Gamma-convergence of Gaussian fractional perimeter

Abstract

We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.

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

“…where γ is the standard Gaussian measure in R N defined in (2.1) and the kernel K s is the jumping kernel defined in (2.3) and study the asymptotics of s P γ s (E; ) as s → 0 + . In this sense this is a parallel study of our previous paper [5], where the -limit of (1 − s)P γ s (E; ) as s → 1 − is studied. In the Euclidean setting the notion of s-fractional perimeter recovers the classical perimeter when s → 1 − in various senses as proved in [1,2,4,7,12,17].…”

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

“…where γ is the standard Gaussian measure in R N defined in (2.1) and the kernel K s is the jumping kernel defined in (2.3) and study the asymptotics of sP γ s (E; Ω) as s → 0 + . In this sense this is a parallel study of our previous paper [5], where the Γ-limit of (1 − s)P γ s (E; Ω) as s → 1 − is studied.…”

The Cheeger problem in abstract measure spaces

“…Properties (P.1) and (P.2) directly follow from the definition, whereas (P.7) is a consequence of the symmetry of the kernel $$K_s(x,y)$$. For the validity of (P.4) and (P.5), we refer to [43–45]. Therefore, Theorem 4.5 and Theorem 5.4 hold true.…”

“…For 𝑥, 𝑦 ∈ ℝ 𝑛 , 𝑡 > 0, let 𝑀 𝑡 (𝑥, 𝑦) ⩾ 0 be the Mehler kernel, see [43][44][45] for the precise definition.…”

“…For $$x,y\in \mathbb {R}^n$$, $$t>0$$, let $$M_t(x,y)\geqslant 0$$ be the Mehler kernel , see [43–45] for the precise definition. For $$s\in (0,1)$$, we set $$$$\begin{equation*} {K}_{\sigma}(x,y)=\displaystyle \int_{0}^{+\infty}\frac{{M}_{t}(x,y)}{{t}^{\frac{s}{2}+1}}\, \mathrm{d}t, \end{equation*}$$$$and we let $$$$\begin{equation} P_s^\gamma (E)=\displaystyle \int _{E} \displaystyle \int _{E^c}K_s(x,y)\, \mathrm{d}\gamma (y)\, \mathrm{d}\gamma (x) \end{equation}$$$$be the fractional Gaussian $$s$$ ‐perimeter of the measurable set $$E\subset \mathbb {R}^n$$, where the measure $$\gamma$$ is as in ().…”