A DeGiorgi type conjecture for minimal solutions to a nonlinear Stokes equation

The aim is to study the symmetry of transition layers in Ginzburg-Landau type functionals for divergence-free maps in R^N. Namely, we determine a class of nonlinear potentials such that the minimal transition layers are one-dimensional.

In particular, this class includes in dimension N=2 the nonlinearities w^2 with w being an harmonic function or a solution to the wave equation, while in dimension N>2, this class contains a perturbation of the standard Ginzburg-Landau potential as well as potentials having N+1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations for divergence-free maps in R^N(similar to the theory of entropies for the Aviles-Giga model when N=2).

This is a joint work with Antonin Monteil (Louvain, Belgium).