The weighted Hardy constant (Q2042703)

Given a domain \(\Omega\subset\mathbb{R}^d\) denote by \(d_\Gamma\) the Euclidean distance to its boundary \(\Gamma\) and let \(\Gamma_r = \{x\in\Omega : d_\Gamma(x) < r\}\). The aim of the paper is to investigate whether the weighted Hardy inequality \[ \|d_\Gamma^{\delta/2-1} \psi \|_2 \leq a \| d_\Gamma^{\delta/2} \nabla \psi \|_2 \tag{1} \] is valid for some \(\delta\geq 0\), all small enough \(r>0\), and all functions \(\psi \in C^1_c(\Gamma_r)\). For fixed \(\delta\) and \(r\), denote by \(a_\delta(\Gamma_r)\) the smallest constant \(a\) with which (1) holds; the \textit{boundary constant} \(a_\delta(\Gamma)\) is defined as the infimum over \(r\) of \(a_\delta(\Gamma_r)\) (note that \( a_\delta(\Gamma_r)\) decreases as \(r\to 0^+\)). One reason for restricting the attention to functions supported in the neighbourhood of the boundary is that -- even for simple domains -- for certain values of \(\delta\) the inequality (1) can fail to hold for all \(\psi\in C^1_c(\Omega)\). Another feature of the inequality is that, for small \(\delta\), it is equivalent to a weak Hardy inequality considered by \textit{E. B. Davies} [Q. J. Math., Oxf. II. Ser. 46, No. 184, 417--431 (1995; Zbl 0857.26005)]. The author establishes that if \(\Omega\) is a uniform domain with a locally uniform Ahlfors regular boundary, then the weighted Hardy inequality (1) is satisfied for all \(\delta \geq 0\) with the exception of \(\delta = 2 - d + d_H\), where \(d_H\) is the Hausdorff dimension of the boundary \(\Gamma\). Moreover, the boundary constant satisfies \(a_\delta(\Gamma) \geq 2/|\delta - 2 + d - d_H|\). According to the author, this theorem has a direct \(L_p\) analogue for \(1<p<\infty\). Then, complementing and extending results known from the literature, the inequality (1) is studied for \(C^{1,1}\)-domains, convex domains, and complements of (closures of) convex domains. In the first two cases, the inequality is satisfied for all \(\delta\geq 0\), \(\delta \neq 1\), and \(a_\delta(\Gamma) = 2/|\delta - 1|\). For complements of convex domains the situation is the same for \(\delta > 1\), but for \(\delta \in [0,1)\) it can happen that \(a_{\delta}(\Gamma) >2/|\delta-1|\) (even for complements of convex polytopes). Finally, the results are used to give self-adjointness criteria for degenerate elliptic diffusion operators of the form \(H =- \operatorname{div}(C \nabla)\) on \(L_2(\Omega)\) (here \(C\) is a strictly positive, symmetric, \(d\times d\) matrix with entries being real Lipschitz continuous functions; \(\Omega\) is a domain of one of the three types listed above).