Optimizing Improved Hardy Inequalities
Optimizing Improved Hardy Inequalities
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DOI：
10.1006/jfan.2001.3900
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发表时间：
2002-06-20
影响因子：
1.7
通讯作者：
Tertikas, A
中科院分区：
文献类型：
--
作者：
Filippas, S;Tertikas, A
设Omega是R-N中的有界域，N大于或等于3，包含原点。受Brezis和Vazquez的一个问题的启发，我们考虑了一个具有最佳常数b的改进的Hardy不等式，我们正式地写成：-Delta大于或等于(N-2/2)(2)i/\x\(2)+Bv(X)。我们首先给出了关于位势V的必要条件，在此条件下，以前的不等式可以或不能进一步改进。我们证明了最佳常数b在H-0(1)(Omega)中永远不会达到，特别是有没有进一步的修正项与b在H-0(1)(Omega)中的不实现无关。我们的分析表明，通过在右侧增加特定位势，可以反复改善原有的不等式。这导致了Hardy不等式的无穷级数展开。得到的序列在某种意义上是最优的。在建立这些结果的过程中，我们得到了各种改进的Hardy-Sobolev不等式。(C)2002年埃尔塞维尔科学公司(美国)。
Let Omega be a bounded domain in R-N, N greater than or equal to 3, containing the origin. Motivated by a question of Brezis and Vazquez, we consider an Improved Hardy Inequality with best constant b, that we formally write as: -Delta greater than or equal to (N-2/2)(2)i/\x\(2) + bV(x). We first give necessary conditions on the potential V, under which the previous inequality can or cannot be further improved. We show that the best constant b is never achieved in H-0(1)(Omega), and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in H-0(1)(Omega). Our analysis reveals that the original inequality can be repeatedly improved by adding on the right-hand side specific potentials. This leads to an infinite series expansion of Hardy's inequality. The series obtained is in some sense optimal. In establishing these results we derive various sharp improved Hardy-Sobolev Inequalities. (C) 2002 Elsevier Science (USA).