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

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 Frechét derivative ${\mathcal{A}}_i^{\prime }$ of the induced nonlinear mapping is continuously invertible and the map ${\mathcal{A}}_i$ is open and injective in chosen spaces.

23 pages