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

Enciso, Alberto; Peralta-Salas, Daniel

doi:10.2140/apde.2016.9.1433

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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

DOI: 10.2140/apde.2016.9.1433

Abstract

We make use of the flexibility of infinite-index solutions to the Allen–Cahn equation to show that, given any compact hypersurface $Σ$ of $ℝ^{d}$ with $d \geq 3$ , 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 $Σ$ . 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

Vol. 9, No. 6, 2016