Journal
COMMUNICATIONS IN MATHEMATICAL PHYSICS
Volume 354, Issue 2, Pages 459-475Publisher
SPRINGER
DOI: 10.1007/s00220-017-2913-y
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Funding
- National Natural Science Foundation of China [11301513]
- the Fundamental Research Funds for the Central Universities
- NSF [DMS-1500943]
- Tang Aoqing visiting professorship in Jilin University
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1500943] Funding Source: National Science Foundation
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We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic transitive models are ground states. The result we prove is very general, and it applies to models with long range and many-body interactions. As an application, we consider several models of networks of interacting particles including models of Frenkel-Kontorova type on and one-dimensional quasi-periodic media. The result above is an analogue of several results in the calculus of variations (fields of extremals) and in PDE's. Since the models we consider are discrete and long range, new proofs need to be given. We also note that the main hypothesis of our result (the existence of foliations of equilibria) is the conclusion (using KAM theory) of several recent papers. Hence, we obtain that the KAM solutions recently established are minimizers when the interaction is ferromagnetic and transitive (these concepts are defined later).
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