*Published Paper*

**Inserted:** 30 nov 2001

**Last Updated:** 6 jul 2002

**Journal:** Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)

**Volume:** 30

**Number:** 2

**Pages:** 311-339

**Year:** 2001

**Abstract:**

We consider the integral functional $\int f(x,Du)\,dx$ under non
standard growth assumptions of $(p,q)$-type: namely, we assume that
$$

z^{{p}(x)}\le f(x,z)\le L(1+

z^{{p}(x)})$$
for some function $p(x) >1$, a condition appearing in several models
from mathematical physics. Under sharp assumptions on the continuous
function $p(x)$ we prove
partial regularity of minimizers in the vector-valued case $u: \Omega \ (\subset
R^{n}) \to R^{N}$,
allowing for quasiconvex energy densities. This is, to our knowledge, the
first regularity theorem for quasiconvex functionals under non
standard growth conditions. The proof relies on a careful, intrisic, blow-up technique
which is dictated by the oscillations of the function $p(x)$ and the
size of $Du$ itself.

**Keywords:**
Partial regularity, quasiconvexity, integral functionals, Nonstandard growth, Minimizers