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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$62 Member Price: US$49.60
Order Code: MMONO/160

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.