Talk:Sobolev inequality

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[edit] Gagliardo--Nirenberg--Sobolev

Hi Igny. I would like your help if possible on the GNS inequality. I am onaraighl and I am the one who edited your article, but I am an applied mathematician and not an expert here.

It's just I have a problem with the GNS inequality if you don't subtract off a constant term: taking u=constant\neq0 on the open unit disc would satisfy the conditions of the theorem, yet gives constant\leq0, which seems to be a contradiction. I have spoken to some of my professors about the GNS inequality and consulted the functional analysis book by Kantorovich, and they all say that you need to subtract off the mean of the function in the GNS inequality.

I would dearly like it if it were the case that you did not have to subtract off the mean, since then some estimates that I am working on for the Cahn--Hilliard equation would work out a lot nicer, yet I just don't see how you can do that.

Also, perhaps the article would benefit from a discussion on the optimal constants for the bound, which clearly depends on the domain in question.

Thanks.Onaraighl 17:57, 13 May 2007 (UTC)