## DG - MP - PDE Seminar: Nassif Ghoussoub (UBC)

- Date: 09/13/2011
- Time: 15:30

University of British Columbia

A self-dual polar factorization for vector fields

Abstract

We show that any non-degenerate vector field u in L^{\infty}(\Omega,

\R^N), where \Omega is a bounded domain in \R^N, can be written as

{equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega},

{equation} where S is a measure preserving point transformation on

\Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to

\R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian.

Moreover, u is a monotone map if and only if S can be taken to be the

identity, which suggests that our result is a self-dual version of

Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi

(S(x)), where \phi is convex and S is a measure preserving

transformation. We also describe how our polar decomposition can be

reformulated as a self-dual mass transport problem.

For further information, please see the event page at: http://www.math.ubc.ca/Dept/Events/index.shtml?period=future&series=all.