Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones

By Frank Pacard, Jun-cheng Wei. 2011

61 citations recorded by the source corpus.

Abstract For all n ⩾ 1 , we are interested in bounded solutions of the Allen–Cahn equation Δ u + u − u 3 = 0 which are defined in all R n + 1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n + 1 ⩾ 8 , we prove the existence of stable solutions of the Allen–Cahn equation whose zero sets are not hyperplanes.

View Paper: https://www.semanticscholar.org/paper/1b442430b497d3b963b839035beae8efd5e319d5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

1. Manuel del Pino
2. Jun-cheng Wei
3. M. Kowalczyk
4. Frank Pacard
5. A. Malchiodi
6. Monica Musso
7. Jaeyoung Byeon
8. Angela Pistoia

Citations in the corpus

• MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, background): “We recall here that the heteroclinic is stable, and from [5] we know that in dimension N = 2 stability of a solution implies that it is necessarily a one dimensional solution (for the related minimality conjecture, see for example [28] and [26] and the reference therein).”