*Published Paper*

**Inserted:** 20 dec 2001

**Last Updated:** 17 dec 2003

**Journal:** J. Convex Anal.

**Volume:** 10

**Number:** 2

**Pages:** 295-324

**Year:** 2003

**Abstract:**

We study the lower semicontinuous envelope of variational functionals given by \,$\int f(x,Du)\,dx$\, for smooth functions $u$, and equal to $+\infty$ elsewhere, under nonstandard growth conditions of $(p,q)$-type:
namely, we assume that
$$
\vert z\vert^{{p}(x)}\leq f(x,z)\leq L(1+\vert z\vert^{{p}(x)})\,.
$$
If the growth exponent is piecewise constant, i.e., $p(x)\equiv p_i$\, on each set of a smooth partition of the domain, we prove measure and representation property of the relaxed functional.
We then extend the previous results by considering $p(x)$ uniformly continuous on each set of the partition.
We finally give an example of energy concentration in the process of relaxation.

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