A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Abstract

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

“…To conclude, we notice that in the recent [14], the authors give a different notion of Gaussian fractional perimeter of a measurable set E in a bounded domain Ω ⊂ R N using a singular integral representation of the form and they prove the Gamma convergence of (1 − s)P γ s (E; Ω) to the Gaussian perimeter as s → 1 − exploiting techniques similar to the ones used in [1]. See also [4], where kernels with faster than L 1 decay at infinity are taken into account.…”

Asymptotics of the s-fractional Gaussian perimeter as $$s\rightarrow 0^+$$

“…In the last Sect. 4 we prove that for the Gaussian fractional perimeter defined and used in [8] the asymptotics for s → 0 + is trivial.…”

Asymptotics of the $s$-fractional Gaussian perimeter as $s\to 0^+$

“…In Section 3 we firstly prove our Main Theorem by stating and proving the ancillary Propositions 3.3 and 3.4 and we show some properties of the limit set function µ. In the last Section 4 we prove that for the Gaussian fractional perimeter defined and used in [7] the asymptotics for s → 0 + is trivial.…”