Analysis & PDE
- Anal. PDE
- Volume 9, Number 6 (2016), 1433-1446.
Bounded solutions to the Allen–Cahn equation with level sets of any compact topology
We make use of the flexibility of infinite-index solutions to the Allen–Cahn equation to show that, given any compact hypersurface of with , there is a bounded entire solution of the Allen–Cahn equation on whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of . More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.
Anal. PDE, Volume 9, Number 6 (2016), 1433-1446.
Received: 5 November 2015
Revised: 27 April 2016
Accepted: 28 May 2016
First available in Project Euclid: 16 November 2017
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Enciso, Alberto; Peralta-Salas, Daniel. Bounded solutions to the Allen–Cahn equation with level sets of any compact topology. Anal. PDE 9 (2016), no. 6, 1433--1446. doi:10.2140/apde.2016.9.1433. https://projecteuclid.org/euclid.apde/1510843327