W^{k,1}中的弱紧性判据及极小元存在定理
泛函分析
2024-12-03 v2
摘要
关于自反Sobolev空间上极小元存在定理已有丰富理论（例如Eberlein-Šmulian定理）。然而，许多定义于非自反Sobolev空间上的变分问题的存在定理仍鲜有探索。本文研究非自反Sobolev空间上泛函的若干例子。为此，我们证明了一个W^{k,1}中的弱紧性判据，它推广了Dunford-Pettis定理——该定理断言L^1中相对弱紧子集等价于等可积族。作为推论，我们也将极小元存在定理从自反Sobolev空间推广到非自反情形。范畴论中的若干概念也使本文工作受益并得以简化。
引用
@article{arxiv.2306.15871,
title = {Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers},
author = {Cheng Chen and Mattie Ji and Yan Tang and Shiqing Zhang},
journal= {arXiv preprint arXiv:2306.15871},
year = {2024}
}
备注
We added Theorem 3, which significantly simplified the proof of Proposition 9, Corollary 5 and 7, a new section in the appendix cataloging all the notations used, and 3 new figures, with minor edits to improve the flow of the writing