Operator Theory: Advances and Applications 196. Basel: Birkhäuser (ISBN 978-3-0346-0170-2/hbk; 978-3-0346-0171-9/ebook). vii, 342 p. EUR 149.00/net; £ 134.00; SFR 239.00 (2010).
The authors of this remarkable monograph “construct a theory of boundary integral equations for plane domains with a finite number of cusps at the boundary”. The work is essentially built up on eleven joint papers of both authors. It contains four chapters each and every of them being self contained.
In the first one the authors provide a theory of linear boundary integral equations (BIEs for short) of the first and second kind in weighted Lebesgue spaces. They are concerned with two main topics, namely, the solvability of BIEs and Fredholm properties of the operators involved. In the second chapter they deal with the same topics but in the context of Hölder-type spaces for a plane, simply connected, bounded domain with a peak at the boundary. In the third chapter the authors carry out some asymptotic results for the solutions of BIEs on contours with first order tangency peaks. They consider Dirichlet as well as Neumann boundary value problems in domains with peaks, study their solvability and provide asymptotic formulae for their solutions near peaks. The last chapter is devoted to integral equations of plane elasticity on contours with inward as well as outward peaks.