A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains

TL;DR

This paper extends the div-curl-lemma to weak Lipschitz domains with mixed boundary conditions in 3D, using advanced tools like the de Rham complex and Maxwell's equations, impacting homogenization and compactness results.

Contribution

It establishes a global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains, a significant advancement in the mathematical analysis of PDEs and homogenization.

Findings

Proves a global div-curl-lemma in 3D weak Lipschitz domains.

Utilizes de Rham complex and Weck's selection theorem.

Enhances understanding of Maxwell's equations in complex domains.

Abstract

We prove a global version of the so-called div-curl-lemma, a crucial result for compensated compactness and in homogenization theory, for mixed tangential and normal boundary conditions in bounded weak Lipschitz domains in 3D and weak Lipschitz interfaces. The crucial tools and the core of our arguments are the de Rham complex and Weck's selection theorem, the essential compact embedding result for Maxwell's equations.