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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.

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