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The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature. 

The Helfrich energy is commonly used to model the elastic bending energy of lipid bilayers in membrane mechanics. The governing differential equations for certain geometric characteristics of the shape of the membrane can be obtained by applying variational methods (minimization principles) to the Helfrich energy functional and are well studied in the axisymmetric framework. However, the Helfrich energy functional and the resulting differential equations involve a number of parameters, and there is little explanation of the choice of parameters in the literature, particularly with respect to the choice of the “spontaneous curvature” term that appears in the functional. In this paper, we present a careful analytical and numerical study of certain aspects of parametric sensitivity of Helfrich’s model. Using simulations of specific model systems, we demonstrate the application of our scheme to the formation of spherical buds and pearled shapes in membrane vesicles.