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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 [[#References|[1]]].

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

====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>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''3''' , Springer  (1985)</TD></TR>
</table>