A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation
(2019) Communications on Pure and Applied Mathematics — Vol. 73, n° 4, p. 771-854 (2019)
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- Abstract
- We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation (Formula presented.) which are periodic in the d − 1 last variables (living on the torus 𝕋d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋d−1, i.e., (Formula presented.). Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x1 → ± ∞ is one-dimensional; i.e., u depends only on the x1-variable. In particular, this class includes in dimension d = 2 the nonlinearities (Formula presented.) 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 x1 → ± ∞.
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Ignat, R., & Monteil, A. (2019). A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation. Communications on Pure and Applied Mathematics, 73(4), 771-854. https://doi.org/10.1002/cpa.21867 (Original work published 2019)