Friedrichs Inequality

Let be an open, bounded, and connected subset of for some and let denote -dimensional Lebesgue measure on . In functional analysis, the Friedrichs inequality says that there exists a constant such that

for all functions in the Sobolev space consisting of those functions in having zero trace on the boundary of whose generalized derivatives are all also square integrable.

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 only 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.