A Morse Index Theorem for Elliptic Operators on Bounded Domains

@article{Cox2014AMI,
  title={A Morse Index Theorem for Elliptic Operators on Bounded Domains},
  author={G. Cox and Christopher Jones and J. Marzuola},
  journal={Communications in Partial Differential Equations},
  year={2014},
  volume={40},
  pages={1467 - 1497}
}
Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ⊂ ℝ n is deformed. For a family of domains {Ω t } t∈[a, b] we prove that the Morse index of L on Ω a differs from the Morse index of L on Ω b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the… Expand
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