*Published Paper*

**Inserted:** 17 dec 2006

**Journal:** J. Math. Pures Appl.

**Volume:** 85

**Pages:** 251-268

**Year:** 2006

**Abstract:**

We prove some existence and regularity results for minimizers of a
class of integral functionals, defined on vector-valued Sobolev
functions $u$ for which the volumes of certain level-sets $\{u=l_i\}$
are prescribed, with $i=1,\dots,m$. More specifically, in the case of
the energy density $W(x,u,D u) =

D u

^2 + \beta F(u)$, we
prove that minimizers exist and are locally Lipschitz, if
the function $F$ and $\{l_1,\dots,l_m\}$ verify suitable
hypotheses.