Friedrichs' inequality

Friedrichs' inequality

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.

tatement of the inequality

Let Ω be a bounded subset of Euclidean space R"n" with diameter "d". Suppose that "u" : Ω → R lies in the Sobolev space W_{0}^{k, p} (Omega) (i.e. "u" lies in "W""k","p"(Ω) and the trace of "u" is zero). Then

:| u |_{L^{p} (Omega)} leq d^{k} left( sum_ u}{partial_{x_{1^{alpha_{1 cdots partial_{x_{n^{alpha_{n }.

A very related result is the Poincaré inequality.


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