The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

  • Pedro GasparEmail author


In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).


Minimal surfaces Allen–Cahn equation Phase transitions 

Mathematics Subject Classification

53C21 (primary) 49Q05 35J60 



This work is partially based on my Ph.D. Thesis at IMPA, Brazil. I would like to thank my Advisor Fernando Codá Marques for his constant encouragement and support. This work was carried out while visiting the Mathematics Department of Princeton University during 2017–2018. I am grateful to this institution for its kind hospitality and support.


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Authors and Affiliations

  1. 1.Instituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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