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Hypersingular integrals: How smooth must the density be? (English) Zbl 0846.65070

This is a very interesting paper that examines the conditions on the density f(t) for the hypersingular integrals

A B f(t) (t-x) n dx,n=1,2,

to exist. It is well known that it is sufficient that f(t) has a Hölder-continuous first derivative. This paper is concerned with finding weaker conditions and it is established that it is sufficient for n=2 (this is a Hadamard finite-part integral) that the even part of f has a Hölder-continuous first derivative. A similar condition is found for n=1 (a Cauchy principal value). The non-trivial consequences of these results are discussed, particularly with regard to collocation at a point x between two boundary elements.

MSC:
65N38Boundary element methods (BVP of PDE)
26A42Integrals of Riemann, Stieltjes and Lebesgue type (one real variable)
35J25Second order elliptic equations, boundary value problems