Abstract
Given a double-well potential F, a \({\mathbb{Z}^n}\)-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:
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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,
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2.
Find u = u ε ,R,δ solving
$$ -\delta\,\Delta u + \frac{F'(u)}{\delta} +\frac{c_0}{2} H_\varepsilon = 0, $$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 .
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Communicated by A. Malchiodi.
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Novaga, M., Valdinoci, E. Bump solutions for the mesoscopic Allen–Cahn equation in periodic media. Calc. Var. 40, 37–49 (2011). https://doi.org/10.1007/s00526-010-0332-4
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DOI: https://doi.org/10.1007/s00526-010-0332-4