The Legendre-Hardy inequality on bounded domains

Abstract:

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.