Given a double-well potential F, a Z(n)-periodic function H, small and with zero average, and epsilon > 0, we find a large R, a small delta and a function H (epsilon) which is epsilon-close to H for which the following two problems have solutions: 1. Find a set E (epsilon) ,R whose boundary is uniformly close to a, B (R) and has mean curvature equal to -H (epsilon) at any point, 2. Find u = u (epsilon) ,R,delta solving -delta Delta u + F'(u)/delta + c(0)/2 H(epsilon) = 0, such that u (epsilon,R,delta) goes from a delta-neighborhood of + 1 in B (R) to a delta-neighborhood of -1 outside B (R) .

Bump solutions for the mesoscopic Allen-Cahn equation in periodic media / M. Novaga, E. Valdinoci. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 40:1-2(2011 Jan), pp. 37-49.

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

#### Abstract

Given a double-well potential F, a Z(n)-periodic function H, small and with zero average, and epsilon > 0, we find a large R, a small delta and a function H (epsilon) which is epsilon-close to H for which the following two problems have solutions: 1. Find a set E (epsilon) ,R whose boundary is uniformly close to a, B (R) and has mean curvature equal to -H (epsilon) at any point, 2. Find u = u (epsilon) ,R,delta solving -delta Delta u + F'(u)/delta + c(0)/2 H(epsilon) = 0, such that u (epsilon,R,delta) goes from a delta-neighborhood of + 1 in B (R) to a delta-neighborhood of -1 outside B (R) .
##### Scheda breve Scheda completa Scheda completa (DC) mixed states; interfaces
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/2434/248025`
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