Vol. 9, No. 6, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Σ of d with d 3, there is a bounded entire solution of the Allen–Cahn equation on 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