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Hardcover ISBN:  9780821805862 
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Product Code:  MMONO/160.B 
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Hardcover ISBN:  9780821805862 
Product Code:  MMONO/160 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470445751 
Product Code:  MMONO/160.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821805862 
eBook ISBN:  9781470445751 
Product Code:  MMONO/160.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 160; 1997; 104 ppMSC: Primary 35
The perturbation theory for the operator div is of particular interest in the study of boundaryvalue 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 boundaryvalue problems for \(L\) and the theory of nonlinear perturbations of \(L\). Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundaryvalue 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 boundaryvalue 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 boundaryvalue 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.
ReadershipGraduate students and research mathematicians interested in partial differential equations.

Table of Contents

Chapters

Chapter 1. Linear perturbations of the operator div

Chapter 2. Nonlinear perturbations of the operator div

Appendix


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The perturbation theory for the operator div is of particular interest in the study of boundaryvalue 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 boundaryvalue problems for \(L\) and the theory of nonlinear perturbations of \(L\). Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundaryvalue 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 boundaryvalue 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 boundaryvalue 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