Approximation of homeomorphisms by piecewise affine homeomorphisms

A solution of a problem in nonlinear Elastostatics is a minimizer of the elastic energy of the deformation of a body. This minimum is searched for among all deformations satisfying certain boundary conditions and, in addition, they must preserve the orientation and be one-to-one (so that interpenetration of matter does not occur). When one approximates by finite elements a solution, it is important that those approximations also preserve the orientation and are one-to-one. This leads to the problem of approximating a homeomorphism by piecewise linear ones. This is a classic problem in Topology that is well understood when the approximation is done in the supremum norm. The results that I present show that, in dimension 2, this approximation is also possible in the Holder norm when the original homeomorphism is Holder continuous.

This is a joint work with J. C. Bellido.