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

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 −u+F(u)+2c0H=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 .