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One of the boundary value problems for partial differential equations (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]). Suppose, for example, that in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926401.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926402.png" />, each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926403.png" /> has a normal, and let the following second-order elliptic equation be given:
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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/t/t092/t092640/t0926404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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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/t/t092/t092640/t0926405.png" /></td> </tr></table>
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One of the boundary value problems for partial differential equations (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]). Suppose, for example, that in a bounded domain  $  \Omega $
 +
with boundary  $  \Gamma $,
 +
each point of  $  \Gamma $
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has a normal, and let the following second-order elliptic equation be given:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926407.png" />. Then a third boundary value problem for (*) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926408.png" /> is the following problem: To select from the set of all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t0926409.png" /> of (*) those that have, at each point of the boundary, derivatives along the interior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264010.png" /> and that satisfy a condition
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$$ \tag{* }
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Lu =
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$$
  
<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/t/t092/t092640/t09264011.png" /></td> </tr></table>
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$$
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= \
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\sum _ {i, j = 1 } ^ { n }  a _ {ij} ( x)
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\frac{\partial  ^ {2} u ( x) }{\partial  x _ {i} \partial  x _ {j} }
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+
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\sum _ {i = 1 } ^ { n }  b _ {i} ( x)
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\frac{\partial  u ( x) }{\partial  x _ {i} }
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+ c ( x) u ( x)  = f ( x),
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264013.png" /> are continuous functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264014.png" />.
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where $  x = ( x _ {1} \dots x _ {n} ) $,
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$  n \geq  2 $.
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Then a third boundary value problem for (*) in  $  \Omega $
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is the following problem: To select from the set of all solutions  $  u ( x) $
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of (*) those that have, at each point of the boundary, derivatives along the interior normal  $  N $
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and that satisfy a condition
  
 +
$$
  
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\frac{\partial  u ( x) }{\partial  N }
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-
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\alpha ( x) u ( x)  = \
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v ( x),\ \
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x \in \Gamma ,
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$$
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 +
where  $  \alpha > 0 $
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and  $  v $
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are continuous functions defined on  $  \Gamma $.
  
 
====Comments====
 
====Comments====
 
The third boundary value problem is sometimes called the Robin problem, after V.G. Robin (1855–1897), and is not to be confused with the problem from potential theory of the same name, discussed in [[Robin problem|Robin problem]].
 
The third boundary value problem is sometimes called the Robin problem, after V.G. Robin (1855–1897), and is not to be confused with the problem from potential theory of the same name, discussed in [[Robin problem|Robin problem]].
  
Quite often, the derivative appearing in a boundary condition of the third type is not necessarily along the interior normal (see, e.g., [[#References|[a1]]]), but along any direction varying continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264015.png" />. If such a direction is nowhere tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092640/t09264016.png" />, the problem is said to be regular.
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Quite often, the derivative appearing in a boundary condition of the third type is not necessarily along the interior normal (see, e.g., [[#References|[a1]]]), but along any direction varying continuously on $  \Gamma $.  
 +
If such a direction is nowhere tangent to $  \Gamma $,  
 +
the problem is said to be regular.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>
 +
<table><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Glibarg, N. Trudinger  "Elliptic partial differential equations of second order", Revised third printing, Springer Verlag  (1997)</TD></TR></table>

Latest revision as of 19:19, 11 September 2020


One of the boundary value problems for partial differential equations (cf. Boundary value problem, partial differential equations). Suppose, for example, that in a bounded domain $ \Omega $ with boundary $ \Gamma $, each point of $ \Gamma $ has a normal, and let the following second-order elliptic equation be given:

$$ \tag{* } Lu = $$

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

where $ x = ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $. Then a third boundary value problem for (*) in $ \Omega $ is the following problem: To select from the set of all solutions $ u ( x) $ of (*) those that have, at each point of the boundary, derivatives along the interior normal $ N $ and that satisfy a condition

$$ \frac{\partial u ( x) }{\partial N } - \alpha ( x) u ( x) = \ v ( x),\ \ x \in \Gamma , $$

where $ \alpha > 0 $ and $ v $ are continuous functions defined on $ \Gamma $.

Comments

The third boundary value problem is sometimes called the Robin problem, after V.G. Robin (1855–1897), and is not to be confused with the problem from potential theory of the same name, discussed in Robin problem.

Quite often, the derivative appearing in a boundary condition of the third type is not necessarily along the interior normal (see, e.g., [a1]), but along any direction varying continuously on $ \Gamma $. If such a direction is nowhere tangent to $ \Gamma $, the problem is said to be regular.

References

[a1] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[a2] D. Glibarg, N. Trudinger "Elliptic partial differential equations of second order", Revised third printing, Springer Verlag (1997)
How to Cite This Entry:
Third boundary value problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Third_boundary_value_problem&oldid=11601
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article