A De Giorgi type conjecture for elliptic systems under the divergence constraint

We study the symmetry of transition layers in Ginzburg-Landau type functionals for divergence-free maps in N-dimensions. Namely, we determine a class of nonlinear potentials such that the minimal transition layers in the periodic strip domain Ω = R × T^{N −1} are one-dimensional where T = R/Z is the 1-torus. In particular, this class includes in dimension N = 2 the nonlinear potentials 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 zeros with prescribed transition cost between the zeros. For that, we develop a theory of calibrations for divergence-free maps in dimension N (similar to the theory of entropies for the Aviles-Giga model when N = 2). We also give a necessary condition for finite energy configurations yielding the boundary condition in L^2 and almost everywhere in T^{N −1} as x_1 → ±∞