On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space
On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space
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欧几里德空间子集支持的分布的最大索博列夫正则性
DOI：
10.1142/s021953051650024x
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Moiola
中科院分区：
文献类型：
--
作者：
D. Hewett;A. Moiola
本文讨论如下问题：设$E$的子集$E$的内部为空，且有一个可积参数$1<；p<；\inty$，则在$E$中所支持的Bessel位势Sobolev空间$H^{S，p}(\mathbb{R}^n)$中存在非零分布的最大正则性$S是什么？对于零勒贝格测度集，我们应用位势理论中关于集合容量的著名结果来刻画关于$E$的Hausdorff维度的最大正则性，改进了以前的结果。此外，我们给出了所有可能的最大正则性作为$p$的函数的完整分类，以及获得最大正则性的$p$的值集，并对每种情况构造了具体的例子。对于极大正则性为非负的正测集，我们给出了某些胖康托集所支持的最大Sobolev正则性的新下界，这些下界既是通过能力论论证得到的，也是通过直接估计特征函数的Sobolev范数得到的。我们收集了几个结果，这些结果表征了在某些特殊的集合类上可以达到的正则性，例如$d$-集合，开集的边界，以及笛卡尔乘积，这些都与微分方程组和积分方程式的应用有关。
This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of $p$, together with the sets of values of $p$ for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as $d$-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.
平面屏幕声散射中的波数显式连续性和矫顽力估计
影响因子：
0.8
作者：
Chandler-Wilde S
通讯作者：
Chandler-Wilde S