New estimates for the Hardy constants of multipolar Schrödinger operators

Abstract

In this paper we study the optimization problem $$μ^\star(Ω):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular potential given by $$V:=\sum _{1\le i<j\le n}\frac{|a_i-a_j{|}^2}{|x-a_i{|}^2|x-a_j{|}^2}$$ and all the singular poles $a_1,\dots ,a_n$, $n\ge 2$, arise either in the interior or at the boundary of a smooth open domain $Ω\subset \rr^N$, with $N\ge 3$ or $N\ge 2$, respectively. For a bounded domain $\Omega$ containing all the singularities in the interior, we prove that $μ^\star(Ω)>μ^\star(\rr^N)$ when $n\ge 3$ and $μ^\star(Ω)=μ^\star(\rr^N)$ when $n=2$ (It is known from \cite{cristi1} that $μ^\star(\rr^N)=(N-2)^2/n^2)$. In the situation when all the poles are located on the boundary we show that ${\mu }^{\star }(\Omega )=N^2/n^2$ if $\Omega$ 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 ${\mu }^{\star }(\Omega )$ is attained in $\hoi$ when $\Omega$ is a ball and $n\ge 3$. Besides, ${\mu }^{\star }(\Omega )$ is attained in $\semi$ when $\Omega$ is the exterior of a ball with $N\ge 3$ and $n\ge 3$ whereas in the case of a half-space ${\mu }^{\star }(\Omega )$ is attained in $\semi$ when $n\ge 3$. We also analyze the critical constants in the so-called \textit{weak} Hardy inequality which characterizes the range of ${\mu }^{\prime }s$ ensuring the existence of a lower bound for the spectrum of the Schrödinger operator $-\Delta -\mu V$. In the context of both interior and boundary singularities we show that the critical constants in the weak Hardy inequality are $(N-2{)}^2/(4n-4)$ and $N^2/(4n-4)$, respectively.

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References in corpus (2)

• Optimal Hardy Weight for Second-Order Elliptic Operator: An Answer to a Problem of Agmon
• Controllability of the heat equation with an inverse-square potential localized on the boundary