Boundary Integral Equations on Contours with Peaks
(Sprache: Englisch)
The purpose of this book is to give a comprehensive exposition of the theory of boundary integral equations for single and double layer potentials on curves with exterior and interior cusps. The theory was developed by the authors during the last twenty...
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The purpose of this book is to give a comprehensive exposition of the theory of boundary integral equations for single and double layer potentials on curves with exterior and interior cusps. The theory was developed by the authors during the last twenty years and the present volume is based on their results.
The first three chapters are devoted to harmonic potentials, and in the final chapter elastic potentials are treated. Theorems on solvability in various function spaces and asymptotic representations for solutions near the cusps are obtained. Kernels and cokernels of the integral operators are explicitly described. The method is based on a study of auxiliary boundary value problems which is of interest in itself.
The first three chapters are devoted to harmonic potentials, and in the final chapter elastic potentials are treated. Theorems on solvability in various function spaces and asymptotic representations for solutions near the cusps are obtained. Kernels and cokernels of the integral operators are explicitly described. The method is based on a study of auxiliary boundary value problems which is of interest in itself.
Klappentext zu „Boundary Integral Equations on Contours with Peaks “
An equation of the form ??(x)? K(x,y)?(y)d?(y)= f(x),x?X, (1) X is called a linear integral equation. Here (X,?)isaspacewith ?-?nite measure ? and ? is a complex parameter, K and f are given complex-valued functions. The function K is called the kernel and f is the right-hand side. The equation is of the ?rst kind if ? = 0 and of the second kind if ? = 0. Integral equations have attracted a lot of attention since 1877 when C. Neumann reduced the Dirichlet problem for the Laplace equation to an integral equation and solved the latter using the method of successive approximations. Pioneering results in application of integral equations in the theory of h- monic functions were obtained by H. Poincar e, G. Robin, O. H older, A.M. L- punov, V.A. Steklov, and I. Fredholm. Further development of the method of boundary integral equations is due to T. Carleman, G. Radon, G. Giraud, N.I. Muskhelishvili,S.G.Mikhlin,A.P.Calderon,A.Zygmundandothers. Aclassical application of integral equations for solving the Dirichlet and Neumann boundary value problems for the Laplace equation is as follows. Solutions of boundary value problemsaresoughtin the formof the doublelayerpotentialW? andofthe single layer potentialV?. In the case of the internal Dirichlet problem and the ext- nal Neumann problem, the densities of corresponding potentials obey the integral equation ???+W? = g (2) and ? ???+ V? = h (3) ?n respectively, where ?/?n is the derivative with respect to the outward normal to the contour.
The purpose of this book is to give a comprehensive exposition of the theory of boundary integral equations for single and double layer potentials on curves with exterior and interior cusps. The theory was developed by the authors during the last twenty years and the present volume is based on their results.
The first three chapters are devoted to harmonic potentials, and in the final chapter elastic potentials are treated. Theorems on solvability in various function spaces and asymptotic representations for solutions near the cusps are obtained. Kernels and cokernels of the integral operators are explicitly described. The method is based on a study of auxiliary boundary value problems which is of interest in itself.
The first three chapters are devoted to harmonic potentials, and in the final chapter elastic potentials are treated. Theorems on solvability in various function spaces and asymptotic representations for solutions near the cusps are obtained. Kernels and cokernels of the integral operators are explicitly described. The method is based on a study of auxiliary boundary value problems which is of interest in itself.
Inhaltsverzeichnis zu „Boundary Integral Equations on Contours with Peaks “
1 Lp -theory of boundary integral equations on a contour with peak1.1 Introduction
1.2 Continuity of boundary integral operators
1.3 Dirichlet and Neumann problems for a domain with peak
1.4 Integral equations of the Dirichlet and Neumann problems
1.5 Direct method of integral equations of the Neumann and Dirichlet problems
- 2 Boundary integral equations in Hölder spaces on a contour with peak
2.1 Weighted Hölder spaces
2.2 Boundedness of integral operators
2.3 Dirichlet and Neumann problems in a strip
2.4 Boundary integral equations of the Dirichlet and Neumann problems in domains with outward peak
2.5 Boundary integral equations of the Dirichlet and Neumann problems in domains with inward peak
2.6 Integral equation of the first kind on a contour with peak
2.7 Appendices
- 3 Asymptotic formulae for solutions of boundary integral equations near peaks
3.1 Preliminary facts
3.2 The Dirichlet and Neumann problems for domains with peaks
3.3 Integral equations of the Dirichlet problem
3.4 Integral equations of the Neumann problem
3.5 Appendices
- 4 Integral equations of plane elasticity in domains with peak
4.1 Introduction
4.2 Boundary value problems of elasticity
4.3 Integral equations on a contour with inward peak
4.4 Integral equations on a contour with outward peak
- Bibliographyquations of the Dirichlet and Neumann problems in domains with outward peak
2.5 Boundary integral equations of the Dirichlet and Neumann problems in domains with inward peak
2.6 Integral equation of the first kind on a contour with peak
2.7 Appendices
-
Autoren-Porträt von Vladimir Maz'ya, Alexander Soloviev
Dr. Alexander Soloviev is an Associate Professor at the NOVA Southeastern University's Oceanographic Center, Dania Beach, Florida. He also worked as a research scientist in the two leading research institutions of the former Soviet Academy of Sciences: P.P. Shirshov Institute of Oceanology and A.M. Oboukhov Institute of Atmospheric Physics.
Bibliographische Angaben
- Autoren: Vladimir Maz'ya , Alexander Soloviev
- 2009, XI, 344 Seiten, Maße: 17,3 x 23,7 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 3034601700
- ISBN-13: 9783034601702
- Erscheinungsdatum: 19.11.2009
Sprache:
Englisch
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