University of Maryland

**Abstract**: When lipid molecules are immersed in aqueous
environment they aggregate spontaneously into 2 mono-molecular layers or
(bio)membranes that form an encapsulating bag called vesicle. This happens
because lipids consist of a hydrophilic head group and a hydrophobic tail,
which isolate itself in the interior of the membrane.

As a first approach, we have studied a model based on geometry assuming that the equilibrium shapes are the minimizers of the Willmore energy under area and volume constraints. In this context, the membrane is the preponderant factor influencing the shape of the vesicle. A gradient flow is established to reach these equilibrium shapes. Then, the effect of the inside (bulk) fluid is taken into account leading to more physical dynamics. The boundary conditions couple Stokes equations to the constrained Willmore force.

A parametric approach is employed, which leads to forth order highly nonlinear PDEs on surfaces and involves large domain deformations. An adaptive finite element method (AFEM), with either piecewise linear or quadratic polynomials, is used for both the geometric and coupled problems.