We compute the precise value of the measure of noncompactness of Sobolev embeddings $W_0^{1,p}\left(D\right)\hookrightarrow L^p\left(D\right)$, $p\in (1,\infty )$, on strip-like domains $D$ of the form ${\mathbb{R}}^k\times \prod _{j=1}^{n-k}(r_j,q_j)$. 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.

我们计算了 Sobolev 嵌入的非紧凑性度量的精确值 ${宽}_0^{1,磷}\left(D\right)\hookrightarrow {升}^{磷}\left(D\right)$, $磷\in (1,\infty )$, 在条状域上 $D$ 形式的 ${\mathbb{电阻}}^{克}\times \prod _{j=1}^{n-克}(r_j,q_j)$. 我们表明这种嵌入总是最大程度的非紧凑性，也就是说，它们的非紧凑性度量与其规范一致。此外，我们表明不仅非紧凑性的度量而且所有严格的$秒$- 所讨论的嵌入数量与其规范一致。我们还证明，即使考虑建立在一般重排不变空间上的 Sobolev 型空间，Sobolev 嵌入在条状域上的最大非紧凑性仍然有效。作为副产品，我们获得了伪函数的第一个特征函数的显式形式$磷$- 拉普拉斯算子 $n$维矩形。