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A method for studying boundary value problems in mathematical physics by reducing them to integral equations; this method consists in representing the solutions of these problems in the form of (generalized) potentials.
 
A method for studying boundary value problems in mathematical physics by reducing them to integral equations; this method consists in representing the solutions of these problems in the form of (generalized) potentials.
  
Let a second-order elliptic partial differential equation be given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741802.png" />,
+
Let a second-order elliptic partial differential equation be given in $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,
  
<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/p/p074/p074180/p0741803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
L u  \equiv  \sum _ {i , j = 1 } ^ { n }
 +
\left ( a _ {ij}
 +
\frac{\partial  ^ {2} u }{\partial  x _ {i} \partial  x _ {j} }
 +
\right ) + \sum
 +
_ {i = 1 } ^ { n }  e _ {i}
 +
\frac{\partial  u }{\partial  x _ {i} }
 +
+ c u  = f ( x),
 +
$$
  
with sufficiently smooth coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741806.png" />, and right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741807.png" />; moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741808.png" /> outside some bounded domain containing in its interior a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p0741809.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418010.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418011.png" />. Then any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418012.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418013.png" /> of (1) can be represented as the sum of three (generalized) potentials: a volume mass potential (cf. [[Newton potential|Newton potential]])
+
with sufficiently smooth coefficients $  a _ {ij} = a _ {ji} = a _ {ij} ( x) $,  
 +
$  e _ {i} = e _ {i} ( x) $,  
 +
$  c ( x) \leq  0 $,  
 +
and right-hand side $  f ( x) $;  
 +
moreover, let $  c ( x) < - k  ^ {2} < 0 $
 +
outside some bounded domain containing in its interior a domain $  D $
 +
with boundary $  S = \partial  D $
 +
of class $  C  ^ {1} $.  
 +
Then any solution $  u ( x) $
 +
of class $  C  ^ {2} ( D \cup S ) $
 +
of (1) can be represented as the sum of three (generalized) potentials: a volume mass potential (cf. [[Newton potential|Newton potential]])
  
<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/p/p074/p074180/p07418014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { D } E ( x , y ) \rho ( y) d y ,
 +
$$
  
 
a single-layer potential (cf. [[Simple-layer potential|Simple-layer potential]])
 
a single-layer potential (cf. [[Simple-layer potential|Simple-layer potential]])
  
<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/p/p074/p074180/p07418015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\int\limits _ { S } E ( x , y ) \sigma ( y) \
 +
d s _ {y} ,
 +
$$
  
 
and a [[Double-layer potential|double-layer potential]]
 
and a [[Double-layer potential|double-layer potential]]
  
<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/p/p074/p074180/p07418016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\int\limits _ { S } Q _ {y} [ E ( x , y ) ] \mu ( y) \
 +
d s _ {y} ,
 +
$$
 +
 
 +
where  $  E ( x , y) $
 +
is a principal [[Fundamental solution|fundamental solution]] of  $  L $,
 +
the symbol  $  Q _ {y} $
 +
denotes the operator
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418017.png" /> is a principal [[Fundamental solution|fundamental solution]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418018.png" />, the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418019.png" /> denotes the operator
+
$$
 +
Q _ {y} \nu  = a  
 +
\frac{\partial  \nu }{\partial  N }
  
<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/p/p074/p074180/p07418020.png" /></td> </tr></table>
+
- b \nu ,
 +
$$
  
acting at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418022.png" /> is a unit co-normal vector at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418023.png" />,
+
acting at a point $  y \in S $,  
 +
$  N $
 +
is a unit co-normal vector at the point $  y \in S $,
  
<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/p/p074/p074180/p07418024.png" /></td> </tr></table>
+
$$
 +
a  ^ {2}  = \sum _ { i= } 1 ^ { n }  \left (
 +
\sum _ { j= } 1 ^ { n }  a _ {ij}  \cos  ( \nu , y _ {j} ) \right )  ^ {2} ,\  b  = \sum _ { i= } 1 ^ { n }  e _ {i}  \cos  ( \nu , y _ {i} ) ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418025.png" /> is the exterior normal vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418027.png" />. The potential densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418030.png" /> are sufficiently-smooth functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418031.png" /> or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418032.png" />.
+
and $  \nu $
 +
is the exterior normal vector to $  S $
 +
at $  y \in S $.  
 +
The potential densities $  \rho ( y) $,  
 +
$  \sigma ( y) $
 +
and $  \mu ( y) $
 +
are sufficiently-smooth functions in $  D $
 +
or on $  S $.
  
All differentiability and boundary properties of harmonic potentials described in the article [[Potential theory|Potential theory]] for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074180/p07418033.png" /> is the [[Laplace operator|Laplace operator]] are valid for the potentials (2)–(4). On the basis of these properties one can reduce boundary value problems for elliptic equations of type (1) to integral equations in the same way as it has been done in the case of the Dirichlet and Neumann problems for harmonic functions in the article [[Potential theory|Potential theory]].
+
All differentiability and boundary properties of harmonic potentials described in the article [[Potential theory|Potential theory]] for the case when $  L $
 +
is the [[Laplace operator|Laplace operator]] are valid for the potentials (2)–(4). On the basis of these properties one can reduce boundary value problems for elliptic equations of type (1) to integral equations in the same way as it has been done in the case of the Dirichlet and Neumann problems for harmonic functions in the article [[Potential theory|Potential theory]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , M. Dekker  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.D. Kupradze,  "The method of potentials in elasticity theory" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.M. Milne-Thomson,  "Theoretical hydrodynamics" , Macmillan  (1949)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , M. Dekker  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.D. Kupradze,  "The method of potentials in elasticity theory" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.M. Milne-Thomson,  "Theoretical hydrodynamics" , Macmillan  (1949)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Král,  "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.G. Simader,  "On Dirichlet's boundary value problem" , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.L. Colton,  R. Kress,  "Integral equation methods in scattering theory" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Jawson,  G. Symm,  "Integral equation methods in potential theory and elastostatics" , Acad. Press  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Král,  "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.G. Simader,  "On Dirichlet's boundary value problem" , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.L. Colton,  R. Kress,  "Integral equation methods in scattering theory" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Jawson,  G. Symm,  "Integral equation methods in potential theory and elastostatics" , Acad. Press  (1977)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


A method for studying boundary value problems in mathematical physics by reducing them to integral equations; this method consists in representing the solutions of these problems in the form of (generalized) potentials.

Let a second-order elliptic partial differential equation be given in $ \mathbf R ^ {n} $, $ n \geq 2 $,

$$ \tag{1 } L u \equiv \sum _ {i , j = 1 } ^ { n } \left ( a _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } \right ) + \sum _ {i = 1 } ^ { n } e _ {i} \frac{\partial u }{\partial x _ {i} } + c u = f ( x), $$

with sufficiently smooth coefficients $ a _ {ij} = a _ {ji} = a _ {ij} ( x) $, $ e _ {i} = e _ {i} ( x) $, $ c ( x) \leq 0 $, and right-hand side $ f ( x) $; moreover, let $ c ( x) < - k ^ {2} < 0 $ outside some bounded domain containing in its interior a domain $ D $ with boundary $ S = \partial D $ of class $ C ^ {1} $. Then any solution $ u ( x) $ of class $ C ^ {2} ( D \cup S ) $ of (1) can be represented as the sum of three (generalized) potentials: a volume mass potential (cf. Newton potential)

$$ \tag{2 } \int\limits _ { D } E ( x , y ) \rho ( y) d y , $$

a single-layer potential (cf. Simple-layer potential)

$$ \tag{3 } \int\limits _ { S } E ( x , y ) \sigma ( y) \ d s _ {y} , $$

and a double-layer potential

$$ \tag{4 } \int\limits _ { S } Q _ {y} [ E ( x , y ) ] \mu ( y) \ d s _ {y} , $$

where $ E ( x , y) $ is a principal fundamental solution of $ L $, the symbol $ Q _ {y} $ denotes the operator

$$ Q _ {y} \nu = a \frac{\partial \nu }{\partial N } - b \nu , $$

acting at a point $ y \in S $, $ N $ is a unit co-normal vector at the point $ y \in S $,

$$ a ^ {2} = \sum _ { i= } 1 ^ { n } \left ( \sum _ { j= } 1 ^ { n } a _ {ij} \cos ( \nu , y _ {j} ) \right ) ^ {2} ,\ b = \sum _ { i= } 1 ^ { n } e _ {i} \cos ( \nu , y _ {i} ) , $$

and $ \nu $ is the exterior normal vector to $ S $ at $ y \in S $. The potential densities $ \rho ( y) $, $ \sigma ( y) $ and $ \mu ( y) $ are sufficiently-smooth functions in $ D $ or on $ S $.

All differentiability and boundary properties of harmonic potentials described in the article Potential theory for the case when $ L $ is the Laplace operator are valid for the potentials (2)–(4). On the basis of these properties one can reduce boundary value problems for elliptic equations of type (1) to integral equations in the same way as it has been done in the case of the Dirichlet and Neumann problems for harmonic functions in the article Potential theory.

References

[1] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[2] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[3] V.S. Vladimirov, "Equations of mathematical physics" , M. Dekker (1971) (Translated from Russian)
[4] V.D. Kupradze, "The method of potentials in elasticity theory" , Moscow (1963) (In Russian)
[5] L.M. Milne-Thomson, "Theoretical hydrodynamics" , Macmillan (1949)

Comments

References

[a1] J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980)
[a2] C.G. Simader, "On Dirichlet's boundary value problem" , Springer (1972)
[a3] D.L. Colton, R. Kress, "Integral equation methods in scattering theory" , Wiley (1983)
[a4] M. Jawson, G. Symm, "Integral equation methods in potential theory and elastostatics" , Acad. Press (1977)
How to Cite This Entry:
Potentials, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potentials,_method_of&oldid=48270
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article