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Low Complexity Solutions of the Allen–Cahn Equation on Three-spheres

Part of: Existence theories

Published online by Cambridge University Press:  07 January 2019

Robert Haslhofer  and
Mohammad N. Ivaki
Robert Haslhofer
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Email: [email protected]@utoronto.ca
Mohammad N. Ivaki
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Email: [email protected]@utoronto.ca
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Abstract

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In this short note, we prove that on the three-sphere with any bumpy metric there exist at least two pairs of solutions of the Allen–Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

Keywords

Allen–Cahn equationphase transitionsmall index

MSC classification

Primary: 49J35: Minimax problems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 287 - 291
Copyright
© Canadian Mathematical Society 2018 

Footnotes

R. H. was partially supported by NSERC grant RGPIN-2016-04331 and a Connaught New Researcher Award. M. I. was supported by a Marsden Postdoctoral Fellowship. Both authors thank the Fields Institute for providing an excellent research environment during the thematic program on Geometric Analysis.

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