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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 \[ \int^B_A {f(t)\over (t- x)^n} dx,\qquad n= 1,2,\dots \] 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:
65N38 Boundary element methods for boundary value problems involving PDEs
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
35J25 Boundary value problems for second-order elliptic equations
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