Calc. Var. (2017) 56:101
Calculus of Variations
An iterative method for generated Jacobian equations
· Cristian E. Gutiérrez
Received: 20 December 2016 / Accepted: 15 June 2017
© Springer-Verlag GmbH Germany 2017
Abstract The purpose of this paper is to present an iterative scheme to ﬁnd approximate
solutions, to any preset degree of accuracy, for a class of generated Jacobian equations
introduced in Trudinger (Discrete Contin Dyn Syst 34(4): 1663–1681 2014). A proof of
an upper bound on the number of iteration steps, without assuming any condition of Ma-
Trudinger-Wang type, is presented. Applications to optimal mass transport and to the parallel
reﬂector problem in optics are also provided.
Mathematics Subject Classiﬁcation 49M25 · 65K15 · 35J96 · 78A05
An iterative scheme for ﬁnding approximate solutions to the far-ﬁeld reﬂector problem in
geometric optics was ﬁrst developed by Caffarelli et al. . These authors consider the global
reﬂector problem with a smooth source density which allows them to show differentiability of
the reﬂector map, which in turn implies that the method converges in a ﬁnite number of steps.
In several settings, the global problem does not make physical sense and one needs to deal
with non-smooth densities. In particular, the differentiability of the transport map strongly
depends on smoothness properties of the domains considered and therefore may not hold in
general. However, it is realized in  for the one source far-ﬁeld refractor problem introduced
in , that the method in  can be simpliﬁed and extended by showing that the refractor
Communicated by L. Ambrosio.
Research partially supported by NSF Grant DMS–1600578.
Cristian E. Gutiérrez
Department of Mathematics, Temple University, Philadelphia, PA 19122, USA