A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one‐dimensional symmetry of solutions to the nonlinear Stokes equation{− Δ u + ∇ W ( u ) = ∇ pin  ℝ d ,∇ ⋅ u = 0in  ℝ d ,which are periodic in the d  − 1 last variables (living on the torus d −1 ) and globally minimize the corresponding energy in Ω = ℝ × d −1 , i.e., E u = ∫ Ω 1 2∇ u 2 + W u dx , ∇ ⋅ u = 0 .Namely, we find a class of nonlinear potentials W  ≥ 0 such that any global minimizer u of E connecting two zeros of W as x 1  →  ± ∞ is one‐dimensional; i.e., u depends only on the x 1 ‐variable. In particular, this class includes in dimension d  = 2 the nonlinearities W = 1 2 w 2 with w being a harmonic function or a solution to the wave equation, while in dimension d  ≥ 3 , this class contains a perturbation of the Ginzburg‐Landau potential as well as potentials W having d  + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x 1  →  ± ∞ . © 2019 Wiley Periodicals, Inc.