  We prove existence and partial regularity of integral rectifiable
$m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and
$\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq
m<n$ and $p>m$, under suitable assumptions on $N$ (in the end of the paper we
give many examples of such ambient manifolds). To this aim we introduce the
following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$
involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of
isoperimetric inequality to bound the mass in terms of the mentioned
functionals.
