论文标题
sobolev嵌入在带状域中的非伴动度的度量
Measure of noncompactness of Sobolev embeddings on strip-like domains
论文作者
论文摘要
我们计算sobolev嵌入的非碰撞度量度的确切价值$ \ mathbb {r}^k \ times \ prod \ limits_ {i = 1}^{n-k}（a_i，b_i）$。我们表明，这种嵌入始终是最大的非绘制，也就是说，它们的非伴侣度量与其规范相吻合。此外，我们不仅表明，不仅衡量了不相同的措施，而且还严格$ s $ numbers涉及的嵌入与其规范相吻合。我们还证明，即使考虑了sobolev型式空间，sobolev嵌入在带状域上的最大非伴侣度仍然有效。作为副产品，我们为$ n $维矩形上的伪$ p $ -laplacian的第一个特征功能获得了明确表格。
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.
