On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866915301929517056 |
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| author | Das, Ujjal Pinchover, Yehuda Devyver, Baptiste |
| author_facet | Das, Ujjal Pinchover, Yehuda Devyver, Baptiste |
| contents | 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
$$\widetilde{W}^{1,p;α}(Ω):= \left\{φ\in {W}^{1,p}_{\mathrm{loc}} (Ω) \mid \left( \| \, |\nabla φ\, |\|_{L^p(Ω;δ_Ω^{-α})}^p + \|φ\|_{L^p(Ω;δ_Ω^{-(α+p)})}^p\right)<\infty \!\right\}.$$
We study the following two variational constants: the weighted Hardy constant \begin{align*}
H_{α,p}(Ω): =\!\inf \left\{\int_Ω |\nabla φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_Ω |φ|^p δ_Ω^{-(α+p)} \mathrm{d}x\!=\!1, φ\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 φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_{Ω\setminus \overline{K}} |φ|^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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03527 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary Das, Ujjal Pinchover, Yehuda Devyver, Baptiste Analysis of PDEs Functional Analysis Spectral Theory 49J40, 35B09, 35J20, 35J92 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 $$\widetilde{W}^{1,p;α}(Ω):= \left\{φ\in {W}^{1,p}_{\mathrm{loc}} (Ω) \mid \left( \| \, |\nabla φ\, |\|_{L^p(Ω;δ_Ω^{-α})}^p + \|φ\|_{L^p(Ω;δ_Ω^{-(α+p)})}^p\right)<\infty \!\right\}.$$ We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{α,p}(Ω): =\!\inf \left\{\int_Ω |\nabla φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_Ω |φ|^p δ_Ω^{-(α+p)} \mathrm{d}x\!=\!1, φ\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 φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_{Ω\setminus \overline{K}} |φ|^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. |
| title | On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary |
| topic | Analysis of PDEs Functional Analysis Spectral Theory 49J40, 35B09, 35J20, 35J92 |
| url | https://arxiv.org/abs/2303.03527 |