Properties and applications of the distance functions on open sets of the Euclidean space (Q2191894)

Let \(\Omega \subset {\mathbb{R}^n}\) with \(\Omega \neq {\mathbb{R}^n}\) (\(n>1\)), \(\delta (x) := \inf_{y\in {\mathbb{R}^n} \setminus \Omega} \{ |x-y|\}\) for \(x \in \Omega\) and \(\nabla u(x)\) be the gradient vector for \(u \in C_0^1 (\Omega)\). Let \(S(\Omega) : = \{ x\in \Omega \mid \delta \text{ is not differentiable at } x \}\) and \(\nabla \delta\) be a vector function defined on \(\Omega \setminus S(\Omega)\) such that \[ \delta (x+h) - \delta (x) = \nabla \delta (x) \cdot h + o(|h|)\] for \(x+h \in \Omega\) and \(|h|\to 0\). The author obtains the following: (1) For a convex domain \(\Omega\) and \(p\in [1, \infty)\), \(s\in (1, \infty)\), the following inequality holds \[ \int_{\Omega} \frac{|\nabla u(x) \cdot \nabla \delta(x) |^p } {\delta(x)^{s-p}} dx \ge \frac{(s-1)^p}{p^p} \int_{\Omega} \frac{|u(x)|^p} {\delta(x)^s} dx \quad \forall u \in C_0^1 (\Omega),\] where the constant \(\frac{(s-1)^p}{p^p}\) is sharp, which generalizes the result given by \textit{M. Marcus} et al. [Trans. Am. Math. Soc. 350, No. 8, 3237--3255 (1998; Zbl 0917.26016)]. (2) Let \(\lambda_0\) be the first positive root of the equation \(J_0(t) -2 t J_1 (t) =0\), where \(J_0\) and \(J_1\) are the Bessel functions of order 0 and 1, respectively. For a convex domain \(\Omega\) such that \(\delta_0(\Omega):= \sup_{x\in \Omega} \delta (x) <\infty\) and for a function \(u \in C_0^1 (\Omega)\), the following inequality holds \[ \int_{\Omega}|\nabla u(x) \cdot \nabla \delta(x) |^2 dx \ge \frac{1}{4} \int_{\Omega} \frac{|u(x)|^2} {\delta(x)^2} dx + \frac{\lambda_0^2}{\delta_0(\Omega)^2} \int_{\Omega} |u(x)|^2 dx ,\] where the constant \(\frac{1}{4}\) is sharp and the constant \(\frac{\lambda_0^2}{\delta_0(\Omega)^2}\) is optimal, which improves the result given by \textit{F. G. Avkhadiev} and \textit{K. J. Wirths} [ZAMM, Z. Angew. Math. Mech. 87, No. 8--9, 632--642 (2007; Zbl 1145.26005)]. (3) For an open set \(\Omega\) and \(p\in [1, \infty)\), \(s\in [n, \infty)\), the following inequality holds \[ \int_{\Omega} \frac{|\nabla u(x) \cdot \nabla \delta(x) |^p } {\delta(x)^{s-p}} dx \ge \frac{(s-n)^p}{p^p} \int_{\Omega} \frac{|u(x)|^p} {\delta(x)^s} dx \quad \forall u \in C_0^1 (\Omega),\] where the constant \(\frac{(s-n)^p}{p^p}\) is optimal, which improves the result given by \textit{F. G. Avkhadiev} [Lobachevskii J. Math. 21, 3--31 (2006; Zbl 1120.26008)]. The results are useful for researchers on Hardy-type inequalities.