A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation. Issue 4 (30th September 2019)

• Record Type: Journal Article
• Authors: Ignat, Radu; Monteil, Antonin
• Is Part Of: Communications on pure and applied mathematics. Volume 73:Issue 4(2020:Apr.)
• Page Start: 771
• Page End: 854
• Publication Date: 2019-09-30
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21867
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract : We study the one‐dimensional symmetry of solutions to the nonlinear Stokes equation { − Δ u + ∇ W ( u ) = ∇ p in ℝ d, ∇ ⋅ 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 = ∫ 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.