Premium
A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation
Author(s) -
Ignat Radu,
Monteil Antonin
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21867
Subject(s) - mathematics , conjecture , nonlinear system , compact space , dimension (graph theory) , torus , mathematical analysis , scalar (mathematics) , type (biology) , class (philosophy) , entropy (arrow of time) , mathematical physics , pure mathematics , quantum mechanics , physics , geometry , ecology , artificial intelligence , computer science , biology
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.