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Orientation and the Leray-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems
Patrick Fitzpatrick and Jacobo Pejsachowicz
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Memoirs of the American Mathematical Society
1993; 131 pp; softcover
Volume: 101
ISBN-10: 0-8218-2544-5
ISBN-13: 978-0-8218-2544-0
List Price: US$30
Individual Members: US$18
Institutional Members: US$24
Order Code: MEMO/101/483
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The aim of this work is to develop an additive, integer-valued degree theory for the class of quasilinear Fredholm mappings. This class is sufficiently large that, within its framework, one can study general fully nonlinear elliptic boundary value problems. A degree for the whole class of quasilinear Fredholm mappings must necessarily accomodate sign-switching of the degree along admissible homotopies. The authors introduce "parity", a homotopy invariant of paths of linear Fredholm operators having invertible endpoints. The parity provides a complete description of the possible changes in sign of the degree and thereby permits use of the degree to prove multiplicity and bifurcation theorems for quasilinear Fredholm mappings. Applications are given to the study of fully nonlinear elliptic boundary value problems.

Readership

Research mathematicians.

Table of Contents

  • Quasilinear Fredholm mappings
  • Orientation and the degree
  • General properties of the degree
  • Mapping theorems
  • The parity of a path of linear Fredholm operators
  • The regular value formula and homotopy dependence
  • Bifurcation and continuation
  • Strong orientability
  • Fully nonlinear elliptic boundary value problems
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