Elliptic operators on non-compact manifolds have closed range

TL;DR

This paper proves that second-order elliptic differential operators on any manifold have closed range in smooth functions, and are surjective if the manifold has no compact components, with applications to Helmholtz decomposition.

Contribution

It establishes the closed range property for elliptic operators on non-compact manifolds and characterizes surjectivity in the absence of compact components.

Findings

Elliptic operators have closed range in $C^ ablafty(M)$.

Operators are surjective on non-compact manifolds without compact components.

Applications to Helmholtz decomposition are discussed.

Abstract

We show that a second-order elliptic differential operator $P$, on any manifold $M$, has closed range in $C^\infty(M)$. If $M$ has no compact components, then $P$ is surjective on $C^\infty(M)$. Applications to Helmholtz decomposition are discussed.