  This article is the first in a series of papers on the analysis of a basic
model for fluid vesicle dynamics. There are two variants of this model, a
parabolic one decribing purely relaxational dynamics and a non-parabolic one
containing the full dynamics. At the heart of both variants lies a linear
elliptic system of Stokes-type. Understanding the mapping properties of this
Stokes-type system is crucial for all further analysis. In this article we give
a basic exposition of the dynamical model and a thorough $L_2$-analysis of the
Stokes-type system that takes into account geometric variations of the fluid
vesicle.
