Let N≥ 1, p∈[1,∞), gamma>0, and let Omega⊂R^N be a bounded open interval if N=1 or a bounded Lipschitz domain if N≥ 2. For lambda>0, define [nG_lambda,p,gamma(u;Omega)=lambda iint_Omega×Omegamathbf1_(x,y):,x≠ y, frac|u(x)-u(y)|^p|x-y|^p+gamma≥lambda|x-y|^gamma-N,dx,dy. ] Identify the Gamma-limit of G_lambda,p,gamma(·;Omega) in L^p(Omega) as lambda→∞, including the precise local limiting energy and its constant. The claimed answer on these domains is [ Psi_p,gamma^cell(u;Omega)= nC_N,p,gamma^celldisplaystyle int_Omega|nabla u|^p,dx,&p∈(1,∞), u∈ W^1,p(Omega),nC_N,1,gamma^cell|Du|(Omega),&p=1, u∈ BV(Omega), +∞,&otherwise, ] where the positive constants are independent of Omega and are characterized by a cell formula.

Brezis Open Problem 9.3

Let $N\ge 1$ , $p\in[1,\infty)$ , $\gamma>0$ , and let $\Omega\subset\mathbb{R}^N$ be a bounded open interval if $N=1$ or a bounded Lipschitz domain if $N\ge 2$ . For $\lambda>0$ , define $

\nG_{\lambda,p,\gamma}(u;\Omega)=\lambda\iint_{\Omega\times\Omega}\mathbf{1}_{\left\{(x,y):\,x\ne y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+\gamma}}\ge\lambda\right\}}|x-y|^{\gamma-N}\,dx\,dy.

$Identify the$ \Gamma $-limit of$ G_{\lambda,p,\gamma}(\cdot;\Omega) $in$ L^p(\Omega) $as$ \lambda\to\infty $, including the precise local limiting energy and its constant. The claimed answer on these domains is$

\Psi_{p,\gamma}^{\mathrm{cell}}(u;\Omega)= \begin{cases}\nC_{N,p,\gamma}^{\mathrm{cell}}\displaystyle\int_\Omega|\nabla u|^p\,dx,&p\in(1,\infty),\ u\in W^{1,p}(\Omega),\nC_{N,1,\gamma}^{\mathrm{cell}}|Du|(\Omega),&p=1,\ u\in BV(\Omega),\\ +\infty,&\text{otherwise}, \end{cases}

$where the positive constants are independent of$ \Omega$ and are characterized by a cell formula.

Progress summary

Solved

A new unrefereed preprint claims to solve the problem on bounded intervals and Lipschitz domains, but the result has not yet been independently verified.

Brezis Open Problem 9.3 concerns identifying the local energy produced by a broad class of thresholded nonlocal functionals.

August 2026 affirmative preprint claim

A preprint dated August 2026 claims the limit is established on bounded intervals and Lipschitz domains, with the constants characterized by cell formulas. If correct, this answers the problem in those settings; the claim is unrefereed and remains unverified.

Current status (as of August 2026): An unrefereed preprint claims an affirmative solution on bounded intervals and Lipschitz domains, while independent verification and the status of any broader formulation remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Γ-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains — arXiv

calculus-of-variations functional-analysis

1 day ago