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One of the boundary value problems (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]) for partial differential equations. For example, let there be given a second-order elliptic equation
 
One of the boundary value problems (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]) for partial differential equations. For example, let there be given a second-order elliptic equation
  
<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/s/s083/s083680/s0836801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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Lu  = \sum _ {i, j = 1 } ^ { n }  a _ {ij} ( x)
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\frac{\partial  ^ {2} u ( x) }{\partial  x _ {i} \partial  x _ {j} }
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+
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\sum _ {i = 1 } ^ { n }  b _ {i} ( x)
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 +
\frac{\partial  u ( x) }{\partial  x _ {i} }
 +
+
 +
$$
 +
 
 +
$$
 +
+
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c ( x) u ( x)  = f ( x),
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$$
  
<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/s/s083/s083680/s0836802.png" /></td> </tr></table>
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where  $  x = ( x _ {1} \dots x _ {n} ) $,
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$  n \geq  2 $,
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in a bounded domain  $  \Omega $,
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with a normal at each point of the boundary  $  \Gamma $.
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The second boundary value problem for equation (*) in  $  \Omega $
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is the following problem: Out of the set of all solutions of equation (*), isolate those solutions which have, at all boundary points, derivatives with respect to the interior normal  $  N $
 +
and which satisfy the condition
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836804.png" />, in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836805.png" />, with a normal at each point of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836806.png" />. The second boundary value problem for equation (*) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836807.png" /> is the following problem: Out of the set of all solutions of equation (*), isolate those solutions which have, at all boundary points, derivatives with respect to the interior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s0836808.png" /> and which satisfy the condition
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$$
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\left .  
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\frac{\partial  u ( x, t) }{\partial  N ( x) }
  
<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/s/s083/s083680/s0836809.png" /></td> </tr></table>
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\right | _ {x \in \Gamma }
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= \phi ( x),
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083680/s08368010.png" /> is a given function. The second boundary value problem is also known as the Neumann problem.
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where $  \phi ( x) $
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is a given function. The second boundary value problem is also known as the Neumann problem.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[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">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[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">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>

Revision as of 08:12, 6 June 2020


One of the boundary value problems (cf. Boundary value problem, partial differential equations) for partial differential equations. For example, let there be given a second-order elliptic equation

$$ \tag{* } Lu = \sum _ {i, j = 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u ( x) }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ( x) \frac{\partial u ( x) }{\partial x _ {i} } + $$

$$ + c ( x) u ( x) = f ( x), $$

where $ x = ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $, in a bounded domain $ \Omega $, with a normal at each point of the boundary $ \Gamma $. The second boundary value problem for equation (*) in $ \Omega $ is the following problem: Out of the set of all solutions of equation (*), isolate those solutions which have, at all boundary points, derivatives with respect to the interior normal $ N $ and which satisfy the condition

$$ \left . \frac{\partial u ( x, t) }{\partial N ( x) } \right | _ {x \in \Gamma } = \phi ( x), $$

where $ \phi ( x) $ is a given function. The second boundary value problem is also known as the Neumann problem.

References

[1] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[2] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[3] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[4] I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)

Comments

References

[a1] P.R. Garabedian, "Partial differential equations" , Wiley (1963)
[a2] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
How to Cite This Entry:
Second boundary value problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_boundary_value_problem&oldid=48639
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article