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 $\mu \le 1/4$. We show that under some smoothness assumption on $\Omega$, the following improved Hardy-type inequality \begin{equation*} \int_Ω|\nabla φ|^2\,\mathrm{d}x -μ\int_Ω \frac{|φ|^2}{δ_Ω^2}\,\mathrm{d}x \geq λ(μ)\int_Ω \frac{|φ|^2}{|x|^2}\,\mathrm{d}x \qquad \forall φ\in C_0^\infty(Ω), \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.

30 pages