*Published Paper*

**Inserted:** 30 nov 2001

**Last Updated:** 6 jul 2002

**Journal:** Arch. Ration. Mech. Anal.

**Volume:** 156

**Number:** 2

**Pages:** 121-140

**Year:** 2001

**Abstract:**

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

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

z^{{p}(x)})\;,$$ a relevant model case
being
the functional
$$\int

Du^{{p}(x)}\,dx\;.$$
Under sharp assumptions on the continuous function $p(x)>1$ we prove
regularity of minimizers. Energies
exhibiting this growth appear in several models from mathematical
physics.

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