**Translations of Mathematical Monographs**

1997;
104 pp;
Hardcover

MSC: Primary 35;
**Print ISBN: 978-0-8218-0586-2
Product Code: MMONO/160**

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# Linear and Nonlinear Perturbations of the Operator \(\operatorname{div}\)

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*V. G. Osmolovskiĭ*

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.