Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers

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 $W^{k,1}$ 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.