Fractional De Giorgi property for the two-component Bose–Einstein condensate system

Establish the De Giorgi property—namely, one-dimensional symmetry under a monotonicity assumption—for the nonlocal two-component Bose–Einstein condensate system involving the fractional Laplacian (−Δ)^s: prove that this property holds in ℝ^3 when s ≥ 1/2 and in ℝ^2 when s < 1/2.

Background

The paper is the first in a series analyzing phase transitions and De Giorgi-type results for a coupled system modeling two-component Bose–Einstein condensates. While the present work treats the local (Laplacian) case, the authors point toward a sequel studying the nonlocal fractional Laplacian counterpart.

In that sequel direction, they put forward a concrete conjecture specifying threshold regimes in dimension and order of the fractional Laplacian under which the De Giorgi property is expected to hold, thus framing a precise open problem for the nonlocal coupled system.