Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals

This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ a_i,j(x) ∂_iu∂_ju + Q(x)χ_(-1,1)(u) dx, for a_i,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and a_i,j is a bounded elliptic matrix. We prove that there exists a universal constant M₀, depending only on n, the bounds on a_i,j and Q stated above, such that: given any ω ∈ ℝⁿ, there exists a class A minimizer u for the functional script J sign for which the set {|u| < 1} is contained in the strip {x s.t. x · ω ∈ [0, M₀|ω|]}. Furthermore, such u enjoys the following property of "quasi- periodicity": if ω ∈ ℚⁿ, then u is periodic (with respect to the identification induced by ω; if ω ∈ ℝⁿ - ℚⁿ, then u can be approximated uniformly on compact sets by periodic class A minimizers. The functional script J sign introduced above has application in models for fluid jets and capillarity. In the second part of the paper, we extend the previous results to the functional script G sign(u) = ∫ a_i,j(x) ∂_iu∂_ju + F(x,u) dx, where a_i,j is Lipschitz and satisfies the same assumption mentioned above and F fulfills suitable conditions (roughly, F is a "double-well potential"). We prove that, for any θ ∈ [0, 1), there exists a constant M ₀, depending only on θ, n, the bounds on a_i,j and F stated above, such that: given any ω ∈ ℝⁿ, there exists a class A minimizer u for the functional script G sign for which the set {|u| ≦ θ} is contained in the strip {x s.t. x · ω ∈ [0, M₀|ω|]}. Also, u enjoys the property of "quasi-periodicity" stated above. In particular, the results apply to the potentials F(x) = Q(x)(1-u²), and F = Q(x)|1 - u ²|², which have application in the Ginzburg-Landau model. In the third part, we show how the results above contain, as a limit case, the recent Theorem of Caffarelli and de la Llave.