*Published Paper*

**Inserted:** 14 dec 2002

**Last Updated:** 9 may 2005

**Journal:** SIAM J. Mathematical Analysis

**Volume:** 36

**Number:** 5

**Pages:** 1540-1579

**Year:** 2005

**Abstract:**

We consider
integral functionals of the type $F(u):=\int_{\Omega} f(x,u,Du)dx$ exhibiting a gap between the coercivity and the growth
exponent:
$$L^{{}-1}

Du^{p\leq} f(x,u,Du)\leq L(1+

Du^{q)\qquad} 1<p< q\,.$$We
give lower semicontinuity results and conditions ensuring that the
relaxed functional $\bar{F}$ is exactly $\int_{\Omega} Qf(x,u,Du)dx$, $Qf$ denoting the usual quasi-convex envelope; our conditions
are sharp. Indeed we also provide counterexamples where such an
integral representation fails, showing that energy concentrations
appear in the relaxation procedure leading a measure
representation of $\bar{F}$ with a non zero singular part, which is
explicitly computed. The main point in our analysis is that such
relaxation results depend in subtle way on the interaction between
the ratio $q/p$ and the degree of regularity of the integrand $f$
with respect to the variable $x$. Our results extend to certain
anisotropic settings, previous theorems for non-convex integrals
due to Fonseca & MalĂ˝ and Kristensen.