Bounded solutions to the Allen–Cahn equation with level sets of any compact topology

Abstract

We make use of the flexibility of infinite-index solutions to the Allen–Cahn equation to show that, given any compact hypersurface $\Sigma$ of ${ℝ}^d$ with $d\ge 3$, there is a bounded entire solution of the Allen–Cahn equation on ${ℝ}^d$ whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of $\Sigma$. More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.