Measure of noncompactness of Sobolev embeddings on strip-like domains
| Autor: | David E. Edmunds, Jan Lang, Zdeněk Mihula |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Numerical Analysis
Pure mathematics Applied Mathematics General Mathematics 010102 general mathematics Of the form 010103 numerical & computational mathematics Eigenfunction 01 natural sciences Measure (mathematics) Functional Analysis (math.FA) Sobolev space Mathematics - Functional Analysis 41A46 46E35 46E30 46B50 FOS: Mathematics Rectangle Mathematics::Differential Geometry 0101 mathematics Value (mathematics) Laplace operator Analysis Mathematics |
| Popis: | 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. 12 pages |
| Databáze: | OpenAIRE |
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