On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,\gamma }$-domains with compact boundary

Abstract

Let $p\in (1,\infty )$, $\alpha \in \mathbb{R}$, and $\Omega ⊊{\mathbb{R}}^N$ be a $C^{1,\gamma }$-domain with a compact boundary $\partial \Omega$, where $\gamma \in (0,1]$. Denote by ${\delta }_{\Omega }(x)$ the distance of a point $x\in \Omega$ to $\partial \Omega$. Let ${\tilde{W}}_0^{1,p;\alpha }(\Omega )$ be the closure of $C_c^{\infty }(\Omega )$ in ${\tilde{W}}^{1,p;\alpha }(\Omega )$, where $${\tilde{W}}^{1,p;\alpha }(\Omega ):=\left\{\phi \in W_{\mathrm{loc}}^{1,p}(\Omega )\mid \left(\Vert |\nabla \phi |{\Vert }_{L^p(\Omega ;{\delta }_{\Omega }^{-\alpha })}^p+\Vert \phi {\Vert }_{L^p(\Omega ;{\delta }_{\Omega }^{-(\alpha +p)})}^p\right)<\infty \right\}.$$ 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 $H_{\alpha ,p}(\Omega )$ is attained if and only if the spectral gap ${\Gamma }_{\alpha ,p}(\Omega ):={\lambda }_{\alpha ,p}^{\infty }(\Omega )-H_{\alpha ,p}(\Omega )$ is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers.

Proof of Theorem D.2 in Appendix D has been corrected for the case p>N. In addition, several typographical errors have been fixed, and Remarks D.3 and D.4 have been added

References in corpus (2)

• Spectral stability of higher order uniformly elliptic operators
• Noncritical Weighted Hardy's Inequalities with compact perturbations