Sharp Hardy-type inequalities with Lamb's constant
Farit Gabidinovich Avkhadiev, Karl Joachim Wirths · Bulletin of the Belgian Mathematical Society - Simon Stevin · 2011
Let $\Omega$ be an $n$-dimensional convex domain with finite inradius \linebreak$\delta _0=\sup_{x\in \Omega} \,\,\delta$, where $\delta = \,\,dist (x, \partial \Omega)$, and let $(p,q)$ be a pair of positive numbers. For functions vanishing at the boundary of the domain and any $ u \in [0, p/q]$ we prove the following Hardy-type inequality $$ \int _{\Omega}\frac{| abla f|^2}{\delta ^{p-1}} \,dx \,\,\geq \,\,h \int_{\Omega}\frac{|f|^2}{\delta ^{p+1}} \,dx + \frac{ \lambda^2}{\delta_0^q} \int_{\Omega}\frac{|f|^2}{\delta ^{p-q+1}} \,dx $$ with two sharp constants $$ h=\frac{p^2- u^2 q^2}{4}\geq 0 \qquad\,\,\,\,\,\,\mbox{and} \qquad\,\,\,\,\,\,\lambda = \frac{q}{2} \lambda_{ u}(2p/q) > 0, $$ where $z = \lambda_{ u}(p)$ is the Lamb constant defined as the first positive root of the equation $ p J_{ u}(z)+ 2 z J_{ u}'(z)=0$ for the Bessel function $J_{ u}$. We prove that $z = \lambda_{ u}(p)$ as a function in $p$ can be found as the solution of an initial value problem for the differential equation $$ \frac{d z}{dp}=\frac{2z}{p^2-4 u^2 +4z^2}. $$ For $n=1$ our inequality is an improvement of the original Hardy inequality for finite intervals. For $n\geq 1$ and $p=q/2=1$ it gives a new sharp form of the Hardy-type inequality due to H. Brezis and M. Marcus. The case $h=0$, $ u =1/2$, $p=1$ and $q=2$ coincides with sharp eigenvalue estimates due to J. Hersch for $n=2$, and L. E. Payne and I. Stakgold for $n\geq 3$.