An open mapping theorem for nonlinear operator equations associated with elliptic complexes

TL;DR

This paper establishes an open mapping theorem for nonlinear equations related to elliptic complexes on smooth manifolds, demonstrating invertibility and openness of the associated nonlinear operators under certain conditions.

Contribution

It introduces an open mapping theorem for nonlinear operator equations tied to elliptic complexes, extending classical results to this geometric setting.

Findings

Frechét derivative invertibility under certain assumptions

Nonlinear map is open and injective in specific spaces

Results applicable to elliptic complexes on Riemannian manifolds

Abstract

Let $ \{A^i,E^i\} $ be the elliptic complex on a $ n $-dimensional smooth closed Riemannian manifold $X$ with the first order differential operators $ A^i $ and smooth vector bundles $ E^i $ over $X$. We consider nonlinear operator equations, associated with the parabolic differential operators $\partial_t + \Delta^{i} $, generated by the Laplacians $\Delta^{i} $ of the complex $ \{A^i,E^i\}$, in special Bochner-Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frech\'et derivative $ \mathcal{A}_i' $ of the induced nonlinear mapping is continuously invertible and the map $ \mathcal{A}_i $ is open and injective in chosen spaces.