Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers
Abstract
There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-\v{S}mulian theorem). However, the existence theorems for many variational problems over non-reflexive Sobolev spaces remain underexplored. In this paper, we investigate various examples of functionals over non-reflexive Sobolev spaces. To do this, we prove a weak compactness criterion in that generalizes the Dunford-Pettis theorem, which asserts that relatively weakly compact subsets of coincide with equi-integrable families. As a corollary, we also extend an existence theorem of minimizers from reflexive Sobolev spaces to non-reflexive ones. This work is also benefited and streamlined by various concepts in category theory.
Keywords
Cite
@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}
}
Comments
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