An open mapping theorem for nonlinear operator equations associated with elliptic complexes
Alexander Polkovnikov · Applicable Analysis · 2021
Let {Ai,Ei} be the elliptic complex on an n-dimensional smooth closed Riemannian manifold X with the first-order differential operators Ai and smooth vector bundles Ei over X. We consider nonlinear operator equations, associated with the parabolic differential operators ∂t+Δi, generated by the Laplacians Δi of the complex {Ai,Ei}, in special Bochner–Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frechét derivative Ai′ of the induced nonlinear mapping is continuously invertible and the map Ai is open and injective in chosen spaces.