On a homotopy formula for generalized steady Stokes' operators, associated with the de Rham complex

TL;DR

This paper develops fundamental solutions for generalized steady Stokes' operators linked to the de Rham complex, leading to a homotopy formula for regular solutions in differential forms.

Contribution

It constructs explicit fundamental solutions for these operators and derives a homotopy formula, extending classical results to a more general setting.

Findings

Constructed left, right, and bilateral fundamental solutions for the operators.

Proved the operators are Douglis-Nirenberg elliptic under certain conditions.

Derived a homotopy formula for regular solutions to the operators.

Abstract

We construct left, right and bilateral fundamental solutions for generalized steady Stokes' operators $S$ with smooth coefficients coefficients, associated with the de Rham complex of differentials on differential forms over a domain $X$ in ${\mathbb R}^n$. The investigated operators are Douglis-Nirenberg elliptic under reasonable assumptions. As an immediate corollary we produce a homotopy formula for regular solutions to this operator.