#### Vol. 9, No. 6, 2016

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

### Alberto Enciso and Daniel Peralta-Salas

Vol. 9 (2016), No. 6, 1433–1446
##### 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.

##### Keywords
Allen–Cahn equation, level sets
##### Mathematical Subject Classification 2010
Primary: 35B05, 35J15, 35B08
##### Milestones
Received: 5 November 2015
Revised: 27 April 2016
Accepted: 28 May 2016
Published: 3 October 2016
##### Authors
 Alberto Enciso Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas Calle Nicolás Cabrera, 13–15 28049 Madrid Spain Daniel Peralta-Salas Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas Calle Nicolás Cabrera, 13–15 28049 Madrid Spain