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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Jian Deng and Christopher Jones
Journal: Trans. Amer. Math. Soc. 363 (2011), 1487-1508
MSC (2000): Primary 35J25, 35P15; Secondary 53D12, 35B05
Published electronically: October 15, 2010
MathSciNet review: 2737274
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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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Additional Information

Jian Deng
Affiliation: CEMA, Central University of Finance and Economics, Beijing, People’s Republic of China, 100085
Email: jdeng@fudan.edu.cn

Christopher Jones
Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599 – and – Warwick Mathematics Institute, University of Warwick, United Kingdom
Email: ckrtj@email.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05129-3
PII: S 0002-9947(2010)05129-3
Keywords: Morse index, Maslov index, elliptic boundary value problem
Received by editor(s): July 3, 2008
Received by editor(s) in revised form: June 8, 2009
Published electronically: October 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.