Reflections on Dubinskii's nonlinear compact embedding theorem

TL;DR

This paper reviews Dubinskii's nonlinear compact embedding theorem, introduces a variant using a seminormed cone, and explores its connections with Maitre's nonlinear compact embedding results, enhancing understanding of compact embeddings in Banach spaces.

Contribution

It presents a new variant of Dubinskii's theorem using seminormed cones and links it to Maitre's nonlinear compact embedding theorem.

Findings

Established a variant of Dubinskii's theorem with seminormed cones.

Connected Dubinskii's results with Maitre's nonlinear embedding theorem.

Enhanced understanding of compact embeddings in Banach spaces.

Abstract

We present an overview of a result by Ju. A. Dubinskii [Mat. Sb. 67 (109) (1965); translated in Amer. Math. Soc. Transl. (2) 67 (1968)], concerning the compact embedding of a seminormed set in $L^p(0,T; \mathcal{A}_0)$, where $\mathcal{A}_0$ is a Banach space and $p \in [1,\infty]$; we establish a variant of Dubinskii's theorem, where a seminormed nonnegative cone is used instead of a seminormed set; and we explore the connections of these results with a nonlinear compact embedding theorem due to E. Maitre [Int. J. Math. Math. Sci. 27 (2003)].