*Published Paper*

**Inserted:** 30 nov 2001

**Last Updated:** 10 dec 2003

**Journal:** Ann. Inst. H. Poincaré Anal. Non Linéaire

**Volume:** 20

**Number:** 3

**Pages:** 359-390

**Year:** 2003

**Abstract:**

It is shown that the relaxed energy
$$
\mathcal F(u,A) :=\inf\Bigl\{\liminf_{{n\to+\infty}} \int_{A}
f(x, \nabla u_{n)} dx:\
\{u_{n\}} \subset \W(A)\, ,\ u_{n} \to u\
\text{in}\ {L}^{1}(A)
\Bigr\},
$$
admits the representation
$$\mathcal F(u,A) = \int_{A} f(x,\nabla u) dx+ \mu_{s}(A)\ ,$$ where
$f$ is a convex, Carathéodory integrand satisfying a non standard
``$\alpha$-$\beta$'' growth hypothesis, $ \beta\in [\alpha, N\alpha/(N-1))$.
Sufficient
conditions
guaranteeing that $\mu_s=0$ are discussed. An example asserting that this
representation may
fail in the quasiconvex case is provided.

**Keywords:**
convexity, Besicovitch covering theorem, Radon-Nikodym derivative, Lavrentiev phenomenon