Inserted: 10 apr 2012
Last Updated: 17 feb 2013
Journal: Discrete Contin. Dyn. Syst.
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.
Keywords: minimal surfaces, fractional Laplacian, fractional Sobolev spaces, Nonlocal perimeter