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Linear and Nonlinear Perturbations of the Operator \(\operatorname{div}\)
V. G. Osmolovskiĭ, St. Petersburg State University, Russia
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Translations of Mathematical Monographs
1997; 104 pp; hardcover
Volume: 160
ISBN-10: 0-8218-0586-X
ISBN-13: 978-0-8218-0586-2
List Price: US$59
Member Price: US$47.20
Order Code: MMONO/160
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The perturbation theory for the operator div is of particular interest in the study of boundary-value problems for the general nonlinear equation \(F(\dot y,y,x)=0\). Taking as linearization the first order operator \(Lu=C_{ij}u_{x_j}^i+C_iu^i\), one can, under certain conditions, regard the operator \(L\) as a compact perturbation of the operator div.

This book presents results on boundary-value problems for \(L\) and the theory of nonlinear perturbations of \(L\). Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundary-value problems for the operator \(L\). An analog of the Weyl decomposition is proved.

The book also contains a local description of the set of all solutions (located in a small neighborhood of a known solution) to the boundary-value problems for the nonlinear equation \(F(\dot y, y, x) = 0\) for which \(L\) is a linearization. A classification of sets of all solutions to various boundary-value problems for the nonlinear equation \(F(\dot y, y, x) = 0\) is given.

The results are illustrated by various applications in geometry, the calculus of variations, physics, and continuum mechanics.

Readership

Graduate students and research mathematicians interested in partial differential equations.

Table of Contents

  • Notation
  • Linear perturbations of the operator div
  • Nonlinear perturbations of the operator div
  • Appendix
  • References
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