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The shape equation for an axisymmetric fluid membrane is derived, assumingaction of an uniform external electric field. The flexoelectric contribution tothe free energy of the membrane, stemming from the latter is accounted withinthe theory by Steigmann and Agrawal. Additionally, we have introduced, in theaforementioned functional, another term associated with a curvature inducedmembrane polarization, as the latter was first hypothesized by A. Petrov. Someexact Naito-type solutions of the studied equation are given, with the freeparameters linked to the model ones.

Flexoelectric fluid membranes in electric field. Shape equations and exact solutions