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Linear and Nonlinear Perturbations of the Operator $\operatorname{div}$

V. G. Osmolovskiĭ St. Petersburg State University, St. Petersburg, Russia
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Hardcover ISBN: 978-0-8218-0586-2
Product Code: MMONO/160
List Price: $66.00 MAA Member Price:$59.40
AMS Member Price: $52.80 Electronic ISBN: 978-1-4704-4575-1 Product Code: MMONO/160.E List Price:$62.00
MAA Member Price: $55.80 AMS Member Price:$49.60
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AMS Member Price: $79.20 Click above image for expanded view Linear and Nonlinear Perturbations of the Operator$\operatorname{div}$V. G. Osmolovskiĭ St. Petersburg State University, St. Petersburg, Russia Available Formats:  Hardcover ISBN: 978-0-8218-0586-2 Product Code: MMONO/160  List Price:$66.00 MAA Member Price: $59.40 AMS Member Price:$52.80
 Electronic ISBN: 978-1-4704-4575-1 Product Code: MMONO/160.E
 List Price: $62.00 MAA Member Price:$55.80 AMS Member Price: $49.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$99.00 MAA Member Price: $89.10 AMS Member Price:$79.20
• Book Details

Translations of Mathematical Monographs
Volume: 1601997; 104 pp
MSC: Primary 35;

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.

Graduate students and research mathematicians interested in partial differential equations.

• Chapters
• Chapter 1. Linear perturbations of the operator div
• Chapter 2. Nonlinear perturbations of the operator div
• Appendix
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Volume: 1601997; 104 pp
MSC: Primary 35;

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.