The second inner variation of energy and the Morse index of limit interfaces

TL;DR

This paper investigates the second variation of energy in the Allen-Cahn equation on closed manifolds, establishing bounds on the Morse index of limit interfaces through spectral analysis, extending previous results without multiplicity or orientability assumptions.

Contribution

It extends existing theories by proving upper semicontinuity of eigenvalues and bounds on Morse index for limit interfaces without requiring multiplicity or orientability.

Findings

Upper semicontinuity of eigenvalues established

Bounds on Morse index for limit interfaces derived

Results extend previous work by removing multiplicity and orientability constraints

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 N. Le and F. Hiesmayr.