Characterization of balls as minimizers of an endpoint Gagliardo seminorm on the boundary
Albert Mas Blesa · arXiv (Cornell University) · 2018
Given a bounded $C^2$ domain $Ω\subset{\mathbb R}^d$ with $d\geq3$, we prove a sharp inequality which relates the perimeter of ${\partialΩ}$ to the endpoint Gagliardo seminorm in $W^{r,2}({\partialΩ})$, corresponding to $r=0$, of the normal vector field on ${\partialΩ}$. The proof of the inequality relies on the use of Bessel potentials and a monotonicity formula; we also show that balls are the unique minimizers. For $1/2