Solving variational problems and partial differential equations mapping into general target manifolds

A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(2) (2001) 759]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system where this embedding function is defined. In the case of variational problems, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we re-write all the equation's geometric characteristics with respect to the embedding function. We then obtain a set of equations that, while defined on the whole Euclidean space, are intrinsic to the implicitly defined target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional hypersurfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is a hypersphere; probability maps, where the target manifold is a hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional manifolds.