The Homotopy Index and Partial Differential Equations
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The Homotopy Index and Partial Differential Equations: (Universitext)

The Homotopy Index and Partial Differential Equations: (Universitext)


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About the Book

The homotopy index theory was developed by Charles Conley for two­ sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi­ cal measure of an isolated invariant set, is defined to be the ho­ motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the "exit ramp" of N . 1 It is shown that the index is independent of the choice of the in­ dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde­ generate critical point p with respect to a gradient flow on a com­ pact manifold. In fact if the Morse index of p is k, then the homo­ topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.

Table of Contents:
I The homotopy index theory.- 1.1 Local semiflows.- 1.2 The no blow-up condition. Convergence of semiflows.- 1.3 Isolated invariant sets and isolating blocks.- 1.4 Admissibility.- 1.5 Existence of isolating blocks.- 1.6 Homotopies and inclusion induced maps.- 1.7 Index and quasi-index pairs.- 1.8 Some special maps used in the construction of the Morse index.- 1.9 The Categorial Morse index.- 1.10 The homotopy index and its basic properties.- 1.11 Linear semiflows. Irreducibility.- 1.12 Continuation of the homotopy index.- II Applications to partial differential equations.- 2.1 Sectorial operators generated by partial differential operators.- 2.2 Center manifolds and their approximation.- 2.3 The index product formula.- 2.4 A one-dimensional example.- 2.5 Asymptotically linear systems.- 2.6 Estimates at zero and nontrivial solution of elliptic equations.- 2.7 Positive heteroclinic orbits of second-order parabolic equations.- 2.8 A homotopy index continuation method and periodic solutions of second-order gradient systems.- III Selected topics.- 3.1 Repeller-attractor pairs and Morse decompositions.- 3.2 Block pairs and index triples.- 3.3 A Morse equation.- 3.4 The homotopy index and Morse theory on Hilbert manifolds.- 3.5 Continuation of the categorial Morse index along paths.- Bibliographical notes and comments.


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Product Details
  • ISBN-13: 9783540180678
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Height: 244 mm
  • No of Pages: 208
  • Returnable: N
  • Width: 170 mm
  • ISBN-10: 3540180672
  • Publisher Date: 24 Aug 1987
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Series Title: Universitext


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