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

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Abstract

This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ ai,j(x) ∂iu∂ju + Q(x)χ(-1,1)(u) dx, for ai,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and ai,j is a bounded elliptic matrix. We prove that there exists a universal constant M0, depending only on n, the bounds on ai,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) = ∫ ai,j(x) ∂iu∂ju + F(x,u) dx, where ai,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 ai,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.

Original languageEnglish
Pages (from-to)147-185
Number of pages39
JournalJournal fur die Reine und Angewandte Mathematik
Issue number574
Publication statusPublished - 1 Jan 2004
Externally publishedYes

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Periodic Media
Jet Flow
Ginzburg-Landau
Minimizer
Quasiperiodicity
Strip
Capillarity
Ginzburg-Landau Model
Double-well Potential
Compact Set
Fluids
Lipschitz
Fluid
Integer
Zero
Theorem
Class
Model

Cite this

@article{e28fb33c78f3483ea8f6f9d1acb574a0,
title = "Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals",
abstract = "This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ ai,j(x) ∂iu∂ju + Q(x)χ(-1,1)(u) dx, for ai,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and ai,j is a bounded elliptic matrix. We prove that there exists a universal constant M0, depending only on n, the bounds on ai,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) = ∫ ai,j(x) ∂iu∂ju + F(x,u) dx, where ai,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 ai,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.",
author = "Enrico Valdinoci",
year = "2004",
month = "1",
day = "1",
language = "English",
pages = "147--185",
journal = "Journal fuer die Reine und Angewandte Mathematik: Crelle's journal",
issn = "0075-4102",
publisher = "Walter de Gruyter GmbH (European Journal of Nanomedicine)",
number = "574",

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TY - JOUR

T1 - Plane-like minimizers in periodic media

T2 - Jet flows and Ginzburg-Landau-type functionals

AU - Valdinoci, Enrico

PY - 2004/1/1

Y1 - 2004/1/1

N2 - This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ ai,j(x) ∂iu∂ju + Q(x)χ(-1,1)(u) dx, for ai,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and ai,j is a bounded elliptic matrix. We prove that there exists a universal constant M0, depending only on n, the bounds on ai,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) = ∫ ai,j(x) ∂iu∂ju + F(x,u) dx, where ai,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 ai,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.

AB - This paper is divided into three parts. In the first part, we consider the functional script J sign(u) = ∫ ai,j(x) ∂iu∂ju + Q(x)χ(-1,1)(u) dx, for ai,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and ai,j is a bounded elliptic matrix. We prove that there exists a universal constant M0, depending only on n, the bounds on ai,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) = ∫ ai,j(x) ∂iu∂ju + F(x,u) dx, where ai,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 ai,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.

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