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Linear and Nonlinear Perturbations of the Operator div

Linear and Nonlinear Perturbations of the Operator div (Hardback)

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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.

Science & MathematicsMathematicsCalculus & mathematical analysisDifferential calculus & equationsScience & MathematicsMathematicsCalculus & mathematical analysisFunctional analysis & transforms Publisher: American Mathematical Society Publication Date: 15/03/1997 ISBN-13: 9780821805862  Details: Type: Hardback Format: Books
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