Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems

Deng, Jian; Jones, Christopher

doi:10.1090/S0002-9947-2010-05129-3

Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems
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by Jian Deng and Christopher Jones PDF

Trans. Amer. Math. Soc. 363 (2011), 1487-1508 Request permission

Abstract:

Morse Index Theorems for elliptic boundary value problems in multi-dimensions are proved under various boundary conditions. The theorems work for star-shaped domains and are based on a new idea of measuring the “oscillation” of the trace of the set of solutions on a shrinking boundary. The oscillation is measured by formulating a Maslov index in an appropriate Sobolev space of functions on this boundary. A fundamental difference between the cases of Dirichlet and Neumann boundary conditions is exposed through a monotonicity that holds only in the former case.

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