带状域上Sobolev嵌入的非紧性度量
Measure of noncompactness of Sobolev embeddings on strip-like domains
摘要：
我们计算的精确值的Sobolev嵌入$W_0^{1,p}(D)\hookrightarrow L^p(D)$，$p\in(1,\infty)$，带状域$D$的形式$\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$的非紧性的措施。我们证明了这样的嵌入总是极大非紧的，也就是说，它们的非紧性测度与它们的范数一致。此外，我们证明了不仅非紧性的措施，而且所有的严格$s$ -数的嵌入问题符合他们的范数。我们还证明了最大非紧性的Sobolev嵌入带状域上仍然有效，即使Sobolev型空间建立在一般的参考不变空间被认为是。作为副产品，我们得到的第一个本征函数的伪$p$拉普拉斯算子上的$n$维矩形的显式形式。
英文摘要：
We compute the precise value of the measure of noncompactness of Sobolev embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on strip-like domains $D$ of the form $\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict $s$-numbers of the embeddings in question coincide with their norms. We also prove that the maximal noncompactness of Sobolev embeddings on strip-like domains remains valid even when Sobolev-type spaces built upon general rearrangement-invariant spaces are considered. As a by-product we obtain the explicit form for the first eigenfunction of the pseudo-$p$-Laplacian on an $n$-dimensional rectangle.
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