  We consider a class of liquid crystal free-boundary problems for which both
the equilibrium shape and internal configuration of a system must
simultaneously be determined, for example in films with air- or fluid-liquid
crystal interfaces and elastomers. We develop a finite element algorithm to
solve such problems with dynamic mesh control, achieved by supplementing the
free energy with an auxiliary functional that promotes mesh quality and is
minimized in the null space of the energy. We apply this algorithm to a
flexible capacitor, as well as to determine the shape of liquid crystal
tactoids as a function of the surface tension and elastic constants. These are
compared with theoretical predictions and experimental observations of tactoids
from the literature.
