论文标题
有限球条件的无界域的分数庞加莱不等式：反示例
Fractional Poincaré Inequality for Unbounded Domains with Finite Ball Condition: Counter Example
论文作者
论文摘要
在本文中，我们调查了无限域中的分数庞加莱不平等。在当地情况下，桑迪普·摩西尼（Sandeep-Mancini）表明，在简单连接的域类别中，当且仅当域不允许任意较大的半径（有限球状态）的球时，庞加莱的不等式才能保持。我们证明，即使有限的球条件被相关的较强条件取代，这种结果即使在“非本地/分数”设置中不可能是正确的。我们进一步提供了一些足够的标准，以使庞加莱的分数不平等现象。最后，解决了长圆柱域上分数迪里奇问题的所有特征值的渐近行为。
In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain does not allow balls of arbitrarily large radius (finite ball condition). We prove that such a result can not be true in the `nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all eigenvalues of fractional Dirichlet problems on long cylindrical domains is addressed.
