  We prove the existence of minimizers for functionals defined over the class
of convex domains contained inside a bounded set D of R^N and with prescribed
volume. Some applications are given, in particular we prove that the
eigenvalues of differential operators of second and fourth order with
non-constant coefficients as well as integral functionals depending on the
solution of an elliptic equation can be minimized over this class of domains.
Another application of this result is related to the famous Newton problem of
minimal resistance. In general, all the results we shall develop hold for
elliptic operators of any even order larger that 0.
