## 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

#### 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 $ℝd$ with $d ≥ 3$, there is a bounded entire solution of the Allen–Cahn equation on $ℝd$ 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.

#### Article information

Source
Anal. PDE, Volume 9, Number 6 (2016), 1433-1446.

Dates
Revised: 27 April 2016
Accepted: 28 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843327

Digital Object Identifier
doi:10.2140/apde.2016.9.1433

Mathematical Reviews number (MathSciNet)
MR3555316

Zentralblatt MATH identifier
1354.35026

Keywords
Allen–Cahn equation level sets

#### Citation

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

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