*Published Paper*

**Inserted:** 12 jan 2012

**Last Updated:** 20 feb 2014

**Journal:** Calc. Var. PDE

**Year:** 2010

**Abstract:**

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.

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