Sharp Poincar\'e Inequalities for Families of Nonlocal Nonconvex Functionals

Abstract

Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincar\'e inequalities for the nonlocal nonconvex functionals $$ \Phi_{\lambda,p}^\gamma(g;Q_0) :=\lambda^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R$ is measurable, $\lambda\in(0,\infty)$, and $\gamma\in\mathbb R$. Specifically, we prove that the estimate $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\,dx\,dy \lesssim\lambda^p+\Phi_{\lambda,p}^\gamma(g;Q_0) $$ holds for any such $g$ and $\lambda$ if and only if $\gamma\in(-\infty,-1]$, where the implicit positive constant depends only on $N$, $p$, and $\gamma$. This extends Nguyen's Poincar\'e inequality for $\gamma=-p$ and provides a useful tool for studying regularity and compactness through these functionals. As applications, we obtain a lower bound for the $\Gamma$-convergence of $\Phi^\gamma_{\lambda,p}$ and related asymptotic characterizations for Sobolev and BV spaces. Moreover, we identify the optimal range of $\gamma$ for which both the Rellich--Kondrachov theorem and a criterion for VMO regularity involving $\Phi^\gamma_{\lambda,p}$ at a fixed threshold hold. These results also answer the question on nonlocal Poincar\'e inequalities posed by H.-M. Nguyen in [C. R. Math. Acad. Sci. Paris 363 (2025)]. For $\gamma\in(-\infty,-1]$, they further answer questions on Sobolev and BV characterizations posed in the same article and by H. Brezis et al. in [Anal. PDE 17 (2024)].