Full-Range Γ-Convergence of Weak-Type Nonlocal Functionals
Abstract
Let N≥1, p∈[1,∞), and Ω⊂ℝN be a bounded open interval when N=1 or a bounded Lipschitz domain when N≥2. For any given γ∈ℝ∖{0} and λ∈(0,∞), consider the weak-type nonlocal functional Fλ,p,γ(u;Ω):=λp underset|u(x)−u(y)|>λ|x−y|1+γ/p ∫Ω∫Ω|x−y|γ−N dy dx. In this article, we prove that, for γ∈(0,∞) as λ→∞ or for γ∈(−∞,−1] as λ→0+, the family Fλ,p,γ(·;Ω) converges, in the sense of Γ-convergence in Lp(Ω), to the functional Ψp,γΩ(u) := CN,p,γcell∫Ω|∇ u|p dx, &p∈(1,∞) and u∈ W1,p(Ω), [2mm] CN,1,γcell|Du|(Ω), &p=1 and u∈ BV(Ω), [1mm] ∞,&otherwise. where the positive constants CN,p,γcell are independent of Ω. As applications, we further show that, for γ∈(0,∞) as λ→∞ or for γ∈(−∞,−1] as λ→0+, the weak-type nonlocal functional Fλ,p,γ(·;ℝN) converges, in the sense of Γ-convergence in L1loc(ℝN), to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].
Related
- Simplifying the computation of weak-⋆ second subderivatives of convex functionals via Γ-convergence
- Harnack-type estimates for nonlocal double phase functionals
- Large deviations for additive functionals of one-dimensional zero-range processes
- Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets
- Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range
BibTeX
@misc{lin2026fullrange,
title = {{Full-Range $Γ$-Convergence of Weak-Type Nonlocal Functionals}},
author = {Xiaosheng Lin and Dachun Yang and Sibei Yang and Wen Yuan and Yangyang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07001},
eprint = {2610.07001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07001},
note = {CC0},
}
RIS
TY - UNPB
TI - Full-Range Γ-Convergence of Weak-Type Nonlocal Functionals
AU - Xiaosheng Lin
AU - Dachun Yang
AU - Sibei Yang
AU - Wen Yuan
AU - Yangyang Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.07001
UR - https://arxiv.org/abs/2610.07001
AB - Let N≥1, p∈[1,∞), and Ω⊂ℝ^N be a bounded open interval when N=1 or a bounded Lipschitz domain when N≥2. For any given γ∈ℝ∖{0} and λ∈(0,∞), consider the weak-type nonlocal functional F_λ,p,γ(u;Ω):=λ^p underset|u(x)−u(y)|>λ|x−y|^1+γ/p ∫_Ω∫_Ω|x−y|^γ−N dy dx. In this article, we prove that, for γ∈(0,∞) as λ→∞ or for γ∈(−∞,−1] as λ→0^+, the family F_λ,p,γ(·;Ω) converges, in the sense of Γ-convergence in L^p(Ω), to the functional Ψ_p,γ^Ω(u) := C_N,p,γ^cell∫_Ω|∇ u|^p dx, &p∈(1,∞) and u∈ W^1,p(Ω), [2mm] C_N,1,γ^cell|Du|(Ω), &p=1 and u∈ BV(Ω), [1mm] ∞,&otherwise. where the positive constants C_N,p,γ^cell are independent of Ω. As applications, we further show that, for γ∈(0,∞) as λ→∞ or for γ∈(−∞,−1] as λ→0^+, the weak-type nonlocal functional F_λ,p,γ(·;ℝ^N) converges, in the sense of Γ-convergence in L^1_loc(ℝ^N), to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].
ER -