Bump solutions for the mesoscopic Allen-Cahn equation in periodic media

Matteo Novaga; Enrico Valdinoci

doi:10.1007/s00526-010-0332-4

Bump solutions for the mesoscopic Allen-Cahn equation in periodic media

Matteo Novaga, Enrico Valdinoci

Research output: Contribution to journal › Article

11 Citations (Scopus)

Abstract

Given a double-well potential F, a ℤⁿ-periodic function H, small and with zero average, and ε > 0, we find a large R, a small δ and a function H_ε which is ε-close to H for which the following two problems have solutions:1. Find a set E_ε,R whose boundary is uniformly close to ∂ B_R and has mean curvature equal to -H_ε at any point, 2. Find u = u_ε,R,δ solving such that u_ε,R,δ goes from a δ-neighborhood of +1 in B_R to a δ-neighborhood of -1 outside B_R.

Original language	English
Pages (from-to)	37-49
Number of pages	13
Journal	Calculus of Variations and Partial Differential Equations
Volume	40
Issue number	1
DOIs	https://doi.org/10.1007/s00526-010-0332-4
Publication status	Published - 1 Jan 2011
Externally published	Yes

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abstract = "Given a double-well potential F, a ℤn-periodic function H, small and with zero average, and ε > 0, we find a large R, a small δ and a function Hε which is ε-close to H for which the following two problems have solutions:1. Find a set Eε,R whose boundary is uniformly close to ∂ BR and has mean curvature equal to -Hε at any point, 2. Find u = uε,R,δ solving such that uε,R,δ goes from a δ-neighborhood of +1 in BR to a δ-neighborhood of -1 outside BR.",

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AB - Given a double-well potential F, a ℤn-periodic function H, small and with zero average, and ε > 0, we find a large R, a small δ and a function Hε which is ε-close to H for which the following two problems have solutions:1. Find a set Eε,R whose boundary is uniformly close to ∂ BR and has mean curvature equal to -Hε at any point, 2. Find u = uε,R,δ solving such that uε,R,δ goes from a δ-neighborhood of +1 in BR to a δ-neighborhood of -1 outside BR.

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