Attainability of the best constant of Hardy–Sobolev inequality with full boundary singularities

IF 1 2区 数学 Q1 MATHEMATICS
Liming Sun, Lei Wang
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引用次数: 0

Abstract

We consider a type of Hardy–Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non-convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non-umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show that the best constant is not achieved if the domain is sufficiently close to a ball in C 2 $C^2$  sense.

考虑一类权函数在整个区域边界上奇异的Hardy-Sobolev不等式。我们关心的是这种不平等的最佳常数的可得性。在二维中,我们利用定义域的共形半径将不等式与一个共形不变不等式联系起来。该不等式在光滑有界区域上的最佳常数当且仅当该区域是非凸的。在更高的维度中,如果区域在某个地方具有负的平均曲率,则可以获得最佳常数。如果平均曲率消失,但在某些地方非脐,我们也建立了一些特殊情况下的可得性。在另一个方向上,我们也证明了如果区域在c2 $C^2$意义上足够接近球,则不会达到最佳常数。
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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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