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

Alberto Enciso and Daniel Peralta-Salas

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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.

Article information

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

Dates
Received: 5 November 2015
Revised: 27 April 2016
Accepted: 28 May 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J15: Second-order elliptic equations 35B08: Entire solutions

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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