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引用次数: 0
摘要
此外,我们还证明,D 上存在这样一个有界的、严格的次谐函数等同于以牛顿容量衡量的 D 的严格半径的有限性。我们还根据这个有界半径的概念,得到了 Dirichlet-Laplacian 最小特征值的尖锐上限。
On the Poincaré Inequality on Open Sets in $$\mathbb {R}^n$$
We show that the Poincaré inequality holds on an open set \(D\subset \mathbb {R}^n\) if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict inradius of D measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet–Laplacian.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.