Best constants for weighted Hardy inequalities in the exterior of balls and circular cylinders

We consider the weighted Hardy inequality ${\int }_{\Omega }\frac{{|\nabla u\left(x\right)|}^2}{d^{s-2}\left(x\right)}dx\ge c_s\left(\Omega \right){\int }_{\Omega }\frac{u^2\left(x\right)}{d^s\left(x\right)}dx,\forall u\in C_c^{\infty }\left(\Omega \right).$For $s>1$, $n\ge 2$, $s\ne n$ we compute the best constant in the case where $\Omega$ is either the complement of a ball or the complement of a circular cylinder. Typically one is able to compute best constants if the domain is weakly mean convex. In our case the domains are not weakly mean convex. The best constant depends on the parameter $s$ in a surprising way. For instance when $n>3$ then $c_s\left(\Omega \right)=\frac{(n-2)(n-s-1)(s-2)}{{(n-3)}^2},\text{if}\frac{3n-5}{n-1}<s<\frac{n^2-3n+4}{n-1},$whereas $c_s\left(\Omega \right)=min\left({\left(\frac{s-1}{2}\right)}^2,{\left(\frac{n-s}{2}\right)}^2\right),\text{otherwise}.$