Linear and Nonlinear Perturbations of the Operator $operatorname{div}$ (Translations of Mathematical Monographs) - Hardcover

V. G. Osmolovskii

Hardcover

ISBN 10: 082180586X ISBN 13: 9780821805862

Publisher: American Mathematical Society, 1997

Synopsis

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.

About this title

Language Notes

Text: English (translation)
Original Language: Russian

Edition details

• Publisher: American Mathematical Society
• Publication date: 1997
• Language: English
• ISBN 10: 082180586X
• ISBN 13: 9780821805862
• Binding: Hardcover
• Edition number: 1
• Number of pages: 104