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''polar singularity''
 
''polar singularity''
  
The unboundedness of an integral kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972301.png" /> (cf. [[Kernel of an integral operator|Kernel of an integral operator]]) when the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972303.png" />, is bounded. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972304.png" /> is a set in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972306.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972309.png" />. The integral operator generated by such a kernel,
+
The unboundedness of an integral kernel $  K ( x, s) $(
 +
cf. [[Kernel of an integral operator|Kernel of an integral operator]]) when the product $  M( x, s)= | x - s |  ^  \alpha  K ( x, s) $,
 +
$  ( x, s) \in \Omega \times \Omega $,  
 +
is bounded. Here, $  \Omega $
 +
is a set in the space $  \mathbf R  ^ {n} $,  
 +
$  | x - s | $
 +
is the distance between two points $  x $
 +
and  $  s $
 +
and  $  0 < \alpha = \textrm{ const } < n $.  
 +
The integral operator generated by such a kernel,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
K \phi ( t)  = \int\limits _  \Omega 
 +
\frac{M ( x, s) }{| x - s |  ^  \alpha  }
 +
\phi ( s) ds,
 +
$$
  
is called an integral operator with a weak singularity (or with a polar singularity). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723011.png" /> be a compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723013.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723014.png" />, the operator (1) is completely continuous (cf. [[Completely-continuous operator|Completely-continuous operator]]) on the space of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723015.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723016.png" /> is bounded, then the operator (1) is completely continuous on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723017.png" />.
+
is called an integral operator with a weak singularity (or with a polar singularity). Let $  \Omega $
 +
be a compact subset of $  \mathbf R  ^ {n} $.  
 +
If $  M ( x, s) $
 +
is continuous on $  \Omega \times \Omega $,  
 +
the operator (1) is completely continuous (cf. [[Completely-continuous operator|Completely-continuous operator]]) on the space of continuous functions $  C ( \Omega ) $,  
 +
and if $  M $
 +
is bounded, then the operator (1) is completely continuous on the space $  L _ {2} ( \Omega ) $.
  
 
The kernel
 
The kernel
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
( K _ {1} \otimes K _ {2} ) ( x, s)  = \
 +
\int\limits _  \Omega  K _ {1} ( x, t) K _ {2} ( t, s) dt
 +
$$
  
is called the convolution of the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723020.png" />. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723021.png" /> have weak singularities, with
+
is called the convolution of the kernels $  K _ {1} $
 +
and $  K _ {2} $.  
 +
Suppose $  K _ {1} , K _ {2} $
 +
have weak singularities, with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723022.png" /></td> </tr></table>
+
$$
 +
| K _ {i} ( x, s) |  \leq  \
 +
 
 +
\frac{\textrm{ const } }{| x- s | ^ {\alpha _ {i} } }
 +
,\ \
 +
\alpha _ {i} = \textrm{ const } < n,\  i = 1, 2;
 +
$$
  
 
then their convolution (2) is bounded or has a weak singularity, and:
 
then their convolution (2) is bounded or has a weak singularity, and:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723023.png" /></td> </tr></table>
+
$$
 +
| K _ {1} \otimes K _ {2} ( x, s) |  \leq  \
 +
\left \{
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723024.png" /> is a constant.
+
where $  c $
 +
is a constant.
  
 
If a kernel has a weak singularity, then all its iterated kernels from some iteration onwards are bounded.
 
If a kernel has a weak singularity, then all its iterated kernels from some iteration onwards are bounded.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Smirnov,   "A course of higher mathematics" , '''5''' , Addison-Wesley (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.S. Vladimirov,   "Equations of mathematical physics" , MIR (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Krasnosel'skii,   et al.,   "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Smirnov, "A course of higher mathematics" , '''5''' , Addison-Wesley (1964) (Translated from Russian) {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) {{MR|}} {{ZBL|0312.47041}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Line 32: Line 76:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.L. Colton,   R. Kress,   "Integral equation methods in scattering theory" , Wiley (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Gorenflo,   S. Vessella,   "Abel integral equations in analysis and applications" , Springer (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Hochstadt,   "Integral equations" , Wiley (1975) pp. Sect. II.4</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.L. Colton, R. Kress, "Integral equation methods in scattering theory" , Wiley (1983) {{MR|0700400}} {{ZBL|0522.35001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Gorenflo, S. Vessella, "Abel integral equations in analysis and applications" , Springer (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Hochstadt, "Integral equations" , Wiley (1975) pp. Sect. II.4 {{MR|1013363}} {{MR|0390680}} {{MR|0190666}} {{ZBL|0718.45001}} {{ZBL|0259.45001}} {{ZBL|0137.08601}} </TD></TR></table>

Latest revision as of 08:28, 6 June 2020


polar singularity

The unboundedness of an integral kernel $ K ( x, s) $( cf. Kernel of an integral operator) when the product $ M( x, s)= | x - s | ^ \alpha K ( x, s) $, $ ( x, s) \in \Omega \times \Omega $, is bounded. Here, $ \Omega $ is a set in the space $ \mathbf R ^ {n} $, $ | x - s | $ is the distance between two points $ x $ and $ s $ and $ 0 < \alpha = \textrm{ const } < n $. The integral operator generated by such a kernel,

$$ \tag{1 } K \phi ( t) = \int\limits _ \Omega \frac{M ( x, s) }{| x - s | ^ \alpha } \phi ( s) ds, $$

is called an integral operator with a weak singularity (or with a polar singularity). Let $ \Omega $ be a compact subset of $ \mathbf R ^ {n} $. If $ M ( x, s) $ is continuous on $ \Omega \times \Omega $, the operator (1) is completely continuous (cf. Completely-continuous operator) on the space of continuous functions $ C ( \Omega ) $, and if $ M $ is bounded, then the operator (1) is completely continuous on the space $ L _ {2} ( \Omega ) $.

The kernel

$$ \tag{2 } ( K _ {1} \otimes K _ {2} ) ( x, s) = \ \int\limits _ \Omega K _ {1} ( x, t) K _ {2} ( t, s) dt $$

is called the convolution of the kernels $ K _ {1} $ and $ K _ {2} $. Suppose $ K _ {1} , K _ {2} $ have weak singularities, with

$$ | K _ {i} ( x, s) | \leq \ \frac{\textrm{ const } }{| x- s | ^ {\alpha _ {i} } } ,\ \ \alpha _ {i} = \textrm{ const } < n,\ i = 1, 2; $$

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

$$ | K _ {1} \otimes K _ {2} ( x, s) | \leq \ \left \{

where $ c $ 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) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[2] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[3] M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) Zbl 0312.47041

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) MR0700400 Zbl 0522.35001
[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 MR1013363 MR0390680 MR0190666 Zbl 0718.45001 Zbl 0259.45001 Zbl 0137.08601
How to Cite This Entry:
Weak singularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_singularity&oldid=14001
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article