Optimal Hardy inequalities in cones

Abstract

Let $\Omega$ be an open connected cone in $\mathbb{R}^n$ with vertex at the origin. Assume that the operator $$P_\mu:=-\Delta-\frac{\mu}{\delta_\Omega^2(x)}$$ is {\em subcritical} in $\Omega$, where $\delta_\Omega$ is the distance function to the boundary of $\Omega$ and $\mu \leq 1/4$. We show that under some smoothness assumption on $\Omega$, the following improved Hardy-type inequality \begin{equation*} \int_{\Omega}|\nabla \varphi|^2\,\mathrm{d}x -\mu\int_{\Omega} \frac{|\varphi|^2}{\delta_\Omega^2}\,\mathrm{d}x \geq \lambda(\mu)\int_{\Omega} \frac{|\varphi|^2}{|x|^2}\,\mathrm{d}x \qquad \forall \varphi\in C_0^\infty(\Omega), \end{equation*} holds true, and the Hardy-weight $\lambda(\mu)|x|^{-2}$ is optimal in a certain definite sense. The constant $\lambda(\mu)>0$ is given explicitly.