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

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Deng, Jian and Jones, C. K. R. T. (Christopher K. R. T.). (2011) Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems. Transactions of the American Mathematical Society, Vol.363 (No.3). pp. 1487-1508. ISSN 0002-9947

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Official URL: http://dx.doi.org/10.1090/S0002-9947-2010-05129-3

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Boundary value problems, Sobolev spaces, Maslov index, Eigenvalues
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	March 2011
Volume:	Vol.363
Number:	No.3
Page Range:	pp. 1487-1508
Identification Number:	10.1090/S0002-9947-2010-05129-3
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Guo jia zi ran ke xue ji jin wei yuan hui (China) [National Natural Science Foundation of China] (NSFC), National Science Foundation (U.S.) (NSF)
Grant number:	10601014 (NSFC), DMS-0410267 (NSF)
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URI:	http://wrap.warwick.ac.uk/id/eprint/39917