Matteo Cozzi
On the variation of the fractional mean curvature under the effect of $C^{1, lpha}$ perturbations
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ABSTRACT.  In this brief note we study how the fractional mean curvature of order $s \in (0, 1)$ varies with respect to $C^{1, lpha}$ diffeomorphisms. We prove that, if $lpha > s$, then the variation under a $C^{1, lpha}$ diffeomorphism $\Psi$ of the $s$-mean curvature of a set $E$ is controlled by the $C^{0, lpha}$ norm of the Jacobian of $\Psi$. 
When $lpha = 1$ we discuss the stability of these estimates as $s 
ightarrow 1^-$ and comment on the consistency of our result with the classical framework.