Variational descriptions for various multiphase level set formulations involving curvature flow are discussed. A representation of $n$ phases using $n−1$ level set functions is introduced having the advantage that constraints preventing overlaps or vacuum are not needed. The representation is then used in conjunction with our variational formulation to deduce a novel level set based algorithm for multiphase flow. In addition, a similar variational formulation is applied to area preserving curvature flow. In this flow, the area (or volume in 3D) enclosed by each level set is preserved. Each algorithm has been implemented numerically and the results of such computations are shown.
"A variational formulation for a level set representation of multiphase flow and area preserving curvature flow." Commun. Math. Sci. 6 (1) 125 - 148, March 2008.