On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary
Ujjal Das, Yehuda Pinchover, Baptiste Devyver
TL;DR
This work establishes a sharp gap criterion for the existence of minimizers of the weighted Hardy functional on $C^{1,γ}$-domains with compact boundary, linking the attainment of $H_{α,p}(Ω)$ to a positive spectral gap $Γ_{α,p}(Ω)=λ^{∞}_{α,p}(Ω)-H_{α,p}(Ω)$. The authors develop a robust criticality-theory framework for the weighted $(α,p)$-Laplacian, including a chain rule, sub-/super-solution constructions via Agmon's method, and weak comparison principles, to derive precise near-boundary and near-infinity asymptotics of minimizers. They compute $λ^{∞}_{α,p}(Ω)$ explicitly in bounded and exterior domains, showing $λ^{∞}_{α,p}(Ω)=c_{α,p,1}$ for bounded domains and $λ^{∞}_{α,p}(Ω)=c_{α,p}$ for exterior domains, with minimizers existing precisely when a spectral gap persists. The results extend known $α=0$ theory to general $α∈ℝ$, clarify the role of domain geometry, and provide tight decay estimates that quantify how minimizers concentrate near the boundary and/or infinity. These findings have implications for weighted Hardy-type inequalities, criticality theory, and potential perturbations in non-smooth geometric settings.
Abstract
Let $p \in (1,\infty)$, $α\in \mathbb{R}$, and $Ω\subsetneq \mathbb{R}^N$ be a $C^{1,γ}$-domain with a compact boundary $\partial Ω$, where $γ\in (0,1]$. Denote by $δ_Ω(x)$ the distance of a point $x\in Ω$ to $\partial Ω$. Let $\widetilde{W}^{1,p;α}_0(Ω)$ be the closure of $C_c^{\infty}(Ω)$ in $\widetilde{W}^{1,p;α}(Ω)$, where $$\widetilde{W}^{1,p;α}(Ω):= \left\{\varphi \in {W}^{1,p}_{\mathrm{loc}} (Ω) \mid \left( \| \, |\nabla \varphi \, |\|_{L^p(Ω;δ_Ω^{-α})}^p + \|\varphi\|_{L^p(Ω;δ_Ω^{-(α+p)})}^p\right)<\infty \!\right\}.$$ We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{α,p}(Ω): =\!\inf \left\{\int_Ω |\nabla \varphi|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_Ω |\varphi|^p δ_Ω^{-(α+p)} \mathrm{d}x\!=\!1, \varphi \in \widetilde{W}^{1,p;α}_0(Ω) \right\} , \end{align*} and the weighted Hardy constant at infinity \begin{align*} λ_{α,p}^{\infty}(Ω) :=\sup_{K\Subset Ω}\, \inf_{W^{1,p}_{c}(Ω\setminus \overline{K})} \left\{\int_{Ω\setminus \overline{K}} |\nabla \varphi|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_{Ω\setminus \overline{K}} |\varphi|^p δ_Ω^{-(α+p)} \mathrm{d}x=1 \right\}. \end{align*} We show that $H_{α,p}(Ω)$ is attained if and only if the spectral gap $Γ_{α,p}(Ω):= λ_{α,p}^{\infty}(Ω)-H_{α,p}(Ω)$ is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers.