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Difference between revisions of "Coerciveness inequality"

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An inequality providing a lower bound for a certain bilinear form, or providing an upper bound for the norm of a solution of a certain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let
 
An inequality providing a lower bound for a certain bilinear form, or providing an upper bound for the norm of a solution of a certain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let
  
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$$ \left( - 1 \right)^m Re \sum_{\left| \alpha \right| = 2 m} a_{\alpha} \left( x \right) \xi^{\alpha} \geqslant c \left| \xi \right|^{2 m}, $$
 
$$ \left( - 1 \right)^m Re \sum_{\left| \alpha \right| = 2 m} a_{\alpha} \left( x \right) \xi^{\alpha} \geqslant c \left| \xi \right|^{2 m}, $$
  
be a uniformly elliptic operator in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229403.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229404.png" />, with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229405.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229406.png" /> be a subregion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229407.png" />, and suppose that in some neighbourhood of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229408.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229409.png" /> one has differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294011.png" />, of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294012.png" />, the characteristics of which are not tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294013.png" /> at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294014.png" />. Then, in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294015.png" />, there exist differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294016.png" /> of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294018.png" />, such that
+
be a uniformly elliptic operator in a region $  \Omega _{1} $
 +
in $  \mathbf R ^{n} $,  
 +
with coefficients $  a _ \alpha  (x) \in C ^ \infty  ( \Omega _{1} ) $;  
 +
let $  \Omega $
 +
be a subregion of $  \Omega _{1} $,  
 +
and suppose that in some neighbourhood of the boundary $  S $
 +
of $  \Omega $
 +
one has differential operators $  M _{j} $,  
 +
$  j = 0 \dots m - 1 $,  
 +
of orders $  j $,  
 +
the characteristics of which are not tangent to $  S $
 +
at any point of $  S $.  
 +
Then, in some neighbourhood of $  S $,  
 +
there exist differential operators $  N _{j} $
 +
of orders $  j $,  
 +
$  j = m \dots 2m -1 $,  
 +
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/c/c022/c022940/c02294019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1}
 +
\sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}}
 +
( \partial ^ \alpha  v ,\  a _{ {\alpha \beta}} \partial ^ \beta  u ) - (v ,\  L (u))\  =
 +
$$
  
<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/c/c022/c022940/c02294020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {j = 0} ^ {m - 1} \int\limits _{S} M _{j} (v) {N _{ {2m - 1 - j}} (u)} bar \  d \sigma
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294022.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294023.png" /> denotes the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294024.png" />.
+
for all $  v $
 +
in $  C ^ \infty  ( \overline \Omega \; ) $.  
 +
Here $  ( \  ,\  ) $
 +
denotes the scalar product in $  L _{2} ( \Omega ) $.
  
 
The form
 
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/c/c022/c022940/c02294025.png" /></td> </tr></table>
+
$$
 +
D (v,\  u ) \  = \
 +
\sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}}
 +
( \partial ^ \alpha  v,\  a _{ {\alpha \beta}} \partial ^ \beta  u )
 +
$$
  
is called a coercive form on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294027.png" />, if there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294029.png" /> such that
+
is called a coercive form on a space $  X $,  
 +
$  W _{2c} ^ {m} ( \Omega ) \subset X \subset W _{2} ^{m} ( \Omega ) $,  
 +
if there exist constants $  C > 0 $
 +
and $  \lambda \geq 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/c/c022/c022940/c02294030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2}
 +
\mathop{\rm Re}\nolimits \  D (u ,\  u) \  \geq \
 +
C \  \| u \| _{ {m,\  \Omega}} ^{2} - \lambda \  \| u \| _{ {0,\  \Omega}} ^{2}  $$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294031.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294032.png" /> is the [[Sobolev space|Sobolev space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294033.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294034.png" /> consisting of all elements with compact support, i.e. elements vanishing in a neighbourhood of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294035.png" />. Inequality (2) is a coerciveness inequality for the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294036.png" />. If (2) remains valid with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294038.png" /> is said to be strongly coercive.
+
for all $  u \in X $.  
 +
Here $  W _{2} ^{m} $
 +
is the [[Sobolev space|Sobolev space]] and $  W _{2c} ^ {m} $
 +
is the subspace of $  W _{2} ^{m} $
 +
consisting of all elements with compact support, i.e. elements vanishing in a neighbourhood of the boundary of $  \Omega $.  
 +
Inequality (2) is a coerciveness inequality for the form $  D (v,\  u) $.  
 +
If (2) remains valid with $  \lambda = 0 $,  
 +
then $  D (v,\  u) $
 +
is said to be strongly coercive.
  
If a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294039.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294040.png" /> satisfies the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294042.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294043.png" />, then one has an inequality
+
If a solution $  u $
 +
of the equation $  L (u) = f $
 +
satisfies the conditions $  M _{j} (u) = 0 $,  
 +
$  j = 0 \dots m - 1 $,  
 +
on $  S $,  
 +
then one has an inequality
  
<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/c/c022/c022940/c02294044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3}
 +
\| u \| _{ {2m,\  \Omega}} \  \leq \  C _{1} \
 +
\| L (u) \| _{ {0,\  \Omega}} + \lambda _{1} \
 +
\| u \| _{ {0,\  \Omega}}  $$
  
for some constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294045.png" />. If a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294046.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294047.png" /> satisfies conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294050.png" />, then instead of (3) one has the inequality
+
for some constants $  C _{1} > 0,\  \lambda _{1} \geq 0 $.  
 +
If a solution $  u $
 +
of the equation $  L (u) = f $
 +
satisfies conditions $  M _{j} (u) = \phi _{j} $
 +
on $  S $,  
 +
$  j = 0 \dots m - 1 $,  
 +
then instead of (3) one has the inequality
  
<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/c/c022/c022940/c02294051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4}
 +
\| u \| _{ {2m,\  \Omega}} \  \leq \  C \  \left \{
 +
\| L (u) \| _{ {0,\  \Omega}} + \sum _ {j = 0} ^ {m - 1}
 +
\| \phi _{j} \| _{ {2m - j,\  s}} +
 +
\| u \| _{ {0,\  \Omega}} \right \} .
 +
$$
  
This inequality provides an estimate for the norm of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294052.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294053.png" /> in the Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294054.png" /> in terms of its norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294055.png" /> and of the norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294058.png" /> in the appropriate spaces. Inequality (4) is a coerciveness inequality for the boundary value problem for an elliptic equation.
+
This inequality provides an estimate for the norm of the solution $  u $
 +
of the equation $  L (u) = f $
 +
in the Sobolev space $  W _{2} ^{2m} ( \Omega ) $
 +
in terms of its norm in $  L _{2} ( \Omega ) $
 +
and of the norms of $  f $
 +
and $  \phi _{j} $,  
 +
$  j = 0 \dots m - 1 $
 +
in the appropriate spaces. Inequality (4) is a coerciveness inequality for the boundary value problem for an elliptic equation.
  
 
Using inequality (4) one obtains the more general inequality
 
Using inequality (4) one obtains the more general inequality
  
<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/c/c022/c022940/c02294059.png" /></td> </tr></table>
+
$$
 +
\| u \| _{ {2m + k,\  \Omega}} \  \leq \  C \  \left \{
 +
\| L (u) \| _{ {k,\  \Omega}} +
 +
\sum _ {j = 0} ^ {m - 1}
 +
\| \phi _{j} \| _{ {2m - j + k,\  s}} +
 +
\| u \| _{ {0,\  \Omega}} \right \} .
 +
$$
  
 
Coerciveness inequalities play an important role in the investigation of coercive boundary value problems and in proofs of the smoothness of solutions of elliptic equations; they are particularly important in analyticity proofs for solutions of analytic elliptic equations [[#References|[2]]].
 
Coerciveness inequalities play an important role in the investigation of coercive boundary value problems and in proofs of the smoothness of solutions of elliptic equations; they are particularly important in analyticity proofs for solutions of analytic elliptic equations [[#References|[2]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Agmon,  "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.B. Morrey,  L. Nirenberg,  "On the analyticity of the solutions of linear elliptic systems of partial differential equations"  ''Comm. Pure Appl. Math.'' , '''10''' :  2  (1957)  pp. 271–290</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Agmon,  "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.B. Morrey,  L. Nirenberg,  "On the analyticity of the solutions of linear elliptic systems of partial differential equations"  ''Comm. Pure Appl. Math.'' , '''10''' :  2  (1957)  pp. 271–290</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Inequalities such as (3), (4), providing upper bounds for elliptic boundary value problems, are better known as boundary estimates for elliptic boundary value problems, instead of coerciveness inequalities. Lower bounds for bilinear forms are frequently encountered in the theory of variational inequalities (see also [[Variational equations|Variational equations]]). See also [[Coercive boundary value problem|Coercive boundary value problem]].
 
Inequalities such as (3), (4), providing upper bounds for elliptic boundary value problems, are better known as boundary estimates for elliptic boundary value problems, instead of coerciveness inequalities. Lower bounds for bilinear forms are frequently encountered in the theory of variational inequalities (see also [[Variational equations|Variational equations]]). See also [[Coercive boundary value problem|Coercive boundary value problem]].
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294060.png" /> in (1) are obtained from the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294061.png" /> by integrating the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294062.png" /> by parts. Clearly, the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294063.png" /> in (2) is inessential.
+
The functions $  a _{ {\alpha \beta}} $
 +
in (1) are obtained from the functions $  a _ \alpha  $
 +
by integrating the expression $  (v,\  Lu) $
 +
by parts. Clearly, the restriction $  \lambda \geq 0 $
 +
in (2) is inessential.

Latest revision as of 22:14, 28 January 2020


An inequality providing a lower bound for a certain bilinear form, or providing an upper bound for the norm of a solution of a certain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let

$$ L = \sum_{\left| \alpha \right| \leqslant 2 m} a_{\alpha} \left( x \right) \partial^{\alpha}, $$ $$ \left( - 1 \right)^m Re \sum_{\left| \alpha \right| = 2 m} a_{\alpha} \left( x \right) \xi^{\alpha} \geqslant c \left| \xi \right|^{2 m}, $$

be a uniformly elliptic operator in a region $ \Omega _{1} $ in $ \mathbf R ^{n} $, with coefficients $ a _ \alpha (x) \in C ^ \infty ( \Omega _{1} ) $; let $ \Omega $ be a subregion of $ \Omega _{1} $, and suppose that in some neighbourhood of the boundary $ S $ of $ \Omega $ one has differential operators $ M _{j} $, $ j = 0 \dots m - 1 $, of orders $ j $, the characteristics of which are not tangent to $ S $ at any point of $ S $. Then, in some neighbourhood of $ S $, there exist differential operators $ N _{j} $ of orders $ j $, $ j = m \dots 2m -1 $, such that

$$ \tag{1} \sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}} ( \partial ^ \alpha v ,\ a _{ {\alpha \beta}} \partial ^ \beta u ) - (v ,\ L (u))\ = $$

$$ = \ \sum _ {j = 0} ^ {m - 1} \int\limits _{S} M _{j} (v) {N _{ {2m - 1 - j}} (u)} bar \ d \sigma $$

for all $ v $ in $ C ^ \infty ( \overline \Omega \; ) $. Here $ ( \ ,\ ) $ denotes the scalar product in $ L _{2} ( \Omega ) $.

The form

$$ D (v,\ u ) \ = \ \sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}} ( \partial ^ \alpha v,\ a _{ {\alpha \beta}} \partial ^ \beta u ) $$

is called a coercive form on a space $ X $, $ W _{2c} ^ {m} ( \Omega ) \subset X \subset W _{2} ^{m} ( \Omega ) $, if there exist constants $ C > 0 $ and $ \lambda \geq 0 $ such that

$$ \tag{2} \mathop{\rm Re}\nolimits \ D (u ,\ u) \ \geq \ C \ \| u \| _{ {m,\ \Omega}} ^{2} - \lambda \ \| u \| _{ {0,\ \Omega}} ^{2} $$

for all $ u \in X $. Here $ W _{2} ^{m} $ is the Sobolev space and $ W _{2c} ^ {m} $ is the subspace of $ W _{2} ^{m} $ consisting of all elements with compact support, i.e. elements vanishing in a neighbourhood of the boundary of $ \Omega $. Inequality (2) is a coerciveness inequality for the form $ D (v,\ u) $. If (2) remains valid with $ \lambda = 0 $, then $ D (v,\ u) $ is said to be strongly coercive.

If a solution $ u $ of the equation $ L (u) = f $ satisfies the conditions $ M _{j} (u) = 0 $, $ j = 0 \dots m - 1 $, on $ S $, then one has an inequality

$$ \tag{3} \| u \| _{ {2m,\ \Omega}} \ \leq \ C _{1} \ \| L (u) \| _{ {0,\ \Omega}} + \lambda _{1} \ \| u \| _{ {0,\ \Omega}} $$

for some constants $ C _{1} > 0,\ \lambda _{1} \geq 0 $. If a solution $ u $ of the equation $ L (u) = f $ satisfies conditions $ M _{j} (u) = \phi _{j} $ on $ S $, $ j = 0 \dots m - 1 $, then instead of (3) one has the inequality

$$ \tag{4} \| u \| _{ {2m,\ \Omega}} \ \leq \ C \ \left \{ \| L (u) \| _{ {0,\ \Omega}} + \sum _ {j = 0} ^ {m - 1} \| \phi _{j} \| _{ {2m - j,\ s}} + \| u \| _{ {0,\ \Omega}} \right \} . $$

This inequality provides an estimate for the norm of the solution $ u $ of the equation $ L (u) = f $ in the Sobolev space $ W _{2} ^{2m} ( \Omega ) $ in terms of its norm in $ L _{2} ( \Omega ) $ and of the norms of $ f $ and $ \phi _{j} $, $ j = 0 \dots m - 1 $ in the appropriate spaces. Inequality (4) is a coerciveness inequality for the boundary value problem for an elliptic equation.

Using inequality (4) one obtains the more general inequality

$$ \| u \| _{ {2m + k,\ \Omega}} \ \leq \ C \ \left \{ \| L (u) \| _{ {k,\ \Omega}} + \sum _ {j = 0} ^ {m - 1} \| \phi _{j} \| _{ {2m - j + k,\ s}} + \| u \| _{ {0,\ \Omega}} \right \} . $$

Coerciveness inequalities play an important role in the investigation of coercive boundary value problems and in proofs of the smoothness of solutions of elliptic equations; they are particularly important in analyticity proofs for solutions of analytic elliptic equations [2].

References

[1] S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)
[2] C.B. Morrey, L. Nirenberg, "On the analyticity of the solutions of linear elliptic systems of partial differential equations" Comm. Pure Appl. Math. , 10 : 2 (1957) pp. 271–290

Comments

Inequalities such as (3), (4), providing upper bounds for elliptic boundary value problems, are better known as boundary estimates for elliptic boundary value problems, instead of coerciveness inequalities. Lower bounds for bilinear forms are frequently encountered in the theory of variational inequalities (see also Variational equations). See also Coercive boundary value problem.

The functions $ a _{ {\alpha \beta}} $ in (1) are obtained from the functions $ a _ \alpha $ by integrating the expression $ (v,\ Lu) $ by parts. Clearly, the restriction $ \lambda \geq 0 $ in (2) is inessential.

How to Cite This Entry:
Coerciveness inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coerciveness_inequality&oldid=44363
This article was adapted from an original article by A.I. Yanushauskas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article