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The coupling of electric fields to the mechanics of lipid membranes givesrise to intriguing electromechanical behavior, as, for example, evidenced bythe deformation of lipid vesicles in external electric fields.Electromechanical effects are relevant for many biological processes, such asthe propagation of action potentials in axons and the activation ofmechanically-gated ion channels. Currently, a theoretical framework describingthe electromechanical behavior of arbitrarily curved and deforming lipidmembranes does not exist. Purely mechanical models commonly treat lipidmembranes as two-dimensional surfaces, ignoring their finite thickness. Whileholding analytical and numerical merit, this approach cannot describe thecoupling of lipid membranes to electric fields and is thus unsuitable forelectromechanical models. In a sequence of articles, we derive an\textit{effective} surface theory of the electromechanics of lipid membranes,named a $(2+\delta)$-dimensional theory, which has the advantages of surfacedescriptions while accounting for finite thickness effects. The present articleproposes a new, generic dimension-reduction procedure relying on low-orderspectral expansions. This procedure is applied to the electrostatics of lipidmembranes to obtain a $(2+\delta)$-dimensional theory that captures potentialdifferences across and electric fields within lipid membranes. This new modelis tested on different geometries relevant for lipid membranes, showing goodagreement with the corresponding three-dimensional electrostatics theory.

The $(2 + \delta)$-dimensional theory of the electromechanics of lipid membranes: I. Electrostatics