Solvability of generalized div-curl system and Friedrichs inequalities

TL;DR

This paper investigates the solvability of generalized div-curl systems, providing compatibility conditions, existence results for solutions in Sobolev spaces, and Friedrichs inequalities, with implications for decompositions of Sobolev spaces.

Contribution

It establishes necessary conditions for solutions, proves existence of $W^{m,p}$ solutions, and introduces new Friedrichs inequalities and Sobolev space decompositions for generalized div-curl systems.

Findings

Full description of compatibility conditions for solutions.

Existence of $W^{m,p}$ solutions under these conditions.

New Friedrichs inequalities relating gradient estimates to div and curl.

Abstract

In this article, we consider the solvability of two generalized div-curl systems. They are referred to as the equations of magnetostatics and electro- , resp.. Necessary compatibility conditions on the data for the existence of solutions are fully described, and existence of $W^{m, p}$-solutions is proved. Moreover, we give some description for the null spaces. As a corollary, we give the estimates of gradient via generalized ${\rm div}$ and ${\rm curl}$ in $L^p$-framework, which can be considered as Friedrichs inequalities. Furthermore, two decompositions of Sobolev spaces are given.