Hardy's inequality in unbounded domains (Q5955008)

Let \(\Omega\) be a domain in \(\mathbb R^N\) \((N \geq 3)\) with non-empty boundary \(\partial \Omega\) and let \(\delta (x) : = \text{ dist} (x, \partial \Omega)\), \(x \in \mathbb R^N\). Denote by \(D^{1, 2}_{\varepsilon} (\Omega)\) the completion of \(D (\Omega)\) with respect to the inner product \((u, v) := \int_{\Omega} \delta^{\varepsilon} \nabla u \cdot \nabla v dx\). The author considers the case when \(\Omega\) is a domain (bounded or unbounded) with compact boundary of class \(C^2\) and investigates the Hardy inequality NEWLINE\[NEWLINE \int_{\Omega} |u|^2 \delta^{\varepsilon - 2} \text{ d} x \leq C \int_{\Omega} |\nabla u|^2 \delta^{\varepsilon} \text{ d} x, \quad u \in D^{1,2}_{\varepsilon} (\Omega), \leqno(1) NEWLINE\]NEWLINE where \(C\) is a positive constant and \(0 \leq \varepsilon < 1\). More precisely, the author considers the connection between the best possible constant NEWLINE\[NEWLINE C = S_{\varepsilon} (\Omega) := \inf \Big\{\int_{\Omega} |\nabla u|^2 \delta^{\varepsilon} \text{ d} x\Big/\int_{\Omega} |u|^2 \delta^{\varepsilon - 2} \text{ d} x;\;u \in D^{1,2}_{\varepsilon} (\Omega)\Big\} NEWLINE\]NEWLINE in \((1)\) and the existence of minimizer for \((1)\). The main result states that \(S_{\varepsilon} (\Omega)\) is achieved provided that \(S_{\varepsilon} (\Omega) < (1 - \varepsilon)^2 /4\). Note that the particular result for \(\varepsilon = 0\) is due to \textit{M. Marcus, V. J. Mizel} and \textit{Y. Pinchover} [Trans. Am. Math. Soc. 350, 3237--3255 (1998; Zbl 0917.26016)].