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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 $f\left(t\right)$ for the hypersingular integrals

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

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.

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