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

Abstract

We study the one‐dimensional symmetry of solutions to the nonlinear Stokes equation {left−Δu+∇W(u)=∇pin ℝd,left∇⋅u=0in ℝd, 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., E()u=∫Ω12∇u2+W()uitalicdx,1em∇⋅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 x1 → ± ∞ is one‐dimensional; i.e., u depends only on the x1‐variable. In particular, this class includes in dimension d = 2 the nonlinearities W=12w2 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 → ± ∞. © 2019 Wiley Periodicals, Inc.

A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips

“…In this context, we have proved in our previous paper [24] that the x ′ -average map ū admits limits u ± as x 1 → ±∞, where u ± 1 = a and they are two wells of W (a, •), see [24,Lemma 3.7]. As in Theorem 1, we will prove that u(x 1 , •) converges to u ± in L 2 and a.e.…”

“…Then the x ′ -average ū : R → R d is continuous and its first component is constant, i.e., there is a ∈ R such that ū1 (x 1 ) = a for every x 1 ∈ R (see [24,Lemma 3.1]). For such maps u, we consider potentials W satisfying the following two conditions:…”

“…Link with micromagnetic models. We have studied Question in the context of divergence-free maps u : R × ω → R N where d = N and ω = T d−1 is the (d − 1)-dimensional torus, see [24]. By developing a theory of calibrations, we have succeeded to give sufficient conditions on the potential W in order that the answer to Question is positive, in particular in the case where (H1) a and (H2) a are satisfied, see [24,Theorem 2.11].…”

Dimension Reduction and Optimality of the Uniform State in a Phase-Field-Crystal Model Involving a Higher-Order Functional

“…Very few analytical results are available: we mention in particular the result in [6] for the one-dimensional symmetry in the extended Fisher-Kolmogorov model in R N . Also, the results in [19] for the one-dimensional symmetry in the Aviles-Giga type models in R N (recall that in 2-dimensions, the standard Aviles-Giga model can be seen as a forth order problem in the stream function corresponding to the order parameter, see [1,2,3,20]). which is nonnegative thanks to (1.10).…”

Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness

“…Jin and Kohn [30] noticed that the divergence of the "Jin-Kohn entropy" For the unit square and boundary conditions u = 0, ∂u ∂n = −1, they proved that the lower bound can be achieved by the "1D" ansatz u ε = ax + f ε (y) when the associated defect set of the limiting map lim ε→0 u ε is parallel to the x axis, corroborating Aviles-Giga's conjecture regarding the onedimensionality of the transition region. Recently, Ignat and Monteil [28] proved that any minimizer of (1.13) on an infinite strip is one-dimensional. By considering the supremum of the divergences of all rotated versions of Σu, Aviles and Giga [6] derived a limiting functional J : W 1,3 (Ω) → [0, ∞) which is lower semicontinuous with respect to strong topology in W 1,3 (Ω) and coincides with F 0 for any u satisfying the eikonal equation with ∇u ∈ BV (Ω).…”