MULTI-DIMENSIONAL MORSE INDEX THEOREMS AND A SYMPLECTIC VIEW OF ELLIPTIC BOUNDARY VALUE PROBLEMS

By Jian Deng, Christopher Jones. 2011

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3933d7ce84608c6f200767f01c18a5699fb33c65