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

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) .