We present complete analytical solutions describing the deformations of both
rectangular and circular lipid membranes subjected to local inflammations and
coordinate-dependent (nonuniform) property distributions. The membrane energy
potential of the Helfrich type is refined to accommodate the coordinate-dependent
responses of the membranes. Within the description of the superposed incremental
deformations and Monge parametrization, a linearized version of the shape equation
describing coordinate-dependent membrane morphology is obtained. The local
inflammation of a lipid membrane is accommodated by the prescribed uniform
internal pressure and/or lateral pressure. This furnishes a partial differential equation
of Poisson type from which a complete analytical solution is obtained by employing
the method of variation of parameters. The solution obtained predicts the
smooth and coordinate-dependent morphological transitions over the domain of
interest and is reduced to those from the classical uniform membrane shape
equation when the equivalent energy potential is applied. In particular, the
obtained model closely assimilated the pressure-induced inflammations of lipid
membranes where only quantitatively equivalent analyses were reported via the
impositions of equivalent edge moments. Lastly, we note that the principle of
superposition remains valid even in the presence of coordinate-dependent membrane
properties.
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