Calculus of Variations and Geometric Measure Theory

A. Mondino

Existence of Integral m-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, p>m, in Riemannian Manifolds

created by mondino on 12 Jan 2012
modified on 20 Feb 2014


Published Paper

Inserted: 12 jan 2012
Last Updated: 20 feb 2014

Journal: Calc. Var. PDE
Year: 2010


We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int
^p$ and $\int
^p$ 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 $\int
^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.