Weak singularity

polar singularity

The unboundedness of an integral kernel (cf. Kernel of an integral operator) when the product , , is bounded. Here, is a set in the space , is the distance between two points and and . The integral operator generated by such a kernel,

(1)

is called an integral operator with a weak singularity (or with a polar singularity). Let be a compact subset of . If is continuous on , the operator (1) is completely continuous (cf. Completely-continuous operator) on the space of continuous functions , and if is bounded, then the operator (1) is completely continuous on the space .

The kernel

(2)

is called the convolution of the kernels and . Suppose have weak singularities, with

then their convolution (2) is bounded or has a weak singularity, and:

where is a constant.

If a kernel has a weak singularity, then all its iterated kernels from some iteration onwards are bounded.

References

[1]	V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)
[2]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[3]	M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian)

Comments

Weakly-singular kernels appear frequently in the boundary integral equation method for solving elliptic equations (see [a1]). Another important integral equation with a weakly-singular kernel is the Abel integral equation ([a2]).

References

[a1]	D.L. Colton, R. Kress, "Integral equation methods in scattering theory" , Wiley (1983)
[a2]	R. Gorenflo, S. Vessella, "Abel integral equations in analysis and applications" , Springer (1991)
[a3]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) pp. Sects. I.1.2; II.6 (Translated from Russian)
[a4]	H. Hochstadt, "Integral equations" , Wiley (1975) pp. Sect. II.4