1D symmetry for semilinear pdes from the limit interface of the solution

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Date
2014
Volume
2024
Issue
Journal
Series Titel
WIAS Preprints
Book Title
Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

We study bounded, entire, monotone solutions of the Allen-Cahn equation. We prove that under suitable assumptions on the limit interface and on the energy growth, the solution is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.

Description
Keywords
Phase transitions, symmetry results, limit interface
Citation
Farina, A., & Valdinoci, E. (2014). 1D symmetry for semilinear pdes from the limit interface of the solution (Vol. 2024). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
License
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