Phd Thesis

Inserted: 10 jul 2014
Last Updated: 9 oct 2014

Year: 2014
Notes:

Defended on July 2, 2014, at Université Paris-Sud (advisor: Prof. Filippo Santambrogio)

Abstract:

The optimal transportation problem, henceforth classical in the calculus of variations, was originally introduced by Monge in the 18th century; it consists in minimizing the total energy of the displacement of a given repartition of mass onto another given repartition of mass. This is mathematically expressed as follows: $ \inf\left\{ \int c(x,T(x)) \, \text{d}\mu(x) \,:\, T_\#\mu=\nu \right\} $ where $c(x,T(x))$ is the cost to send $x$ onto $T(x)$, and the measure $\mu$ and $\nu$ represent the repartitions of source and target masses.

This thesis is devoted to similar variational problems, which involve the Jacobian matrix of the transport map, namely $ \inf\left\{ \int L(x,T(x),DT(x)) \, \text{d}\mu(x) \,:\, T_\#\mu = \nu \right\} \,; $ we typically add $\int
DT
^2$ to the transport functional $\int c(x,T(x)) \, \text{d}\mu(x)$ in order to obtain a Sobolev-type penalization. This kind of constraints finds its motivations in continuum mechanics, incompressible elasticity or shape analysis, and a quite different mathematical approach than in the usual theory of optimal transportation is needed.

We consider the following questions:

- proper definition of the problem, in particular of the term $\int
DT
^2\, \text{d}\mu$, thanks to the theory of Sobolev spaces with respect to a measure, and existence results;

- characterizations of these minimizers: optimality of the monotone transport map on the real line, and Euler-Lagrange-like approach in any dimension;

- selection of a minimizer via a $\Gamma$-convergence-like penalization procedure (namely adding $\varepsilon\int
DT
^2$ to the transport cost, where $\varepsilon$ is a vanishing positive parameter) where the transport cost is the Monge cost $c(x,y)=
x-y
$ (for which the optimal transport map is not unique);

- other related problems and perspectives: dynamic Benamou-Brenier-like formulation, and dual Kantorovich-like formulation.