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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|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 [[#References|[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 [[#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

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

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

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

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 [[#References|[1]]]).

====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>