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An inequality of the form
 
An inequality of 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/g/g043/g043320/g0433201.png" /></td> </tr></table>
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$$
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\| u \| _ {m}  ^ {2}  \leq  \
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c _ {1}  \mathop{\rm Re}  B [ u, u]  = \
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c _ {2}  \| u \| _ {0}  ^ {2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433202.png" /> is a complex-valued function with compact support (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433203.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433204.png" /> is a bounded domain and
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where $  u \in C _ {0}  ^  \infty  ( G) $
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is a complex-valued function with compact support (in $  G $),  
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$  G \subset  \mathbf R  ^ {n} $
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is a bounded domain 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/g/g043/g043320/g0433205.png" /></td> </tr></table>
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$$
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B [ u, u]  = \
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\sum _ {| s |, | t | \leq  m } \
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\int\limits _ { G } a _ {st} D  ^ {s} u \overline{ {D  ^ {t} u }}\; dx
 +
$$
  
is a quadratic integral form with complex continuous coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433206.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433207.png" />. A sufficient condition for the Gårding inequality to be valid for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433208.png" /> is the existence of a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g0433209.png" /> such that
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is a quadratic integral form with complex continuous coefficients $  a _ {st} $
 +
in $  \overline{G}\; $.  
 +
A sufficient condition for the Gårding inequality to be valid for any function $  u \in C _ {0}  ^  \infty  ( G) $
 +
is the existence of a positive constant $  c _ {0} $
 +
such that
  
<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/g/g043/g043320/g04332010.png" /></td> </tr></table>
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$$
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\mathop{\rm Re}  \sum _ {| s |, | t | \leq  m }
 +
a _ {st} \xi  ^ {s} \xi  ^ {t}  \geq  \
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c _ {0}  | \xi |  ^ {2m} ,
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g04332011.png" /> and all real vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043320/g04332012.png" />. Formulated and proved by L. Gårding [[#References|[1]]].
+
for any $  x \in G $
 +
and all real vectors $  \xi = ( \xi  ^ {1} \dots \xi  ^ {n)} $.  
 +
Formulated and proved by L. Gårding [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Gårding,  "Dirichlet's problem for linear elliptic partial differential equations"  ''Math. Scand.'' , '''1'''  (1953)  pp. 55–72</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Gårding,  "Dirichlet's problem for linear elliptic partial differential equations"  ''Math. Scand.'' , '''1'''  (1953)  pp. 55–72</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:42, 5 June 2020


An inequality of the form

$$ \| u \| _ {m} ^ {2} \leq \ c _ {1} \mathop{\rm Re} B [ u, u] = \ c _ {2} \| u \| _ {0} ^ {2} , $$

where $ u \in C _ {0} ^ \infty ( G) $ is a complex-valued function with compact support (in $ G $), $ G \subset \mathbf R ^ {n} $ is a bounded domain and

$$ B [ u, u] = \ \sum _ {| s |, | t | \leq m } \ \int\limits _ { G } a _ {st} D ^ {s} u \overline{ {D ^ {t} u }}\; dx $$

is a quadratic integral form with complex continuous coefficients $ a _ {st} $ in $ \overline{G}\; $. A sufficient condition for the Gårding inequality to be valid for any function $ u \in C _ {0} ^ \infty ( G) $ is the existence of a positive constant $ c _ {0} $ such that

$$ \mathop{\rm Re} \sum _ {| s |, | t | \leq m } a _ {st} \xi ^ {s} \xi ^ {t} \geq \ c _ {0} | \xi | ^ {2m} , $$

for any $ x \in G $ and all real vectors $ \xi = ( \xi ^ {1} \dots \xi ^ {n)} $. Formulated and proved by L. Gårding [1].

References

[1] L. Gårding, "Dirichlet's problem for linear elliptic partial differential equations" Math. Scand. , 1 (1953) pp. 55–72
[2] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1

Comments

A sharp form of this inequality has been given by L. Hörmander. See [a1], Sects. 18.1, 18.6, and the literature quoted there.

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)
How to Cite This Entry:
Gårding inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A5rding_inequality&oldid=47152
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article