Measure of noncompactness of Sobolev embeddings on strip-like domains

We compute the precise value of measure noncompactness Sobolev embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on strip-like domains $D$ form $\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. show that such are always maximally noncompact, is, their coincides with norms. Furthermore, we not only but also all strict $s$-numbers in question coincide prove maximal remains valid even when Sobolev-type spaces built upon general rearrangement-invariant considered. As a by-product obtain explicit for first eigenfunction pseudo-$p$-Laplacian an $n$-dimensional rectangle.