Hodge-Helmholtz Decompositions of Weighted Sobolev Spaces in Irregular Exterior Domains with Inhomogeneous and Anisotropic Media

TL;DR

This paper investigates Hodge-Helmholtz decompositions in irregular exterior domains with complex media, providing foundational tools for electromagnetic theory and translating results to classical vector analysis.

Contribution

It offers detailed decompositions of differential forms in weighted Sobolev spaces within non-smooth, inhomogeneous, and anisotropic exterior domains, extending classical frameworks.

Findings

Decomposition of differential forms into irrotational and solenoidal parts.

Application to electromagnetic theory in complex media.

Translation of results to classical vector analysis.

Abstract

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.