Boundary Integral Equations on Contours with Peaks

ISBN-13: 9783034601702 / Angielski / Twarda / 2009 / 344 str.

Vladimir Maz'ya; Alexander Soloviev

Opis

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.

Spis treści

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.

O autorze

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.