The Legendre-Hardy inequality on bounded domains

There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C^2-domain in \mathbb {R}^n of the following form \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d(x) is the distance from x \in \Omega to the boundary \partial \Omega and \alpha ,\beta \in \mathbb {R}. We classify all (\alpha ,\beta ) \in \mathbb {R}^2 for which C(\alpha ,\beta ) > 0. Then, we study whether an optimal constant C(\alpha ,\beta ) is attained or not. Our study on C(\alpha ,\beta ) for general (\alpha ,\beta ) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version.