*Submitted Paper*

**Inserted:** 27 apr 2009

**Year:** 2009

**Abstract:**

Given a double-well potential $F$, a 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 and for which the following two problems have solutions:

1) Find a set whose boundary is uniformly close to the boundary of $B_R$, and has mean curvature equal to $H_\epsilon$ at any point;

2) Find a function $u$ solving an Allen-Cahn type equation with forcing term $H_\epsilon$, and such that such that $u$ goes from a $\delta$-neighborhood of $+1$ in $B_R$ to a $\delta$-neighborhood of minus $1$ outside $B_R$.

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