Improved Multipolar Hardy Inequalities

Abstract

In this paper we prove optimal Hardy-type inequalities for Schrödinger operators with positive multi-singular inverse square potentials of the form

$$A_{\lambda } := -\Delta- \lambda \displaystyle\sum _{1\leq i<j\leq n} \frac{\vert x_{i} - x_{j}{\vert }^{2}} {\vert x - x_{i}{\vert }^{2}\vert x - x_{j}{\vert }^{2}},\quad \lambda> 0.$$

More precisely, we show that A _λ is nonnegative in the sense of L ² quadratic forms in ${\mathbb{R}}^{N}$, if and only if $\lambda\leq{(N - 2)}^{2}/{n}^{2}$, independently of the number n and location of the singularities $x_{i} \in{\mathbb{R}}^{N}$, where N ≥ 3 denotes the space dimension. This aims to complement some of the results in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) obtained by the “expansion of the square” method. Due to the interaction of poles, our optimal result provides a singular quadratic potential behaving like $(n - 1){(N - 2)}^{2}/({n}^{2}\vert x - x_{i}{\vert }^{2})$ at each pole x _i . Besides, the authors in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) showed optimal Hardy inequalities for Schrödinger operators with a finite number of singular poles of the type $B_{\lambda } := -\Delta-\sum _{i=1}^{n}\lambda /\vert x - x_{i}{\vert }^{2}$, up to lower order L ²-reminder terms. By means of the optimal results obtained for A _λ, we also build some examples of bounded domains Ω in which these lower order terms can be removed in H ₀ ¹(Ω). In this way we obtain new lower bounds for the optimal constant in the standard multi-singular Hardy inequality for the operator B _λ in bounded domains. The best lower bounds are obtained when the singularities x _i are located on the boundary of the domain.

2010 Mathematics Subject Classification: 35J10, 26D10, 46E35, 35Q40, 35J75.