Hypersingular Integrals: How Smooth Must the Density Be?

Hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satisfies certain conditions in a neighbourhood of x. It is well known that a sufficient condition is that f has a Hölder-continuous first derivative. This is a stringent condition, especially when it is incorporated into boundary-element methods for solving hypersingular integral equations. This paper is concerned with finding weaker conditions for the existence of one-dimensional Hadamard finite-part integrals: it is shown that it is sufficient for the even part of f (with respect to x) to have a Hölder-continuous first derivative—the odd part is allowed 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 sufficient conditions, it is reaffirmed that, for hypersingular integral equations, collocation at a point x at the junction between two standard conforming boundary elements is not permissible, theoretically. Various modifications to the definition of finite-part integral are explored.