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''at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355201.png" />''
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A partial differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355202.png" />,
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{{TEX|auto}}
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{{TEX|done}}
  
<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/e/e035/e035520/e0355203.png" /></td> </tr></table>
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''at a given point  $  x $''
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355204.png" /> is a differential operator of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355205.png" />, whose characteristic equation at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355206.png" />,
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A partial differential equation of order $  m $,
  
<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/e/e035/e035520/e0355207.png" /></td> </tr></table>
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$$
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\sum a _ {i _ {1}  \dots i _ {n} } ( x)
  
has no real roots except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e0355208.png" />.
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\frac{\partial  ^ {m} u }{\partial  x _ {1} ^ {i _ {1} } \dots
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\partial  x _ {r} ^ {i _ {n} } }
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+ L _ {1} u  =  f ,\ \
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\sum_{j=1}^ { n }  i _ {j}  =  m ,
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$$
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such that  $  L _ {1} $
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is a differential operator of order less than  $  m $,
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whose characteristic equation at  $  x $,
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$$
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K ( \lambda _ {1} \dots \lambda _ {n} )  =  \sum a _ {i _ {1}  \dots
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i _ {n} } ( x) \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} }  =  0 ,\  \sum _ { j=1}^ { n }  i _ {j}  =  m ,
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$$
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has no real roots except $  \lambda _ {1} = 0 \dots \lambda _ {n} = 0 $.
  
 
For second-order equations the characteristic form is quadratic,
 
For second-order equations the characteristic form is quadratic,
  
<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/e/e035/e035520/e0355209.png" /></td> </tr></table>
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$$
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Q ( \lambda _ {1} \dots \lambda _ {n} )  = \sum _
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{i , j = 1 } ^ { n }  A _ {ij} ( x) \lambda _ {i} \lambda _ {j} ,
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$$
  
 
and can be brought to the form
 
and can be brought to the form
  
<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/e/e035/e035520/e03552010.png" /></td> </tr></table>
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$$
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= \sum_{i=1}^ { n }  \alpha _ {i} \xi _ {i}  ^ {2}
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$$
  
by a non-singular affine transformation of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552012.png" />.
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by a non-singular affine transformation of the variables $  \lambda _ {i} = \lambda _ {i} ( \xi _ {1} \dots \xi _ {n} ) $,  
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$  i = 1 \dots n $.
  
When all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552013.png" /> or all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552014.png" />, the equation is said to be of elliptic type.
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When all $  \alpha _ {i} = 1 $
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or all $  \alpha _ {i} = - 1 $,  
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the equation is said to be of elliptic type.
  
 
A partial differential equation is said to be of elliptic type in its domain of definition if it is elliptic at every point of this domain.
 
A partial differential equation is said to be of elliptic type in its domain of definition if it is elliptic at every point of this domain.
  
An elliptic partial differential is called uniformly elliptic if there are positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035520/e03552016.png" /> such that
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An elliptic partial differential is called uniformly elliptic if there are positive numbers $  k _ {0} $
 
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and $  k _ {1} $
<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/e/e035/e035520/e03552017.png" /></td> </tr></table>
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such that
 
 
For references see [[Differential equation, partial|Differential equation, partial]].
 
 
 
  
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$$
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k _ {0} \sum_{i=1}^ { n }  \lambda _ {i}  ^ {2}  \leq  Q ( \lambda _ {1} \dots
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\lambda _ {n} )  \leq  k _ {1} \sum_{i=1}^ { n }  \lambda _ {i}  ^ {2} .
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$$
  
====Comments====
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For references see [[Differential equation, partial]].
  
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983) {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>

Latest revision as of 20:22, 17 January 2024


at a given point $ x $

A partial differential equation of order $ m $,

$$ \sum a _ {i _ {1} \dots i _ {n} } ( x) \frac{\partial ^ {m} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {r} ^ {i _ {n} } } + L _ {1} u = f ,\ \ \sum_{j=1}^ { n } i _ {j} = m , $$

such that $ L _ {1} $ is a differential operator of order less than $ m $, whose characteristic equation at $ x $,

$$ K ( \lambda _ {1} \dots \lambda _ {n} ) = \sum a _ {i _ {1} \dots i _ {n} } ( x) \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} } = 0 ,\ \sum _ { j=1}^ { n } i _ {j} = m , $$

has no real roots except $ \lambda _ {1} = 0 \dots \lambda _ {n} = 0 $.

For second-order equations the characteristic form is quadratic,

$$ Q ( \lambda _ {1} \dots \lambda _ {n} ) = \sum _ {i , j = 1 } ^ { n } A _ {ij} ( x) \lambda _ {i} \lambda _ {j} , $$

and can be brought to the form

$$ Q = \sum_{i=1}^ { n } \alpha _ {i} \xi _ {i} ^ {2} $$

by a non-singular affine transformation of the variables $ \lambda _ {i} = \lambda _ {i} ( \xi _ {1} \dots \xi _ {n} ) $, $ i = 1 \dots n $.

When all $ \alpha _ {i} = 1 $ or all $ \alpha _ {i} = - 1 $, the equation is said to be of elliptic type.

A partial differential equation is said to be of elliptic type in its domain of definition if it is elliptic at every point of this domain.

An elliptic partial differential is called uniformly elliptic if there are positive numbers $ k _ {0} $ and $ k _ {1} $ such that

$$ k _ {0} \sum_{i=1}^ { n } \lambda _ {i} ^ {2} \leq Q ( \lambda _ {1} \dots \lambda _ {n} ) \leq k _ {1} \sum_{i=1}^ { n } \lambda _ {i} ^ {2} . $$

For references see Differential equation, partial.


References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001
How to Cite This Entry:
Elliptic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_partial_differential_equation&oldid=12485
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article