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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/f/f041/f041740/f0417401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _  \Omega  f ^ { 2 } \
 +
d \Omega  \leq  C \left \{
 +
\int\limits _  \Omega
 +
\sum _ {i = 1 } ^ { n }
 +
\left (
 +
 
 +
\frac{\partial  f }{\partial  x _ {i} }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417402.png" /> is a bounded domain of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417403.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417404.png" />-dimensional Euclidean space with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417405.png" />-dimensional boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417406.png" /> satisfying a local Lipschitz condition, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417407.png" /> (a [[Sobolev space|Sobolev space]]).
+
\right )  ^ {2} \
 +
d \Omega + \int\limits _  \Gamma
 +
f ^ { 2 }  d \Gamma \right \} ,
 +
$$
  
The right-hand side of the Friedrichs inequality gives an equivalent norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417408.png" />. Using another equivalent norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f0417409.png" />, one obtains (see [[#References|[2]]]) a modification of the Friedrichs inequality of the form
+
where  $  \Omega $
 +
is a bounded domain of points  $  x = x ( x _ {1} \dots x _ {n} ) $
 +
in an  $  n $-
 +
dimensional Euclidean space with an  $  ( n - 1) $-
 +
dimensional boundary  $  \Gamma $
 +
satisfying a local Lipschitz condition, and the function  $  f \equiv f ( x) \in W _ {2}  ^ {1} ( \Omega ) $(
 +
a [[Sobolev space|Sobolev space]]).
  
<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/f/f041/f041740/f04174010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
The right-hand side of the Friedrichs inequality gives an equivalent norm in  $  W _ {2}  ^ {1} ( \Omega ) $.  
 +
Using another equivalent norm in  $  W _ {2}  ^ {1} ( \Omega ) $,
 +
one obtains (see [[#References|[2]]]) a modification of the Friedrichs inequality of the form
  
There are generalizations (see [[#References|[3]]]–[[#References|[5]]]) of the Friedrichs inequality to weighted spaces (see [[Weighted space|Weighted space]]; [[Imbedding theorems|Imbedding theorems]]). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174011.png" /> and that the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174014.png" /> are real, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174015.png" /> being a natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174016.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174017.png" /> if the norm
+
$$ \tag{2 }
 +
\int\limits _  \Omega  f ^ { 2 } \
 +
d \Omega  \leq  C \left \{
 +
\int\limits _  \Omega
 +
\sum _ {i = 1 } ^ { n }
 +
\left (
  
<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/f/f041/f041740/f04174018.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  x _ {i} }
 +
 
 +
\right )  ^ {2}  d \Omega + \left (
 +
\int\limits _  \Gamma  f  d \Gamma \right )
 +
^ {2} \right \} .
 +
$$
 +
 
 +
There are generalizations (see [[#References|[3]]]–[[#References|[5]]]) of the Friedrichs inequality to weighted spaces (see [[Weighted space|Weighted space]]; [[Imbedding theorems|Imbedding theorems]]). Suppose that  $  \Gamma \subset  C  ^ {(} l) $
 +
and that the numbers  $  r $,
 +
$  p $
 +
and  $  \alpha $
 +
are real, with  $  r $
 +
being a natural number and  $  1 \leq  p < \infty $.
 +
One says that  $  f \in W _ {p, \alpha }  ^ {r} ( \Omega ) $
 +
if the norm
 +
 
 +
$$
 +
\| f \| _ {W _ {p, \alpha }  ^ {r} ( \Omega ) }  = \
 +
\| f \| _ {L _ {p}  ( \Omega ) } +
 +
\| f \| _ {\omega _ {p, \alpha }  ^ {r} ( \Omega ) }
 +
$$
  
 
is finite, where
 
is finite, where
  
<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/f/f041/f041740/f04174019.png" /></td> </tr></table>
+
$$
 +
\| f \| _ {L _ {p}  ( \Omega ) }  = \
 +
\left ( \int\limits _  \Omega  | f |  ^ {p} \
 +
d \Omega \right )  ^ {1/p} ,
 +
$$
 +
 
 +
$$
 +
\| f \| _ {\omega _ {p, \alpha }  ^ {r} ( \Omega ) }
 +
= \sum _ {| k | = r } \| \rho  ^  \alpha  f ^ {(} k) \| _ {L _ {p}  ( \Omega ) } ,
 +
$$
  
<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/f/f041/f041740/f04174020.png" /></td> </tr></table>
+
$$
 +
f ^ { ( k) }  =
 +
\frac{\partial  ^ {| k | } f }{\partial  x _ {1} ^ {k _ {1} } \dots \partial  x _ {n} ^ {k _ {n} } }
 +
,\ \
 +
| k |  = \sum _ {i = 1 } ^ { n }  k _ {i} ,
 +
$$
  
<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/f/f041/f041740/f04174021.png" /></td> </tr></table>
+
and  $  \rho = \rho ( x) $
 +
is distance function from  $  x \in \Omega $
 +
to  $  \Gamma $.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174022.png" /> is distance function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174024.png" />.
+
Suppose that  $  s _ {0} $
 +
is a natural number such that
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174025.png" /> is a natural number such that
+
$$
 +
r - \alpha - {
 +
\frac{1}{p}
 +
}  \leq  \
 +
s _ {0}  < r - \alpha + 1 - {
 +
\frac{1}{p}
 +
} .
 +
$$
  
<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/f/f041/f041740/f04174026.png" /></td> </tr></table>
+
Then, if  $  \Gamma \subset  C ^ {( s _ {0} + 1) } $,
 +
$  - p  ^ {-} 1 < \alpha < r - p  ^ {-} 1 $,
 +
$  r/2 \leq  s _ {0} $,
 +
for  $  f \in W _ {p, \alpha }  ^ {r} ( \Omega ) $
 +
the following inequality holds:
  
Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174029.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174030.png" /> the following inequality holds:
+
$$
 +
\| f \| _ {L _ {p}  ( \Omega ) }  \leq  \
 +
C \left \{
 +
\sum _ {l + s < r/2 }
 +
\left \| \left ( \left .
  
<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/f/f041/f041740/f04174031.png" /></td> </tr></table>
+
\frac{\partial  ^ {s} f }{\partial  n  ^ {s} }
 +
\
 +
\right | _  \Gamma  \right )  ^ {(} l) \
 +
\right \| _ {L _ {p}  ( \Gamma ) } +
 +
\| f \| _ {\omega _ {p, \alpha }  ^ {r} ( \Omega ) }
 +
\right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174032.png" /> is the derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174033.png" /> with respect to the interior normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174034.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174035.png" />.
+
where $  ( \partial  ^ {s} f/ \partial  n  ^ {s} ) \mid  _  \Gamma  $
 +
is the derivative of order $  s $
 +
with respect to the interior normal to $  \Gamma $
 +
at the points of $  \Gamma $.
  
 
One can also obtain an inequality of the type (2), which has in the simplest case the form
 
One can also obtain an inequality of the type (2), which has in the simplest case 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/f/f041/f041740/f04174036.png" /></td> </tr></table>
+
$$
 +
\| f \| _ {L _ {p}  ( \Omega ) }  ^ {p}  \leq  \
 +
C \left (
 +
\| f \| _ {\omega _ {p, \alpha }  ^ {1} ( \Omega ) }  ^ {p} +
 +
\left | \int\limits _  \Gamma
 +
u \tau  d \Gamma \
 +
\right |  ^ {p} \right ) ,
 +
$$
  
 
where
 
where
  
<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/f/f041/f041740/f04174037.png" /></td> </tr></table>
+
$$
 +
p , \gamma  > 1,\ \
 +
-
 +
\frac{1}{p}
 +
  < \alpha  < 1 -  
 +
\frac{1}{p}
 +
-
 +
\frac{1} \gamma
 +
,
 +
$$
  
<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/f/f041/f041740/f04174038.png" /></td> </tr></table>
+
$$
 +
\tau  \in  L _  \gamma  ( \Gamma ),\  \int\limits _  \Gamma  \tau  d \Gamma  \neq  0.
 +
$$
  
The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174039.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174040.png" /> throughout.
+
The constant $  C $
 +
is independent of f $
 +
throughout.
  
The inequality is named after K.O. Friedrichs, who obtained it for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174042.png" /> (see [[#References|[1]]]).
+
The inequality is named after K.O. Friedrichs, who obtained it for $  n = 2 $,  
 +
f \in C  ^ {(} 2) ( \overline \Omega \; ) $(
 +
see [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K.O. Friedrichs,  "Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz"  ''Math. Ann.'' , '''98'''  (1927)  pp. 566–575</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  P.I. Lizorkin,  "On some inequalities for weight-class functions and boundary-value problems with a strong degeneracy at the boundary"  ''Soviet Math. Dokl.'' , '''5'''  (1964)  pp. 1535–1539  ''Dokl. Akad. Nauk SSSR'' , '''159''' :  3  (1964)  pp. 512–515</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.F. Kalinichenko,  "Some properties of functions in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174044.png" />"  ''Mat. Sb.'' , '''64''' :  3  (1964)  pp. 436–457  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Nirenberg,  "On elliptic partial differential equations"  ''Ann. Scuola Norm. Sup. Pisa Ser. 3'' , '''13''' :  2  (1959)  pp. 115–162</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. Sandgren,  "A vibration problem"  ''Medd. Lunds Univ. Mat. Sem.'' , '''13'''  (1955)  pp. 1–84</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K.O. Friedrichs,  "Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz"  ''Math. Ann.'' , '''98'''  (1927)  pp. 566–575</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  P.I. Lizorkin,  "On some inequalities for weight-class functions and boundary-value problems with a strong degeneracy at the boundary"  ''Soviet Math. Dokl.'' , '''5'''  (1964)  pp. 1535–1539  ''Dokl. Akad. Nauk SSSR'' , '''159''' :  3  (1964)  pp. 512–515</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.F. Kalinichenko,  "Some properties of functions in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041740/f04174044.png" />"  ''Mat. Sb.'' , '''64''' :  3  (1964)  pp. 436–457  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Nirenberg,  "On elliptic partial differential equations"  ''Ann. Scuola Norm. Sup. Pisa Ser. 3'' , '''13''' :  2  (1959)  pp. 115–162</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. Sandgren,  "A vibration problem"  ''Medd. Lunds Univ. Mat. Sem.'' , '''13'''  (1955)  pp. 1–84</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


An inequality of the form

$$ \tag{1 } \int\limits _ \Omega f ^ { 2 } \ d \Omega \leq C \left \{ \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} \ d \Omega + \int\limits _ \Gamma f ^ { 2 } d \Gamma \right \} , $$

where $ \Omega $ is a bounded domain of points $ x = x ( x _ {1} \dots x _ {n} ) $ in an $ n $- dimensional Euclidean space with an $ ( n - 1) $- dimensional boundary $ \Gamma $ satisfying a local Lipschitz condition, and the function $ f \equiv f ( x) \in W _ {2} ^ {1} ( \Omega ) $( a Sobolev space).

The right-hand side of the Friedrichs inequality gives an equivalent norm in $ W _ {2} ^ {1} ( \Omega ) $. Using another equivalent norm in $ W _ {2} ^ {1} ( \Omega ) $, one obtains (see [2]) a modification of the Friedrichs inequality of the form

$$ \tag{2 } \int\limits _ \Omega f ^ { 2 } \ d \Omega \leq C \left \{ \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} d \Omega + \left ( \int\limits _ \Gamma f d \Gamma \right ) ^ {2} \right \} . $$

There are generalizations (see [3][5]) of the Friedrichs inequality to weighted spaces (see Weighted space; Imbedding theorems). Suppose that $ \Gamma \subset C ^ {(} l) $ and that the numbers $ r $, $ p $ and $ \alpha $ are real, with $ r $ being a natural number and $ 1 \leq p < \infty $. One says that $ f \in W _ {p, \alpha } ^ {r} ( \Omega ) $ if the norm

$$ \| f \| _ {W _ {p, \alpha } ^ {r} ( \Omega ) } = \ \| f \| _ {L _ {p} ( \Omega ) } + \| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) } $$

is finite, where

$$ \| f \| _ {L _ {p} ( \Omega ) } = \ \left ( \int\limits _ \Omega | f | ^ {p} \ d \Omega \right ) ^ {1/p} , $$

$$ \| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) } = \sum _ {| k | = r } \| \rho ^ \alpha f ^ {(} k) \| _ {L _ {p} ( \Omega ) } , $$

$$ f ^ { ( k) } = \frac{\partial ^ {| k | } f }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } ,\ \ | k | = \sum _ {i = 1 } ^ { n } k _ {i} , $$

and $ \rho = \rho ( x) $ is distance function from $ x \in \Omega $ to $ \Gamma $.

Suppose that $ s _ {0} $ is a natural number such that

$$ r - \alpha - { \frac{1}{p} } \leq \ s _ {0} < r - \alpha + 1 - { \frac{1}{p} } . $$

Then, if $ \Gamma \subset C ^ {( s _ {0} + 1) } $, $ - p ^ {-} 1 < \alpha < r - p ^ {-} 1 $, $ r/2 \leq s _ {0} $, for $ f \in W _ {p, \alpha } ^ {r} ( \Omega ) $ the following inequality holds:

$$ \| f \| _ {L _ {p} ( \Omega ) } \leq \ C \left \{ \sum _ {l + s < r/2 } \left \| \left ( \left . \frac{\partial ^ {s} f }{\partial n ^ {s} } \ \right | _ \Gamma \right ) ^ {(} l) \ \right \| _ {L _ {p} ( \Gamma ) } + \| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) } \right \} , $$

where $ ( \partial ^ {s} f/ \partial n ^ {s} ) \mid _ \Gamma $ is the derivative of order $ s $ with respect to the interior normal to $ \Gamma $ at the points of $ \Gamma $.

One can also obtain an inequality of the type (2), which has in the simplest case the form

$$ \| f \| _ {L _ {p} ( \Omega ) } ^ {p} \leq \ C \left ( \| f \| _ {\omega _ {p, \alpha } ^ {1} ( \Omega ) } ^ {p} + \left | \int\limits _ \Gamma u \tau d \Gamma \ \right | ^ {p} \right ) , $$

where

$$ p , \gamma > 1,\ \ - \frac{1}{p} < \alpha < 1 - \frac{1}{p} - \frac{1} \gamma , $$

$$ \tau \in L _ \gamma ( \Gamma ),\ \int\limits _ \Gamma \tau d \Gamma \neq 0. $$

The constant $ C $ is independent of $ f $ throughout.

The inequality is named after K.O. Friedrichs, who obtained it for $ n = 2 $, $ f \in C ^ {(} 2) ( \overline \Omega \; ) $( see [1]).

References

[1] K.O. Friedrichs, "Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz" Math. Ann. , 98 (1927) pp. 566–575
[2] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[3] S.M. Nikol'skii, P.I. Lizorkin, "On some inequalities for weight-class functions and boundary-value problems with a strong degeneracy at the boundary" Soviet Math. Dokl. , 5 (1964) pp. 1535–1539 Dokl. Akad. Nauk SSSR , 159 : 3 (1964) pp. 512–515
[4] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[5] D.F. Kalinichenko, "Some properties of functions in the spaces and " Mat. Sb. , 64 : 3 (1964) pp. 436–457 (In Russian)
[6] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[7] L. Nirenberg, "On elliptic partial differential equations" Ann. Scuola Norm. Sup. Pisa Ser. 3 , 13 : 2 (1959) pp. 115–162
[8] L. Sandgren, "A vibration problem" Medd. Lunds Univ. Mat. Sem. , 13 (1955) pp. 1–84
How to Cite This Entry:
Friedrichs inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Friedrichs_inequality&oldid=46991
This article was adapted from an original article by D.F. KalinichenkoN.V. Miroshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article