In mathematics, Friedrichs' inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent.
Let Ω be a bounded subset of Euclidean space Rn with diameter d. Suppose that u : Ω → R lies in the Sobolev space (i.e. u lies in Wk,p(Ω) and the trace of u is zero). Then
In the above
A very closely related result is the Poincaré inequality.