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

Abstract

Let $\{A^i,E^i\}$ be the elliptic complex on an 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 ${\mathrm{\partial }}_t+{\mathrm{\Delta }}^i$, generated by the Laplacians ${\mathrm{\Delta }}^i$ 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.

Keywords:

• Elliptic differential complexes
• parabolic nonlinear equations
• open mapping theorem

2010 Mathematics Subject Classifications:

• 58J10
• 35K45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘BASIS’.