Hypersingular integrals: How smooth must the density be?

*(English)*Zbl 0846.65070This is a very interesting paper that examines the conditions on the density $f\left(t\right)$ for the hypersingular integrals

$${\int}_{A}^{B}\frac{f\left(t\right)}{{(t-x)}^{n}}dx,\phantom{\rule{2.em}{0ex}}n=1,2,\cdots $$

to exist. It is well known that it is sufficient that $f\left(t\right)$ 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.

Reviewer: B.Burrows (Stafford)

##### MSC:

65N38 | Boundary element methods (BVP of PDE) |

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type (one real variable) |

35J25 | Second order elliptic equations, boundary value problems |