Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929609771057152 |
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| author | Chen, Cheng Ji, Mattie Tang, Yan Zhang, Shiqing |
| author_facet | Chen, Cheng Ji, Mattie Tang, Yan Zhang, Shiqing |
| contents | There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-Šmulian 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 $W^{k,1}$ that generalizes the Dunford-Pettis theorem, which asserts that relatively weakly compact subsets of $ L^1 $ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_15871 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers Chen, Cheng Ji, Mattie Tang, Yan Zhang, Shiqing Functional Analysis 28A20, 46E30, 46E35, 46N20, 49A99 There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-Šmulian 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 $W^{k,1}$ that generalizes the Dunford-Pettis theorem, which asserts that relatively weakly compact subsets of $ L^1 $ 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. |
| title | Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers |
| topic | Functional Analysis 28A20, 46E30, 46E35, 46N20, 49A99 |
| url | https://arxiv.org/abs/2306.15871 |