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

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## Abstract

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).

## Keywords

Minimal surfaces Allen–Cahn equation Phase transitions## Mathematics Subject Classification

53C21 (primary) 49Q05 35J60## Notes

### Acknowledgements

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.

## References

- 1.Cahn, J., Allen, S.: A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics. J. Phys. Colloq.
**38**, C7–C51 (1977)CrossRefGoogle Scholar - 2.Chodosh, O., Mantoulidis, C.: Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates. arXiv:1803.02716v2 preprint [math.DG] (2018)
- 3.Gaspar, P., Guaraco, M.A.: The Allen-Cahn equation on closed manifolds. Calc. Var. Partial Differ. Equ.
**57**, 101 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Guaraco, M.A.: Min-max for phase transitions and the existence of embedded minimal hypersurfaces. J. Differ. Geom.
**108**, 91–133 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Hiesmayr, F.: Spectrum and index of two-sided Allen–Cahn minimal hypersurfaces. arXiv:1704.07738 preprint [math.DG] (2017)
- 6.Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differ. Equ.
**10**, 49–84 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom.
**38**, 417–461 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Kohn, R.V., Sternberg, P.: Local minimisers and singular perturbations. Proc. R. Soc. Edinb. A
**111**, 69–84 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Le, N.Q.: On the second inner variation of the Allen-Cahn functional and its applications. Indiana Univ. Math. J.
**60**, 1843–1856 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Le, N.Q.: On the second inner variations of Allen-Cahn type energies and applications to local minimizers. J. Math. Pures Appl.
**103**, 1317–1345 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Mantoulidis, C.: Allen–Cahn min–max on surfaces. arXiv:1706.05946 preprint [math.AP] (2017)
- 12.Marques, F.C., Neves, A.: Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math.
**4**, 463–511 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Marques, F.C., Neves, A.: Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. Invent. Math.
**209**, 577–616 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal.
**98**, 123–142 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Modica, L., Mortola, S.: Il limite nella \(\Gamma \)-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5)
**14**, 526–529 (1977)MathSciNetzbMATHGoogle Scholar - 16.Pacard, F.: The role of minimal surfaces in the study of the Allen–Cahn equation. In: Geometric Analysis: Partial Differential Equations and Surfaces, Contemporary Mathematics, vol. 570, pp. 137–163. American Mathematical Society, Providence (2012)Google Scholar
- 17.Reshetnyak, Y.G.: Weak convergence of completely additive vector functions on a set. Sib. Math. J.
**9**, 1039–1045 (1968)CrossRefzbMATHGoogle Scholar - 18.Savin, O.: Phase transitions, minimal surfaces and a conjecture of de Giorgi. Curr. Dev. Math.
**2009**, 59–114 (2009)CrossRefzbMATHGoogle Scholar - 19.Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math.
**34**, 741–797 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Simon, L.: Lectures on Geometric Measure Theory. The Australian National University, Mathematical Sciences Institute, Centre for Mathematics and Its Applications (1983)Google Scholar
- 21.Smith, G.: Bifurcation of solutions to the Allen-Cahn equation. J. Lond. Math. Soc.
**94**, 667–687 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal.
**101**, 209–260 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Tonegawa, Y.: On stable critical points for a singular perturbation problem. Commun. Anal. Geom.
**13**, 439–459 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Tonegawa, Y., Wickramasekera, N.: Stable phase interfaces in the van der Waals-Cahn-Hilliard theory. J. reine angew. Math. (Crelles J.)
**2012**, 191–210 (2012)MathSciNetzbMATHGoogle Scholar - 25.Wang, K., Wei, J.: Finite Morse index implies finite ends. arXiv:1705.06831 preprint [math.AP] (2017)
- 26.Wickramasekera, N.: A general regularity theory for stable codimension 1 integral varifolds. Ann. Math.
**179**, 843–1007 (2014)MathSciNetCrossRefzbMATHGoogle Scholar