Gamma-limit for p equal to one and positive gamma

Determine whether, for p=1 and γ>0, the Gamma-limit of the functionals F_{γ,1,λ} equals (C_{N,1}/(γ+1))F_1(u,ℝ^N) for every u∈L^1_loc(ℝ^N).

Background

In the p=1, γ>0 regime, the paper describes distinct contributions from absolutely continuous, jump, and Cantor parts in the pointwise liminf behavior and gives an area-strict Gamma-limit. The standard L1_loc Gamma-limit and the proposed constant remain unresolved.

References

Let $p=1$ and $\gamma>0$. Is it true that \begin{equation}\label{open_p=1} \Gamma-\lim_{\lambda\to +\infty} F_{\gamma,1,\lambda}(u,RN)=C_{N,1}\cdot \frac{1}{\gamma+1} \cdot F_p(u,RN) \qquad \forall u\in\ L1_{loc}(RN)? \end{equation}

open_p=1:

Γ−lim⁡λ→+∞Fγ,1,λ(u,RN)=CN,1⋅1γ+1⋅Fp(u,RN)∀u∈ Lloc1(RN)?\Gamma-\lim_{\lambda\to +\infty} F_{\gamma,1,\lambda}(u,R^N)=C_{N,1}\cdot \frac{1}{\gamma+1} \cdot F_p(u,R^N) \qquad \forall u\in\ L^1_{loc}(R^N)?

— Lower bounds and Poincaré inequality for the Brezis-Seeger-Van Schaftingen-Yung functionals  (2610.05929 - Picenni, 5 Oct 2026) in Section 5, fourth Open problem