New estimates for the Hardy constants of multipolar Schr\"odinger operators

In this paper we study the optimization problem $${\mu }^{\star }(\mathrm{\Omega }):=\underset{u\in \text{semi}}{\inf }\frac{\text{into}\mathrm{\mid }\text{n}u{\mathrm{\mid }}^2\text{dx}}{\text{into}V{u}^2\text{dx}}$$ in a suitable functional space $\text{semi}$. Here, $V$ is the multi-singular potential given by $$V:=\sum _{1\le i<j\le n}\frac{\mathrm{\mid }{a}_i-a_j{\mathrm{\mid }}^2}{\mathrm{\mid }x-a_i{\mathrm{\mid }}^2\mathrm{\mid }x-a_j{\mathrm{\mid }}^2}$$ and all the singular poles $a_1, \ldots, a_n$,$n\geq 2$, arise either in the interior or at the boundary of a smooth open domain $\mathrm{\Omega }\subset {\text{rr}}^N$, with $N\ge 3$ or $N\ge 2$, respectively. For a bounded domain $\mathrm{\Omega }$ containing all the singularities in the interior, we prove that ${\mu }^{\star }(\mathrm{\Omega })>{\mu }^{\star }({\text{rr}}^N)$ when $n\ge 3$ and ${\mu }^{\star }(\mathrm{\Omega })={\mu }^{\star }({\text{rr}}^N)$ when $n=2$ (It is known from \cite{cristi1} that ${\mu }^{\star }({\text{rr}}^N)=(N-2{)}^2\mathrm{/}{n}^2)$. In the situation when all the poles are located on the boundary we show that ${\mu }^{\star }(\mathrm{\Omega })=N^2\mathrm{/}{n}^2$ if $\mathrm{\Omega }$ is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between the poles. In addition, in the case of boundary singularities we obtain that ${\mu }^{\star }(\mathrm{\Omega })$ is attained in $\text{hoi}$ when $\mathrm{\Omega }$ is a ball and $n\geq 3$\mathrm{.}Besides,$\mu^\star(\Omega)$isattainedin$\semi$when$\Omega$ is the exterior of a ball with $N\ge 3$ and $n\ge 3$ whereas in the case of a half-space ${\mu }^{\star }(\mathrm{\Omega })$ is attained in $\text{semi}$ when $n\ge 3$. We also analyze the critical constants in the so-called $\mathit{\text{weak}}$ Hardy inequality which characterizes the range of ${\mu }^{\mathrm{\prime }}s$ ensuring the existence of a lower bound for the spectrum of the Schr"{o}dinger operator $-\mathrm{\Delta }-\mu V$. In the context of both interior and boundary singularities we show that the critical constants in the weak Hardy inequality are $(N-2{)}^2\mathrm{/}(4n-4)$ and $N^2\mathrm{/}(4n-4)$, respectively.