then

f^{∞}(x_{0}, u_{0}, ξ) = \{ \begin{matrix}+∞ & if ξ ≠ 0 \\ 0 & if ξ = 0. \end{matrix}

(iii)(Jump part) We proceed as in Theorem 1.1(iii) up to (2.14). By (2.14) and (1.10)

(1 + ε)\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) + ε ≥ lim_{k→∞} ∫_{Q_{ν}} ε_{k}f (x_{0}, \frac{1}{ε_{k}} ∇w_{k}(y)) dy

Now we continue exactly as in the proof of the Cantor part in Theorem 1.7, starting from (6.5), with the vector a in place of u^{+}(x_{0}) − u^{−}(x_{0}). □

§7. Proof of Theorems 1.8–1.10.

Proof of Theorem 1.8 (Lebesgue part). We proceed as in Theorem 1.1(i) until (2.7). If f(x_{0}, u(x_{0}), ξ) ≡ 0 for all ξ then there is nothing to prove. Thus, we assume that (1.13) and (1.14) hold, we fix ε > 0, and let δ > 0 be given by (1.13) and (1.14).

Step 1. We prove first the theorem under the additional hypothesis that there exists M > 0 such that

0 ≤ f(x, u, ξ) ≤ M(1 + ξ)  (7.1)

for all x ∈ Ω with |x − x_{0}| ≤ δ, u ∈ ℝ^{d}, and ξ ∈ ℝ^{dN}. As in [17, Proposition 2.6 ], we may find w_{k} ∈ C_{0}^{∞}(ℝ^{N}; ℝ^{d}), with w_{k} → w_{0} in L^{1}, w_{0}(x) := ∇u(x_{0}) x, such that by (2.7), (1.14), and for k sufficiently large,

\frac{dμ}{dL^{N}} (x_{0}) ≥ ∫_{Q∩\{|w_{k}|≤δ/ε_{k}\}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy ≥ C_{1} ∫_{Q∩\{|w_{k}|≤δ/ε_{k}\}} |∇w_{k}(y)| dy − C_{2};

thus there exists a constant K > 0 such that

∫_{Q∩{|w_{k}|≤δ/ε_{k}}} |∇w_{k}(y))| dy ≤ K.  (7.2)

In order to truncate w_{k}, fix s_{k} > ||w_{0}||_{L^{∞}(Q;ℝ^{d})} + 1, L_{k} > s_{k}, and construct a smooth cut–off function g_{k} : ℝ^{d} → ℝ^{d} such that

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

with |g_{k}(u)| ≤ |u| and |Dg_{k}(u)| ≤ C L_{k}/(L_{k} − s_{k}) for all u ∈ ℝ^{d}. 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 → ∞,

Lower semicontinuity and relaxation 25

because

\begin{matrix}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{matrix}  (7.3)

Moreover

\begin{matrix}∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy = ∫_{E_{k}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy \\ + ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u(x_{0}), 0) dy + ∫_{E_{k}^{−}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy. \end{matrix}  (7.4)

We claim that the last two integrals are infinitesimal as k → ∞. Indeed, by (7.1) and (7.3), 0 ≤ ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u(x_{0}), 0) dy ≤ ML^{N}(E_{k}^{+}) → 0,

while, from (7.1) and the coarea formula,

\begin{matrix}∫_{E_{k}^{−}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy ≤ M ∫_{E_{k}^{−}} (1 + |∇g_{k}(w_{k})∇w_{k}|) dy \\ ≤ M (L^{N}(E_{k}^{−}) + \frac{CL_{k}}{L_{k} − s_{k}} ∫_{E_{k}^{−}} |∇w_{k}| dy) \\ = M (L^{N}(E_{k}^{−}) + \frac{CL_{k}}{L_{k} − s_{k}} ∫_{s_{k}}^{L_{k}} H^{N−1}(\{y ∈ Q : |w_{k}(y)| = t\}) dt) . \end{matrix}  (7.5)

By Theorem 7.10 of [28] and (7.2), for L^{1} a.e. L ≤ δ/ε_{k} we have

lim_{s→L} \frac{1}{L − s} ∫_{s}^{L} H^{N−1}(\{y ∈ Q : |w_{k}(y)| = t\}) dt = H^{N−1}(\{y ∈ Q : |w_{k}(y)| = L\}).  (7.6)

Moreover, by Lemma 2.6 in [17] and (7.2), for any 0 < α < β < δ/ε_{k} we obtain

essinf_{L∈(α,β)} LH^{N−1}(\{y ∈ Q : |w_{k}(y)| = L\}) ≤ \frac{K}{log(β/α)}.

Set α := δ/ε_{k}^{1/4} and β := δ/ε_{k}^{1/2}, and find L_{k} ∈ (δ/ε_{k}^{1/4}, δ/ε_{k}^{1/2}) such that (7.6) holds and

L_{k}H^{N−1}(\{y ∈ Q : |w_{k}(y)| = L_{k}\}) ≤ \frac{2K}{log(1/ε_{k}^{1/4})}.

Choose s_{k} ≥ L_{k}/2 so that

\frac{L_{k}}{L_{k} − s_{k}} ∫_{s_{k}}^{L_{k}} H^{N−1}(\{y ∈ Q : |w_{k}(y)| = t\}) dt ≤ \frac{2K}{log (1/ε_{k}^{1/4})} + \frac{1}{k} .

Then the integral in the right hand side of (7.5) approaches zero as k → ∞, and so, from (7.4),

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy.

Since ε_{k} → 0, by (1.13) we obtain

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇v_{k}(y)) dy.

We can now continue as in the proof of Theorem 1.7(i), and the result is established if f satisfies (7.1).

Step 2. In the general case, let ψ ∈ C_{0}^{∞}(ℝ^{d}; ℝ) be a cut–off function, with 0 ≤ ψ ≤ 1, and such that ψ ≡ 1 on B(u(x_{0}), δ/2), ψ ≡ 0 outside B(u(x_{0}), δ). From (2.7)

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

and by (1.13),

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} ψ(u(x_{0}) + ε_{k}w_{k}(y))f(x_{0}, u(x_{0}), ∇w_{k}(y)) dy.

If f(x_{0}, u(x_{0}), ·) is convex (resp. quasiconvex with q > 1 in (1.11)), we use Proposition 9.1 (resp. Proposition 1.9 of Kristensen [22]) to approximate f(x_{0}, u(x_{0}), ξ) by an increasing sequence g_{j}(ξ) of convex (resp. quasiconvex) functions such that

0 ≤ g_{j}(ξ) ≤ C_{j}(|ξ| + 1).  (7.7)

If f(x_{0}, u(x_{0}), ·) is quasiconvex with q = 1 in (1.11), we simply take g_{j}(ξ) ≡ f(x_{0}, u(x_{0}), ξ) for all j. For any fixed j

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} ψ(u(x_{0}) + ε_{k}w_{k}(y))g_{j}(∇w_{k}(y)) dy,

and f_{j}(x, u, ξ) := ψ(u)g_{j}(ξ) satisfies (7.1). Moreover, by (7.7), (7.2) continues to hold, provided we replace δ with δ/2. Finally, (1.13) is still satisfied at the point (x_{0}, u(x_{0})). Therefore we can apply the first part of the proof to get

\frac{dμ}{dL^{N}} (x_{0}) ≥ ψ(u(x_{0}))g_{j}(∇u(x_{0})) = g_{j}(∇u(x_{0})).

It suffices to take the supremum in j. □

Proof of Theorem 1.8 (Cantor part). We proceed as in Theorem 1.1(ii) until (2.11). We can now truncate the sequence w_{k} using an argument similar to that of the Lebesgue part of Theorem 1.8 (note that the only property of w_{0}(y) which has been used is the fact that it is bounded), and then continue as in the Cantor part of Theorem 1.7, using (1.13) in place of (1.11). We omit the details. □

Proof of Theorem 1.9. The proofs of Theorem 1.3(i)–(ii) and of the first part of Theorem 1.6 continue to hold. We observe that in Theorem 1.3(ii), since a ⊗ ν has rank one, the function g(t) = f^{∞}(x, u, t a ⊗ ν) is convex and thus we can still use Proposition 9.1, (9.1).

If f^{∞} = f^{∞}(x, ξ) then the proof of Theorem 1.3(iii) is still valid with some obvious modifications. □

Proof of Theorem 1.10. We proceed as in Theorem 1.1(iii) until (2.14). Fix ε > 0 and let k be so large that ε_{k} < min{δ, 1/L}, where δ and L are provided by (1.15)–(1.16). Then by (1.16) and (1.15), in this order,

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) = lim_{k→∞} ∫_{Q_{ν}} ε_{k}f (x_{0} + ε_{k}y, w_{k}(y), \frac{1}{ε_{k}} ∇w_{k}(y)) dy

≥ \frac{1}{1 + ε} lim inf_{k→∞} ∫_{Q_{ν}} f^{∞}(x_{0} + ε_{k}y, w_{k}(y), ∇w_{k}(y)) dy − \frac{ε}{1 + ε}

≥ \frac{1}{(1 + ε)^{2}} lim inf_{k→∞} ∫_{Q_{ν}} f^{∞}(x_{0}, w_{k}(y), ∇w_{k}(y)) dy − \frac{ε}{(1 + ε)^{2}} − \frac{ε}{1 + ε}

≥ \frac{1}{(1 + ε)^{2}} lim sup_{k→∞} ∫_{Q_{ν}} f^{∞}(x_{0}, v_{k}(y), ∇v_{k}(y)) dy − \frac{ε}{(1 + ε)^{2}} − \frac{ε}{1 + ε},

(7.8) where we have used Lemma 2.6 and Remark 2.7(1) of [7] to obtain a new sequence v_{k} ∈ W^{1,1}(Q_{ν}; ℝ^{d}) which converges to w_{0} in L^{1}(Q_{ν}; ℝ^{d}) and such that v|_{∂Q_{ν}} = w_{0}. It now follows from (7.8) and the definition of the function h in (1.18) that

\frac{dμ}{dH^{N−1}⌊S(u)}(x_{0}) ≥ \frac{1}{(1 + ε)^{2}} h(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν) − \frac{ε}{(1 + ε)^{2}} − \frac{ε}{1 + ε},

and we obtain the first part of the theorem upon letting ε → 0^{+}.

We first prove the reverse inequality to (1.17) under the additional coercivity assumption that there exists C > 0 such that

f(x, u, ξ) ≥ C|ξ|  for all (x, u, ξ) ∈ Ω × ℝ^{d} × ℝ^{dN}.  (7.9)

Fix ε_{0} > 0, and define

u_{0}(x) := \{ \begin{matrix}u^{+}(x_{0}) & if y · ν > 0, \\ u^{−}(x_{0}) & if y · ν ≤ 0. \end{matrix}

By Lemma 4.1.3 and (3.17) of [7] for H^{N−1} a.e. x_{0} ∈ S(u)

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0})

= lim sup_{ε→0^{+}} \frac{1}{ε^{N−1}} inf \{∫_{Q_{ν}(x_{0},ε)}f(x, v(x), ∇v(y))dy : v ∈ W^{1,1}(Q_{ν}(x_{0}, ε); ℝ^{d}), v|_{∂Q_{ν}(x_{0},ε)} = u_{0}(· − x_{0})\}

= lim sup_{ε→0^{+}} inf \{∫_{Q_{ν}} ε f (x_{0} + ε y, w(y), \frac{1}{ε}∇w(y)) dy : w ∈ W^{1,1}(Q_{ν}; ℝ^{d}), w|_{∂Q_{ν}} = u_{0}\}

≤ \frac{1}{1 − ε_{0}} lim sup_{ε→0^{+}} inf \{∫_{Q_{ν}} f^{∞}(x_{0} + ε y, w(y), ∇w(y)) dy : w ∈ W^{1,1}(Q_{ν}; ℝ^{d}), w|_{∂Q_{ν}} = u_{0}\} + \frac{ε_{0}}{1 − ε_{0}}

≤ \frac{1}{1 − ε_{0}} lim sup_{ε→0^{+}} ∫_{Q_{ν}} f^{∞}(x_{0} + ε y, w_{1}(y), ∇w_{1}(y)) dy + \frac{ε_{0}}{1 − ε_{0}} ,

for any w_{1} ∈ W^{1,1}(Q_{ν}; ℝ^{d}), with w_{1}|_{∂Q_{ν}} = u_{0}, and where we have used (1.16)′. We now take w_{1} in the previous inequality such that

∫_{Q_{ν}} f^{∞}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy ≤ h(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) + ε_{0}.  (7.10)

By (3.5), (3.6), and Lebesgue Dominated Convergence Theorem,

lim_{λ→∞} ∫_{Q_{ν}} f_{λ}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy = ∫_{Q_{ν}} f^{∞}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy,  (7.11)

where the Yosida transforms f_{λ} were introduced in (3.4); thus, for fixed λ sufficiently large, by (7.10), and by (7.11)

∫_{Q_{ν}} f_{λ}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy ≤ h(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) + 2ε_{0}.  (7.12)

Consequently, also from (3.5) and (3.8),

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ \frac{1}{1 − ε_{0}} lim sup_{ε→0^{+}} ∫_{Q_{ν}} f_{λ}(x_{0} + ε y, w_{1}(y), ∇w_{1}(y)) dy + \frac{ε_{0}}{1 − ε_{0}}
≤ \frac{1}{1 − ε_{0}} lim sup_{ε→0^{+}} (∫_{Q_{ν}} f_{λ}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy + ε λ) + \frac{ε_{0}}{1 − ε_{0}}
≤ \frac{1}{1 − ε_{0}} (h(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) + 2ε_{0}) + \frac{ε_{0}}{1 − ε_{0}}

by (7.12). Letting ε_{0} → 0^{+} in the previous inequality yields the desired result when (7.9) holds. In the general case, it suffices to consider the family of perturbed energy densities

f_{ρ}(x, u, ξ) := f(x, u, ξ) + ρ |ξ|, ρ > 0.

Then, since F ≤ F_{ρ}, it follows that

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ \frac{dF_{ρ}(u, ·)}{dH^{N−1}⌊S(u)}(x_{0})

≤ inf {∫_{Q_{ν}} f^{∞}(x_{0}, w(y), ∇w(y)) dy + ρ ∫_{Q_{ν}} |∇w(y))| dy : w ∈ W^{1,1}(Q_{ν}; ℝ^{d}), w|_{∂Q_{ν}(x_{0},ε)} = u_{0}}

≤ ∫_{Q_{ν}} f^{∞}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy + ρ ∫_{Q_{ν}} |∇w_{1}(y))| dy

for any fixed w_{1} ∈ W^{1,1}(Q_{ν}; ℝ^{d}), such that w_{1}|_{∂Q_{ν}(x_{0},ε)} = u_{0}. Letting ρ → 0^{+} yields

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ ∫_{Q_{ν}} f^{∞}(x_{0}, w_{1}(y), ∇w_{1}(y)) dy,

and since w_{1} is arbitrary, by taking the infimum over all functions w_{1} we obtain

\frac{dF(u, ·)}{dH^{N−1}⌊S(u)}(x_{0}) ≤ h(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})).

□

§8. Further results.

As in Theorems C and D of Dal Maso [9], some of our results continue to hold if the regularity conditions on the integrand f are required everywhere except at most on “small” sets. In order to establish the main result of this section, Theorem 8.5, we prove first three lemmas.

Lemma 8.1. Let N_{0} be a Borel subset of Ω × ℝ, with H^{N}(N_{0}) = 0, let u ∈ BV (Ω; ℝ), and define

A := {x ∈ Ω\(S(u) ∪ M(u)) : (x, u(x)) ∈ N_{0}}, B := {x ∈ M(u) : (x, u(x)) ∈ N_{0}},

where M(u) is the support of C(u). Then

(i) L^{N}(A) = 0; (ii) |C(u)|(B) = 0.

Proof. (i) If L^{N}(A) > 0, then by Corollary 1 in Sec. 2.4.1 of [15] we obtain

0 = H^{N}(N_{0}) ≥ H^{N}({(x, u(x)) : x ∈ A}) ≥ H^{N}(A) = L^{N}(A) > 0,

which is clearly a contradiction.

(ii) Let

\begin{aligned}G_{+} & := \{(x, s) ∈ Ω × ℝ : s < u^{+}(x)\}, \\ G_{±} & := \{(x, s) ∈ Ω × ℝ : u^{−}(x) ≤ s ≤ u^{+}(x)\}. \end{aligned}

Then χ_{G_{+}} ∈ BV_{loc}(Ω × ℝ) (see [25]), and for any Borel set K ⊂ Ω × ℝ and D ⊂ Ω we have

|Dχ_{G_{+}}|(K) = H^{N}(K ∩ G_{±}),  ∫_{D×ℝ} |Dχ_{G_{+}}| = ∫_{D} |ζ(u)|,  (8.1)

where ζ(u) := (Du, −L^{N}) (see [9], Lemma 2.2). Take K := B × ℝ. If x ∈ M(u) then u^{+}(x) = u^{−}(x), and thus

|Dχ_{G_{+}}|(K) = H^{N}({(x, u(x)) : x ∈ B}) ≤ H^{N}(N_{0}) = 0.

In turn, by (8.1), ∫_{B} |ζ(u)| = 0. Since |ζ(u)| coincides with |Du| on M(u) (recall that L^{N}(M(u)) = 0) it follows that |Du|(B) = 0. □

The following Lemma is a generalization of Theorem 3 in Sec. 2.4.3 of [15] Lemma 8.2. Let h ∈ L_{loc}^{1}(Ω; ℝ), let μ be a positive Radon measure, and define

B_{0} := \{x_{0} ∈ Ω : lim sup_{ε→0^{+}} \frac{1}{μ(Q_{ν}(x_{0}, ε))} ∫_{Q_{ν}(x_{0},ε)} |h(x)| dx > 0\} ,  (8.2)

where in the limsup we consider only those ε > 0 such that μ(∂Q_{ν}(x, ε)) = 0. Then μ_{s}(B_{0}) = 0, where

μ := \frac{dμ}{dL^{N}} L^{N} + μ_{s}.

Proof. Without loss of generality, we can assume that Ω is bounded, h ≥ 0 and h ∈ L^{1}(Ω; ℝ). Given η > 0 there exists δ > 0 such that

∫_{U} h(x) dx < η whenever L^{N}(U) < δ.

Let E be the support of the measure μ_{s}. Since μ_{s}(X) = μ_{s}(X ∩ E), we consider B′_{0} = B_{0} ∩ E. Clearly L^{N}(B′_{0}) = 0, and

B′_{0} = ⋃_{r∈ℚ^{+}} B_{r}, where B_{r} := \{x_{0} ∈ E : lim sup_{ε→0^{+}} \frac{1}{μ(Q_{ν}(x_{0}, ε))} ∫_{Q_{ν}(x_{0},ε)} h(x) dx > r\} .

We claim that μ(B_{r}) = 0, from what will follow that μ(B′_{0}) = 0. Let U be an open set such that B_{r} ⊂ U and L^{N}(U) < δ. Fix ρ > 0 and consider

F_{1}^{ρ} : = \{Q_{ν}(x, ε) : x ∈ B_{r}, 0 < ε < ρ, μ(∂Q_{ν}(x, ε)) = 0, \overline{Q_{ν}(x, ε)} ⊂ U, ∫_{Q_{ν}(x,ε)} h(y)dy > rμ(Q_{ν}(x, ε))\}

and consider the Borel sets

U^{ρ} := ⋃ {Q_{ν}(x, ε) : Q_{ν}(x, ε) ∈ F_{1}^{ρ}} , U_{0} := ⋂_{ρ>0} U^{ρ}.

Since B_{r} ⊂ U_{0}, it suffices to prove that μ(U_{0}) = 0. Fix a compact set K ⊂ U_{0}, ρ_{0} > 0, and let

F_{2}^{ρ_{0}} : = \{Q_{ν}(x, ε) : x ∈ B_{r}, 0 < ε < ρ, μ(∂Q_{ν}(x, ε)) = 0, \overline{Q_{ν}(x, ε)} ⊂ U^{ρ_{0}}\K\} .

Then U^{ρ_{0}} admits a fine covering

U^{ρ_{0}} := ( ⋃_{Q_{ν}(x,ε)∈F_{1}^{ρ_{0}}} Q_{ν}(x, ε)) ∪ ( ⋃_{Q_{ν}(x,ε)∈F_{2}^{ρ_{0}}} Q_{ν}(x, ε)) ,

and by Morse’s version of Besicovitch’s Covering Theorem (see [27], Theorem 5.11) we may find a subcovering of U^{ρ_{0}} such that

U^{ρ_{0}} := (⋃_{i∈I} Q_{i}) ∪ (⋃_{j∈I} Q_{j}) ∪ N, K ⊂ (⋃_{i∈I} \overline{Q_{i}}) ∪ N,

I and J are countable, Q_{i} ∈ F_{1}^{ρ_{0}}, Q_{j} ∈ F_{2}^{ρ_{0}}, the sets \overline{Q_{i}} and \overline{Q_{j}} are mutually disjoint, and μ(N) = 0. Then

η > ∫_{U} h(x) dx ≥ ∑_{i∈I} ∫_{Q_{i}} h(x) dx ≥ r ∑_{i∈I} μ(Q_{i}) = r ∑_{i∈I} μ(\overline{Q_{i}}) ≥ rμ(K).

By letting η → 0 we obtain μ(K) = 0, and by the inner regularity of μ we conclude that μ(U_{0}) = 0. □

Remark 8.3. Since for μ_{s} a.e. x_{0} ∈ Ω

lim_{ε→0^{+}} \frac{μ(Q_{ν}(x_{0}, ε))}{ε^{N}} = ∞,

it is clear that if x_{0} ∈ B_{0} then μ_{s} a.e. x_{0} is not a Lebesgue point for |h|, otherwise

lim sup_{ε→0^{+}} \frac{1}{μ(Q_{ν}(x_{0}, ε))} ∫_{Q_{ν}(x_{0},ε)} |h(x)| dx = lim sup_{ε→0^{+}} \frac{ε^{N}}{μ(Q_{ν}(x_{0}, ε))}|h(x_{0})| = 0.

Using Lemma 8.2, it is possible in some cases to weaken (1.8) in Proposition 2.1. Indeed, assume that for L^{1} a.e. u ∈ ℝ

f(·, u, 0) ∈ L_{loc}^{1}(Ω; ℝ).  (8.3)

Then there exists a countable set R_{0} := {r_{j}}_{j}, dense in ℝ, such that f(·, r_{j}, 0) ∈ L_{loc}^{1}(Ω; ℝ) for all j. Let Ω_{j} be the set of Lebesgue points of f(·, r_{j}, 0) and set A_{j} := Ω\Ω_{j}. Then

L^{N}(∪_{j=1}^{∞}A_{j}) = 0.

Let B_{j} be the set of points corresponding to the set B_{0} in Lemma 8.2 when h := f(·, r_{j}, 0) and μ := |Du|. Then

|D_{s}u|(∪_{j=1}^{∞}B_{j}) = 0.

Lemma 8.4. Proposition 2.1 is still valid provided we replace (1.8) by (8.3), and we take

x_{0} ∈ Ω\ ( ⋃_{i,j=1}^{∞} A_{j} ∪ B_{i}) , t_{k} := \{ \begin{matrix}1 & if x_{0} ∈ ⋂_{j=1}^{∞} Ω_{j} \\ |Du|(Q_{ν}(x_{0}, ε_{k}))/ε_{k}^{N} & otherwise, \end{matrix}

where |Du|(∂Q_{ν}(x_{0}, ε_{k})) = 0, t_{k} → T ∈ (0, ∞], and u_{0} + α_{1}, u_{0} + α_{2} ∈ R_{0}.

Proof. The only change is in (2.2). Considering first the case where x_{0} ∈ Ω\(∪_{j=1}^{∞}A_{j}) = ∩_{j=1}^{∞}Ω_{j}, then t_{k} = 1 and (2.2) becomes

\begin{aligned}0 ≤ & ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u_{0} + α_{2}, 0) dy = \frac{1}{ε_{k}^{N}} ∫_{Q_{ν}(x_{0},ε_{k})∩(x_{0}+ε_{k}E_{k}^{+})} f(x, u_{0} + α_{2}, 0) dx \\ & ≤ \frac{1}{ε_{k}^{N}} ∫_{Q_{ν}(x_{0},ε_{k})} |f(x, u_{0} + α_{2}, 0) − f(x_{0}, u_{0} + α_{2}, 0)| dx + f(x_{0}, u_{0} + α_{2}, 0)L^{N}(E_{k}^{+}). \end{aligned}

Since x_{0} is a Lebesgue point for f(x, u_{0} + α_{2}, 0) (recall that u_{0} + α_{2} ∈ R_{0} and that x_{0} ∈ ∩_{j=1}^{∞}Ω_{j}) the first integral on the right hand side approaches zero as k → ∞. Moreover L^{N}(E_{k}^{+}) → 0 as before.

If x_{0} ∈ Ω\(∪_{i=1}^{∞}B_{i}) and x_{0} ∉ ∩_{j=1}^{∞}Ω_{j} then (2.2) may be estimated as follows:

0 ≤ lim_{k→∞} \frac{1}{t_{k}} ∫_{E_{k}^{+}} f(x_{0} + ε_{k}y, u_{0} + α_{2}, 0) dy ≤ lim_{k→∞} \frac{1}{|Du|(Q(x_{0}, ε_{k}))} ∫_{Q(x_{0},ε_{k})} f(x, u_{0} + α_{2}, 0) dx = 0,

where we have used the fact that t_{k} = |Du|(Q_{ν}(x_{0}, ε_{k}))/ε_{k}^{N} and the definition of B_{j} as in (8.3), with h := f(·, r_{j}, 0). □

We are now ready to state the main result of this section.

Theorem 8.5. Assume that f : Ω × ℝ × ℝ^{N} → [0, ∞) is a Borel integrand, f(x, u, ·) is convex in ℝ^{N}, and f satisfies (8.3). Suppose also that (1.7) holds for all (x_{0}, u_{0}) ∈ (Ω × ℝ)\N_{0}, where N_{0} is a Borel subset of Ω × ℝ. Let u ∈ BV_{loc}(Ω; ℝ), and let {u_{n}} be a sequence of functions in W_{loc}^{1,1}(Ω; ℝ) converging to u in L_{loc}^{1}(Ω; ℝ).

(i) If either H^{N}(N_{0}) = 0 or N_{0} = M_{0} × ℝ with L^{N}(M_{0}) = 0 then

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

(ii) If either H^{N}(N_{0}) = 0 or N_{0} = M_{0} × ℝ with H^{N−1}(M_{0}) < ∞ then

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

(iii) If N_{0} = M_{0} × ℝ and either H^{N}(N_{0}) = 0 or H^{N−1}(M_{0}) = 0 and we assume that for all (x_{0}, u_{0}) ∈ (Ω × ℝ)\N_{0} and ε > 0 there exists δ > 0 such that

f(x_{0}, u, ξ) − f(x, u, ξ) ≤ ε(1 + f(x, u, ξ))  (8.4)

for all (x, u) ∈ Ω × ℝ with |x − x_{0}| + |u − u_{0}| ≤ δ and for all ξ ∈ ℝ^{N}, then

∫_{S(u)∩Ω} (∫_{u^{−}(x)}^{u^{+}(x)} f^{∞}(x, s, ν_{u}) ds) dH^{N−1}(x) ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx.

Proof. (i) We proceed as in Theorem 1.1(i) starting from (2.4). If H^{N}(N_{0}) = 0 then in (2.6) we take x_{0} ∈ Ω\A, where A is the set given in Lemma 8.1(i), otherwise take x_{0} ∈ Ω\M_{0}. Using the notation introduced in Lemma 8.1 and thereafter, we may assume, in addition, that x_{0} is also a Lebesgue point for all the functions f(·, r_{j}, 0)’s, precisely

x_{0} ∈ Ω\ (∪_{j=1}^{∞}A_{j}) .

We can now continue with the same argument as in the proof of the Lebesgue part in Theorem 1.1, except that we invoke Lemma 8.4 instead of Proposition 2.1 to justify the truncation step.

(ii) If H^{N}(N_{0}) = 0 then take x_{0} ∈ Ω\B, where B is the set given in Lemma 8.1(ii), otherwise take x_{0} ∈ Ω\M_{0}. As before, let B_{j} be the set introduced in Lemma 8.2 and corresponding to f(·, r_{j}, 0) ∈ L_{loc}^{1}(Ω; ℝ) (see (8.2)). Since

|C(u)|(∪_{j=1}^{∞}B_{j}) = 0,

we may assume that x_{0} ∈ Ω\(∪_{j=1}^{∞}B_{j}). Now we continue as in Theorem 1.1(ii), using Lemma 8.4 in place of Proposition 2.1, but now in order to apply Lemma 2.5 of Ambrosio and Dal Maso [4] we first need to approximate g(ξ) := f(x_{0}, u(x_{0}), ξ) from below by a nondecreasing sequence of convex functions which grow at most linearly. This can be done by virtue of Proposition 9.1.

(iii) Since N_{0} = M_{0} × ℝ and H^{N} = H^{N−1} × L^{1} on S(u) × ℝ, it follows that H^{N−1}(M_{0} ∩ S(u)) = 0. Moreover, by Lemma 8.2,

H^{N−1} (S(u) ∩ (∪_{j=1}^{∞}B_{j})) = 0.

Take

x_{0} ∈ S(u)\ (M_{0} ∪ ⋃_{j=1}^{∞} B_{j}) .

We pursue the proof of the jump part as in Theorem 1.1(iii), using Lemma 8.4 instead of Proposition 2.1, but now in order to apply the density result of Ambrosio we first need to approximate h(u, ξ) := f(x_{0}, u, ξ) from below by a nondecreasing sequence of continuous functions which grow at most linearly. For this purpose we invoke Proposition 9.3 below. □

Remark 8.6. (i) The hypothesis placed on part (iii) above, i.e. in the jump part, ensuring that the set N_{0} is of the form N_{0} = M_{0} × ℝ, is used heavily to apply the compactness argument leading to (2.15).

(ii) Theorems 1.3, 1.7, 1.8 may be improved similarly to Theorem 8.5 versus Theorem 1.1. We leave this to the interested reader.

(iii) As in Theorem D of Dal Maso [9], in the special case where f(x, u, 0) ≡ 0 condition (1.9) can be weakened as follows:

Assume that there exists a set P_{0} ⊂ ℝ with L^{1}(P_{0}) = 0 such that for all x_{0} ∈ Ω and ε > 0 there exists δ > 0 such that

|f(x_{0}, u, ξ) − f(x, u, ξ)| ≤ ε(1 + f(x, u, ξ))  (1.9)′

for all x ∈ Ω with |x − x_{0}| ≤ δ and for all (u, ξ) ∈ (ℝ\P_{0}) × ℝ^{N}.

The proof of Theorem 1.5 should now be modified accordingly, using the fact that, if A := {x ∈ Ω\S(u) : u(x) ∈ P_{0}} then ∇u ≡ 0 for L^{N} a.e. x ∈ A and |C(u)| = 0 a.e. x ∈ A. We omit the details.

§9. Approximation of convex functions.

Let g : ℝ^{N} → [0, ∞) be a convex function. Then

t ↦ \frac{g(t ξ) − g(0)}{t}

is increasing, and we define the recession function

g^{∞}(ξ) := lim_{t→∞} \frac{g(t ξ)}{t} = sup_{t>0} \frac{g(t ξ) − g(0)}{t}.

Proposition 9.1. Let g : ℝ^{N} → [0, ∞) be a convex function. Then

\frac{g(t ξ)}{t} ≤ g^{∞}(ξ) + \frac{g(0)}{t}  (9.1)

for t > 0, and there exists an increasing sequence {g_{j}}_{j} of nonnegative convex functions such that: (i) g(ξ) = sup_{j} g_{j}(ξ) for all ξ ∈ ℝ^{N}; (ii) g^{∞}(ξ) = sup_{j} g_{j}^{∞}(ξ) for all ξ ∈ ℝ^{N}; (iii) g_{j} is Lipschitz continuous with Lipschitz constant j; (iv) if g(ξ) ≥ C(|ξ| − 1) for some C > 0, then g_{j} satisfies the same growth condition for j ≥ [C] + 1.

Proof. Inequality (9.1) follows immediately from the definition of the g^{∞}. Define

g_{j}(ξ) := sup_{|ξ^{∗}|≤j}(ξ^{∗} · ξ − g^{∗}(ξ^{∗})),

where g^{∗} is the Young–Fenchel conjugate of g. Since g is convex, g = g^{∗∗}; hence g(ξ) = sup_{j} g_{j}(ξ). Also, g_{j} are convex and

g_{j}(ξ) ≥ −g^{∗}(0) ≥ inf g ≥ 0.

This proves (i).

Since g(ξ) ≥ g_{j}(ξ) for each j, it follows that g^{∞}(ξ) ≥ sup_{j} g_{j}^{∞}(ξ). Conversely, and by (9.1),

\frac{g(t ξ)}{t} = sup_{j} \frac{g_{j}(t ξ)}{t} = sup_{j} [\frac{g_{j}(t ξ) − g_{j}(0)}{t} + \frac{g_{j}(0)}{t}] ≤ sup_{j} g_{j}^{∞}(ξ) + sup_{j} \frac{g_{j}(0)}{t} = sup_{j} g_{j}^{∞}(ξ) + \frac{g(0)}{t}.

Letting t → ∞ we conclude that g^{∞}(t) ≤ sup_{j} g_{j}^{∞}(ξ).

Property (iii) is straightforward. We prove (iv). If g(ξ) ≥ C(|ξ| − 1) then g^{∗}(ξ^{∗}) ≤ [C(| · | − 1)]^{∗}(ξ^{∗}); hence

g_{j}(ξ) = sup_{|ξ^{∗}|≤j}(ξ^{∗} · ξ − g^{∗}(ξ^{∗})) ≥ sup_{|ξ^{∗}|≤j} (ξ^{∗} · ξ − [C(| · | − 1)]^{∗}(ξ^{∗})) .

Since

[C(| · | − 1)]^{∗}(ξ^{∗}) = \{ \begin{matrix}C & if |ξ^{∗}| ≤ C, \\ ∞ & otherwise, \end{matrix}

we conclude that if j ≥ [C] + 1 then

g_{j}(ξ) ≥ sup_{|ξ^{∗}|≤C}(ξ^{∗} · ξ − C) = C(|ξ| − 1).

□

The last proposition of this section uses a corollary of Lindelöf Theorem which allows us to select a countable collection of functions yielding the supremum function of a noncountable family. For convenience, we include the proof below.

Lemma 9.2. Let X be a σ-compact metric space, let G ⊂ C(X; ℝ), and let f(x) := sup_{g∈G} g(x), for all x ∈ X. Then there exists a countable collection {g_{n}}_{n∈ℕ} ⊂ G, such that

f(x) = sup_{n} g_{n}(x) for all x ∈ X.

Proof. It is clear that f is lower semicontinuous. Therefore, for every x ∈ X there exist 0 < r(x) < 1 and g_{x}^{1} ∈ G such that

f(y) ≥ f(x) − \frac{1}{2} for all y ∈ B(x, r^{1}(x)), g_{x}^{1}(x) ≥ f(x) − \frac{1}{2}.

Let ρ^{1}(x) < r(x) be such that

|g_{x}^{1}(y) − g_{x}^{1}(y′)| ≤ \frac{1}{2} for all y, y′ ∈ B(x, ρ^{1}(x)).

Since {B(x, ρ^{1}(x))}_{x∈X} is an open covering of X, by Lindelöf Theorem we may extract a countable subcovering \{B(x_{n}^{1}, ρ_{n}^{1})\}. Recursively, we may find an open covering of X, \{B(x, ρ^{k}(x))\}_{x∈X}, ρ^{k}(x) < \frac{1}{2^{k}}, and functions g_{x}^{k} ∈ G, such that for all x ∈ X

f(y) ≥ f(x)− \frac{1}{2^{k}} for all y ∈ B(x, ρ^{k}(x)), g_{x}^{k}(x) ≥ f(x)− \frac{1}{2^{k}} , |g_{x}^{k}(y)−g_{x}^{k}(y′)| ≤ \frac{1}{2^{k}} for all y, y′ ∈ B(x, ρ^{k}(x)).

Again by Lindelöf Theorem, X is covered by a countable family {B(x_{n}^{k}, ρ_{n}^{k})}. We claim that

f(x) = sup_{m,k} g_{x_{m}^{k}}^{k}(x).

Let x ∈ X, k ∈ ℕ, and choose n ∈ ℕ such that x ∈ B(x_{n}^{k}, ρ_{n}^{k}). Then

f(x) ≥ g_{x_{n}^{k}}^{k}(x) ≥ g_{x_{n}^{k}}^{k}(x_{n}^{k}) − \frac{1}{2^{k}} ≥ f(x_{n}^{k}) − \frac{2}{2^{k}} ≥ inf_{B(x; \frac{1}{2^{k}})} f − \frac{2}{2^{k}} .

As f is lower semicontinuous, lim inf_{ε→0} inf_{B(x,ε)} f = f(x), and we conclude that

f(x) ≤ lim inf_{k→∞} inf_{B(x; \frac{1}{2^{k}})} f − \frac{2}{2^{k}} ≤ lim inf_{k→∞} g_{x_{n}^{k}}^{k}(x) ≤ sup_{m,k} g_{x_{m}^{k}}^{k}(x) ≤ f(x).

□

Proposition 9.3. ( [10]) Let A be an open set of ℝ^{N} and let h : A × ℝ × ℝ^{N} → [0, ∞) be a function convex in the ξ variable, and such that for every (x_{0}, u_{0}) ∈ A × ℝ and ε > 0 there exists δ > 0 such that

h(x, u, ξ) ≥ (1 − ε)h(x_{0}, u_{0}, ξ)  (9.2)

for all (x, u) ∈ A × ℝ with |x − x_{0}| ≤ δ, |u − u_{0}| ≤ δ and for all ξ ∈ ℝ^{N}. Then there exists an increasing sequence {h_{j}}_{j} of nonnegative continuous functions, convex in the ξ variable, satisfying (9.2) and such that:

(i) h(x, u, ξ) = sup_{j} h_{j}(x, u, ξ) for all (x, u, ξ) ∈ A × ℝ × ℝ^{N};

(ii) h^{∞}(x, u, ξ) = sup_{j} h_{j}^{∞}(x, u, ξ) for all (x, u, ξ) ∈ A × ℝ × ℝ^{N};

(iii) h_{j}(x, u, ξ) ≤ C_{j}(|ξ| + 1) for all (x, u, ξ) ∈ A × ℝ × ℝ^{N}and for some C_{j} > 0; (iv) for every (x_{0}, u_{0}) ∈ A × ℝ and ε > 0 there exists δ_{j} > 0 such that

h_{j}(x, u, ξ) ≥ (1 − ε)h_{j}(x_{0}, u, ξ), h_{j}(x, u, ξ) ≥ (1 − ε)h_{j}(x, u_{0}, ξ)

for all (x, u) ∈ A × ℝ with |x − x_{0}| ≤ δ_{j}, |u − u_{0}| ≤ δ_{j}, and for all ξ ∈ ℝ^{N}.

Proof. Let G be the class of all continuous functions g : A × ℝ × ℝ^{N} → [0, ∞), convex in the ξ variable, and such that:

(1) g(x, u, ξ) ≤ h(x, u, ξ) for all (x, u, ξ) ∈ A × ℝ × ℝ^{N};

(2) for every (x_{0}, u_{0}) ∈ A × ℝ and ε > 0 there exists δ > 0 such that

g(x, u, ξ) ≥ (1 − ε)g(x_{0}, u_{0}, ξ)

and

g(x, u, ξ) ≥ (1 − ε)g(x_{0}, u, ξ), g(x, u, ξ) ≥ (1 − ε)g(x, u_{0}, ξ)

for all (x, u) ∈ A × ℝ with |x − x_{0}| ≤ δ, |u − u_{0}| ≤ δ, and for all ξ ∈ ℝ^{N};

(3) there exists C > 0 such that

g(x, u, ξ) ≤ C(|ξ| + 1) for all (x, u, ξ) ∈ A × ℝ × ℝ^{N}.

Clearly G ≠ ∅, as 0 ∈ G. Following [10], we claim that

h(x_{0}, u_{0}, ξ) = sup_{g∈G} g(x_{0}, u_{0}, ξ)  for all (x_{0}, u_{0}, ξ) ∈ A × ℝ × ℝ^{N}.  (9.3)

By definition of G, it follows immediately that h ≥ sup_{g∈G} g. Conversely, fix (x_{0}, u_{0}) ∈ A × ℝ, ε > 0, and let δ be such that (9.2) is satisfied. Consider two cut–off functions φ ∈ C_{0}^{∞}(A), ψ ∈ C_{0}^{∞}(ℝ), with 0 ≤ φ ≤ 1, 0 ≤ ψ ≤ 1, φ ≡ 1 on B(x_{0}, δ/2), φ ≡ 0 outside B(x_{0}, δ), and, similarly, ψ ≡ 1 on B(u_{0}, δ/2), ψ ≡ 0 outside B(u_{0}, δ). We can write

h(x_{0}, u_{0}, ξ) = sup_{j} h_{j}(ξ),

where h_{j} are convex functions satisfying the properties stated in Proposition 9.1. Consider

h_{j}^{ε}(x, u, ξ) := (1 − ε)φ(x)ψ(u)h_{j}(ξ).

Clearly h_{j}^{ε} ∈ G (in particular property (1) follows from (9.2)). Letting j → ∞ we get

(1 − ε)h(x_{0}, u_{0}, ξ) = sup_{j} h_{j}^{ε}(x_{0}, u_{0}, ξ) ≤ sup_{g∈G} g(x_{0}, u_{0}, ξ);

hence the claim follows by letting ε → 0^{+}.

By Lemma 9.2 and (9.3) there exist a sequence h_{j} in G such that h(x, u, ξ) = sup_{j} h_{j}(x, u, ξ) for all (x, u, ξ) in A × ℝ × ℝ^{N}. Due to the stability properties of the class G, we can assume the sequence {h_{j}}_{j} increasing. Indeed it is easy to see that if g_{1}, g_{2} ∈ G then g_{1} ∨ g_{2} ∈ G (while in general g_{1} ∧ g_{2} ∉ G, since we may loose convexity). This proves (i). Clearly, properties (iii) and (iv) follow immediately from the definition of G.

(ii) follows easily from Proposition 9.1, (9.1), and the fact that clearly h^{∞}(x, u, ξ) ≥ sup_{j} h_{j}^{∞}(x, u, ξ). Indeed,

\frac{h(x, u, t ξ)}{t} = sup_{j} [\frac{h_{j}(x, u, t ξ) − h_{j}(x, u, 0)}{t} + \frac{h_{j}(x, u, 0)}{t}] ≤ sup_{j} h_{j}^{∞}(x, u, ξ) + \frac{h(x, u, 0)}{t},

so letting t → ∞ we obtain (ii). □

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 both authors by the Max–Planck Institute for Mathematics in the Sciences, Leipzig (Germany).

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