arXiv 1402.59332014-02math.AP

多极薛定谔算子Hardy常数的新估计

New estimates for the Hardy constants of multipolar Schrödinger operators

Cristian Cazacu

备注：: revisions
发表信息：: Commun. Contemp. Math. 18 (2016), no. 5, 1550093, 28 pp
DOI：: 10.1142/S0219199715500935

摘要：

本文研究了在适当的函数空间$\semi$中的优化问题$$μ^\star(Ω):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$.这里，$V$是由$$V:=\sum_{1\leq i<j\leq n} \frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$给出的多奇异势，并且所有奇异极点$a_1, \ldots, a_n$、$n\geq 2$分别出现在光滑开域$Ω\subset \rr^N$的内部或边界处，其中$N\geq 3$或$N \geq 2$。对于一个内部包含所有奇点的有界域$Ω$，我们证明了$μ^\star(Ω)>μ^\star(\rr^N)$当$n\geq 3$和$μ^\star(Ω)=μ^\star(\rr^N)$当$n=2$（从\cite{cristi1}可知，$μ^\star(\rr^N)=(N-2)^2/n^2)$ .在所有极点都位于边界上的情况下，我们证明了如果$Ω$是球，球的外部或半空间，则$μ^\star(Ω)=N^2/n^2$。我们的结果不依赖于极点之间的距离。此外，在边界奇异的情况下，我们得到了$μ^\star(Ω)$达到$\hoi$时，$Ω$是一个球和$n\geq 3$。此外，当$Ω$是具有$N\geq 3$和$n\geq 3$的球的外部时，在$\semi$中获得$μ^\star(Ω)$，而在半空间的情况下，当$n\geq 3$时，在$\semi$中获得$μ^\star(Ω)$。我们还分析了临界常数在所谓的\textit{weak}哈代不等式，其特征的范围$μ's$确保薛定谔算子$-Δ-μV$的谱的下界的存在。在内部和边界奇点的情况下，我们证明了弱Hardy不等式的临界常数分别是$(N-2)^2/(4n-4)$和$N^2/(4n-4)$。

英文摘要：

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 $$V:=\sum_{1\leq i<j\leq n} \frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$ 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 $Ω\subset \rr^N$, with $N\geq 3$ or $N \geq 2$, respectively. For a bounded domain $Ω$ containing all the singularities in the interior, we prove that $μ^\star(Ω)>μ^\star(\rr^N)$ when $n\geq 3$ 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 $μ^\star(Ω)=N^2/n^2$ if $Ω$ 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 $μ^\star(Ω)$ is attained in $\hoi$ when $Ω$ is a ball and $n\geq 3$. Besides, $μ^\star(Ω)$ is attained in $\semi$ when $Ω$ is the exterior of a ball with $N\geq 3$ and $n\geq 3$ whereas in the case of a half-space $μ^\star(Ω)$ is attained in $\semi$ when $n\geq 3$. We also analyze the critical constants in the so-called \textit{weak} Hardy inequality which characterizes the range of $μ's$ ensuring the existence of a lower bound for the spectrum of the Schrödinger operator $-Δ-μV$. 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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