The Second Inner Variation of Energy and the Morse Index of Limit Interfaces
- 19 Downloads
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).
KeywordsMinimal surfaces Allen–Cahn equation Phase transitions
Mathematics Subject Classification53C21 (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.
- 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)
- 5.Hiesmayr, F.: Spectrum and index of two-sided Allen–Cahn minimal hypersurfaces. arXiv:1704.07738 preprint [math.DG] (2017)
- 11.Mantoulidis, C.: Allen–Cahn min–max on surfaces. arXiv:1706.05946 preprint [math.AP] (2017)
- 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
- 20.Simon, L.: Lectures on Geometric Measure Theory. The Australian National University, Mathematical Sciences Institute, Centre for Mathematics and Its Applications (1983)Google Scholar
- 25.Wang, K., Wei, J.: Finite Morse index implies finite ends. arXiv:1705.06831 preprint [math.AP] (2017)