## Abstract

This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ a_{i,j}(x) ∂_{i}u∂_{j}u + 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 M_{0}, depending only on n, the bounds on a_{i,j} and Q stated above, such that: given any ω ∈ ℝ^{n}, 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, M_{0}|ω|]}. Furthermore, such u enjoys the following property of "quasi- periodicity": if ω ∈ ℚ^{n}, then u is periodic (with respect to the identification induced by ω; if ω ∈ ℝ^{n} - ℚ^{n}, 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) ∂_{i}u∂_{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 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 ∈ ℝ^{n}, 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, M_{0}|ω|]}. 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.

Original language | English |
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Pages (from-to) | 147-185 |

Number of pages | 39 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 574 |

Publication status | Published - 1 Jan 2004 |

Externally published | Yes |