A Hardy type inequality for \(W_0^{2,1}(\Omega)\) functions (Q639592)

The main result in this paper is Theorem 1.1, where the authors prove the following: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary. Given \(x\in \Omega ,\) define \(\delta (x):=\mathrm{dist}(x,\partial \Omega ).\) Let \( d:\Omega \rightarrow (0,\infty )\) be a smooth function such that \( d(x)=\delta (x)\) near \(\partial \Omega\). Then for every \(u\in W_{0}^{2,1}(\Omega )\), we have \(\frac{u(x)}{d(x)}\in W_{0}^{1,1}(\Omega )\) with \[ \left\| \nabla \left( \frac{u(x)}{d(x)}\right) \right\| _{L^{1}(\Omega )}\leq C\left\| u\right\| _{W_{0}^{2,1}(\Omega )}, \] where \(C\) is a constant depending only on \(d\) and \(\Omega\).