Optimal Hardy inequalities in cones

Let $\mathrm{\Omega }$ be an open connected cone in ${\mathbb{R}}^n$ with vertex at the origin. Assume that the operator $$P_{\mu }:=-\mathrm{\Delta }-\frac{\mu }{{\delta }_{\mathrm{\Omega }}^2(x)}$$ is {\em subcritical} in $\mathrm{\Omega }$, where ${\delta }_{\mathrm{\Omega }}$ is the distance function to the boundary of $\mathrm{\Omega }$ and $\mu \le 1\mathrm{/}4$. We show that under some smoothness assumption on $\mathrm{\Omega }$, 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 $\lambda (\mu )\mathrm{\mid }x{\mathrm{\mid }}^{-2}$ is optimal in a certain definite sense. The constant $\lambda (\mu )>0$ is given explicitly.