The Friedrichs inequality in functional analysis states that, for an open, bounded,
and connected subset 
of 
, there exists a constant 
such that
 |
for all real-valued functions 
in the Sobolev
space 
,
where 
denotes 
-dimensional Lebesgue measure.
In general, 
is the topological closure in the 
norm of the C∞-functions
with compact support in 
. When the boundary is locally
describable, after suitable rotations, as graphs of
Lipschitz functions, with 
locally on one side of each graph, it is equivalently
the functions in 
having zero trace
on 
(Hu 2023).
This inequality plays an important role in the study of both function spaces and partial
differential equations. As such, a number of generalizations have been established
to domains 
and functions 
which are less well-behaved, e.g., to polyhedral domains 
and to functions 
which have desirable behavior only piecewise
on 
.
In some literature, the Friedrichs inequality is unfortunately referred to as the Poincaré inequality though it should be differentiated from the closely-related (mean) Poincaré inequality. Inequalities which are qualitatively similar to those of Friedrichs and Poincaré are sometimes referred to collectively as Poincaré-Friedrichs inequalities.
." Numer. Func. Anal. and Opt. 26, 925-952,
2005.