A De Giorgi type conjecture for minimal solutions to a nonlinear Stokes equation

Radu Ignat (1) , Antonin Monteil (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Département de mathématique [Louvain]
3 UCL IRMP - Institut de Recherche en Mathématiques et Physique

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation −∆u + ∇W (u) = ∇p, ∇ · u = 0 in R^d , which are periodic in the d−1 last variables (living on the torus T^{d−1}) and globally minimize the corresponding energy in Ω = R×T^{d−1} , i.e., E(u)=\int_Ω |∇u|^2+W (u) dx, with ∇ · u = 0. Namely, we determine 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=w^2 with w being an 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 → ±∞.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Radu Ignat, Antonin Monteil. A De Giorgi type conjecture for minimal solutions to a nonlinear Stokes equation. Communications on Pure and Applied Mathematics, In press, 73 (4), pp.771-854. ⟨10.1002/cpa.21867⟩. ⟨hal-01772618⟩