Talk:Friedrichs' inequality

From Wikipedia, the free encyclopedia

Hm. I may be missing something, but doesn't the function u have to be zero on at least a part of the boundary (in the sense of a trace operator)? Else we could just take the constant 1 function, which has all zero derivatives but a non-zero L^p norm, and would therefore clearly violate this estimation. Eriatarka 21:57, 19 March 2007 (UTC)

Yup, that's a typo: this holds only for functions u ∈ W^k,p(Ω) with zero trace. In the proof of Friedrichs' ineuaqlity that I know, a key step is to extend u to be 0 outside Ω, and this extension by zero is also k-times weakly-differentiable with derivatives in L^p. The required estimates are then proved for the extension, and then you restrict back to the original u. I will amend the article accordingly. Sullivan.t.j 01:04, 20 March 2007 (UTC)

Thanks for the quick correction. In fact I'm now sure the assumptions can be weakened such that the trace of u has to be zero only on a part of the boundary (with measure > 0); a proof follows easily from Sobolev's norm equivalence theorem. Hence this theorem is applicable also to mixed boundary value problems, and not only pure Dirichlet boundary problems. However, I'm not sure which version is usually called "Friedrichs' inequality", so I'll leave it as it is for now. --Eriatarka 20:53, 20 March 2007 (UTC)