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 J(u) = integral a(i,j)(x)partial derivative(i)upartial derivative(j) u+Q(x) chi((-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 M-0, depending only on n, the bounds on a(i,j) and Q stated above, such that: given any omega is an element of R-n, there exists a class A minimizer u for the functional J for which the set {\u\ < 1} is contained in the strip {x s: t: x omega is an element of [0, M-0\omega\]}. Furthermore, such u enjoys the following property of "quasi-periodicity'': if omega is an element of Q(n), then u is periodic (with respect to the identification induced by omega); if omega is an element of R-n - Q(n), then u can be approximated uniformly on compact sets by periodic class A minimizers. The functional J 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 G(u) = integral a(i,j)(x)partial derivative(i)u partial derivative(j)u + 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 theta is an element of [0, 1), there exists a constant M-0, depending only on theta, n, the bounds on a(i,j) and F stated above, such that: given any omega is an element of R-n, there exists a class A minimizer u for the functional G for which the set {\u\<=theta} is contained in the strip {x s: t: x center dot omega is an element of [0, M-0\omega\]}. Also, u enjoys the property of "quasi-periodicity'' stated above. In particular, the results apply to the potentials F(x) = Q(x)(1 - u(2)), 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.