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An infinite-dimensional Evans function theory for elliptic boundary value problems

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
244
Issue
4
Identifiers
DOI: 10.1016/j.jde.2007.10.037
Keywords
  • Evans Function
  • Stability Index
  • Fredholm GraßMannian
  • Elliptic Eigenvalue Problems
Disciplines
  • Mathematics

Abstract

Abstract An infinite-dimensional Evans function E ( λ ) and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain Ω ⊂ R m . The number of zero points of the Evans function in a bounded, simply connected complex domain D is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in D. When the domain Ω is star-shaped, an associated unstable bundle E ( D ) based on D is constructed, and the first Chern number of E ( D ) also gives the number of eigenvalues of the elliptic operator inside D.

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