Calculus of Variations and Geometric Measure Theory

S. Dipierro - A. Figalli - G. Palatucci - E. Valdinoci

Asymptotics of the $s$-perimeter as $s \searrow 0$

created by palatucci on 10 Apr 2012
modified on 17 Feb 2013


Published Paper

Inserted: 10 apr 2012
Last Updated: 17 feb 2013

Journal: Discrete Contin. Dyn. Syst.
Volume: 33
Number: 7
Pages: 2777-2790
Year: 2013
Doi: doi:10.3934/dcds.2013.33.2777


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