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 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.