Measure of noncompactness of Sobolev embeddings on strip-like domains

TL;DR

This paper precisely calculates the measure of noncompactness for Sobolev embeddings on strip-like domains, showing they are maximally noncompact and extending results to general rearrangement-invariant spaces.

Contribution

It provides explicit formulas for the measure of noncompactness and all strict s-numbers of Sobolev embeddings on strip-like domains, demonstrating their maximal noncompactness.

Findings

Sobolev embeddings are maximally noncompact on strip-like domains

Measure of noncompactness equals the operator norm for these embeddings

All strict s-numbers coincide with the norm

Abstract

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.