The only book dedicated to boundary integral equations for non-Lipschitz domains
New method, different from the traditional approach based on the theories of Fredholm and singular integral operators
Detailed study of both functional analytic and asymptotic properties of solutions
1 Lp -theory of boundary integral equations on a contour with peak.- 1.1 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.- Bibliography.
This book is a comprehensive exposition of the theory of boundary integral equations for single and double layer potentials on curves with exterior and interior cusps. Three chapters cover harmonic potentials, and the final chapter treats elastic potentials.
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.
Lp-theory of Boundary Integral Equations on a Contour with Peak.- Boundary Integral Equations in Hölder Spaces on a Contour with Peak.- Asymptotic Formulae for Solutions of Boundary Integral Equations Near Peaks.- Integral Equations of Plane Elasticity in Domains with Peak.
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.
Inhaltsverzeichnis
1 Lp -theory of boundary integral equations on a contour with peak.- 1.1 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.- Bibliography.
Klappentext
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 only book dedicated to boundary integral equations for non-Lipschitz domains
New method, different from the traditional approach based on the theories of Fredholm and singular integral operators
Detailed study of both functional analytic and asymptotic properties of solutions
