论文标题
在sobolev-slobodecki \uı空间中尖锐的不平等
On the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces
论文作者
论文摘要
我们研究了在欧几里得空间的开放子集上定义的分数Sobolev空间的耐力不等式的急剧常数。我们首先列出了这样常数以及相关的变分问题的一些属性。然后，我们将讨论限制为打开凸集，并通过距离函数构造合适的超溶液来计算如此锐利的常数。这种证明方法仅适用于$ s \，p \ ge 1 $或$ω$的半空间。我们展示了一个简单的示例，表明此方法与$ s \，p <1 $和$ω$不同，与半空间不同。对于普通凸组的情况，$ s \，p <1 $是一个有趣的开放问题，除了在希尔伯特式设置中（即$ p = 2 $）：在这种情况下，我们可以计算整个范围内的急剧常数$ 0 <s <1 $。这完成了文献中保持开放的结果。
We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for $s\,p\ge 1$ or for $Ω$ being a half-space. We exhibit a simple example suggesting that this method can not work for $s\,p<1$ and $Ω$ different from a half-space. The case $s\,p<1$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for $p=2$): in this case we can compute the sharp constant in the whole range $0<s<1$. This completes a result which was left open in the literature.
