Ebook: The Homotopy Index and Partial Differential Equations

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.

---

The book presents an extension, due to the present author, of Conley's homotopy index theory to certain (one-sided) semiflows on general (not necessarily locally compact) metric spaces. This permits direct applications to say, parabolic partial differential equations, or functional differential equations. The presentation is self-contained. The subject of the book was previously presented by the author in a series of published papers.

---

The book presents an extension, due to the present author, of Conley's homotopy index theory to certain (one-sided) semiflows on general (not necessarily locally compact) metric spaces. This permits direct applications to say, parabolic partial differential equations, or functional differential equations. The presentation is self-contained. The subject of the book was previously presented by the author in a series of published papers.
Content:
Front Matter....Pages I-XII
The homotopy index theory....Pages 1-71
Applications to partial differential equations....Pages 72-139
Selected topics....Pages 140-194
Back Matter....Pages 195-208

---

The book presents an extension, due to the present author, of Conley's homotopy index theory to certain (one-sided) semiflows on general (not necessarily locally compact) metric spaces. This permits direct applications to say, parabolic partial differential equations, or functional differential equations. The presentation is self-contained. The subject of the book was previously presented by the author in a series of published papers.
Content:
Front Matter....Pages I-XII
The homotopy index theory....Pages 1-71
Applications to partial differential equations....Pages 72-139
Selected topics....Pages 140-194
Back Matter....Pages 195-208
....