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Morse Theory

In: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

Author

Listed:
  • Dumitru Motreanu

    (University of Perpignan, Department of Mathematics)

  • Viorica Venera Motreanu

    (Ben-Gurion University of the Negev, Department of Mathematics)

  • Nikolaos Papageorgiou

    (National Technical University, Department of Mathematics)

Abstract

This chapter represents a self-contained presentation of basic results and techniques of Morse theory that are useful for studying the multiplicity of solutions of nonlinear elliptic boundary value problems with a variational structure. The first section of the chapter contains the needed preliminaries of algebraic topology. The second section focuses on the Morse lemma and the splitting and shifting theorems. The third section is devoted to the Morse relations, including the Poincaré–Hopf formula, which involve the critical groups and critical groups at infinity. The fourth section sets forth efficient results for the computation of critical groups that are powerful tools in the study of multiple solutions. Here an original approach is developed, and improvements of known results are shown. Notes on related literature and comments are provided in a remarks section.

Suggested Citation

  • Dumitru Motreanu & Viorica Venera Motreanu & Nikolaos Papageorgiou, 2014. "Morse Theory," Springer Books, in: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, edition 127, chapter 0, pages 141-179, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-9323-5_6
    DOI: 10.1007/978-1-4614-9323-5_6
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