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

TL;DR

This paper extends the theory of minimizer existence to non-reflexive Sobolev spaces by establishing a weak compactness criterion in $W^{k,1}$, generalizing classical results and employing category theory concepts.

Contribution

It introduces a weak compactness criterion in $W^{k,1}$ that generalizes the Dunford-Pettis theorem and extends minimizer existence theorems to non-reflexive Sobolev spaces.

Findings

Proved a weak compactness criterion in $W^{k,1}$.

Extended existence theorems for minimizers to non-reflexive Sobolev spaces.

Utilized concepts from category theory to streamline the analysis.

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 $W^{k,1}$ 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.