A modified least action principle allowing mass concentrations for the early universe reconstruction problem

Brenier, Yann

Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 361-385.

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [46]. By substituting the fully nonlinear Monge–Ampère equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model ("Monge-Ampère gravitation/MAG"), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developed by Ghoussoub [29]. A fully discrete algorithm is introduced for the EUR problem in one space dimension.

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