HYPERSINGULAR INTEGRALS: HOW SMOOTH MUST THE DENSITY BE?

Abstract

Hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x. It is well known that a su- cient condition is that f has a Holder-continuous rst derivative. This is a stringent condition, especially when it is incorporated into boundary-element methods for solving hypersingular in- tegral equations. This paper is concerned with nding weaker conditions for the existence of one-dimensional Hadamard nite-part integrals: it is shown that it is sucien t for the even part of f (with respect to x) to have a Holder-continuous rst derivative { the odd part is al- lowed to be discontinuous. A similar condition is obtained for Cauchy principal-value integrals. These simple results have non-trivial consequences. They are applied to the calculation of the tangential derivative of a single-layer potential and to the normal derivative of a double-layer potential. Particular attention is paid to discontinuous densities and to discontinuous boundary conditions. Also, despite the weaker sucien t conditions, it is rearmed that, for hypersingular integral equations, collocation at a point x at the junction between two standard conforming boundary elements is not permissible, theoretically. Various modications to the denition of nite-part integral are explored.