On the variation of the fractional mean curvature under the effect of C1,α perturbations

In this brief note we study how the fractional mean curvature of order s ∈ (0, 1) varies with respect to C1,α diffeomorphisms. We prove that, if α>s, then the variation under a C1,α diffeomorphism ψ of the s-mean curvature of a set E is controlled by the C0,α norm of the Jacobian of ψ. When α = 1 we discuss the stability of these estimates as s → 1- and comment on the consistency of our result with the classical framework.