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If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the inequality holds ?

Thank you for your help.

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    Usually such questions can be answered quickly by means of a dimensional analysis.2012-05-30
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    @GiuseppeNegro: is it true that the follwoing condition should hold $q=(np)/(n-p)$ ? Is it sufficient ?2012-05-30
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    [Gagliardo–Nirenberg–Sobolev inequality](http://en.wikipedia.org/wiki/Sobolev_inequality#Gagliardo.E2.80.93Nirenberg.E2.80.93Sobolev_inequality)2012-05-30
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    Here is the 'dimensional analysis' I was referring to: [Remark 10](http://books.google.it/books?id=GAA2XqOIIGoC&lpg=PP1&hl=it&pg=PA278#v=onepage&q&f=false). The text refers to it as a 'scaling argument'.2012-06-03

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This result is a special case of what is known as Friedrich's Inequality, also sometimes known as Poincare's inequality. It is always true with $n$ arbitrary and $p = q$, although $C$ will depend on $n$ and the size and shape of the support of $u$. However, if you fix the supports of all your $u$'s to lie inside of some fixed set $\Omega$ (i.e. $ u \in C_c(\Omega)$), then you can choose $C$ depending only on $n$ and $\Omega$. Of course, since the support is compact and hence of finite Lebesgue measure, the inequality will also be true with $q \geq p$ simply because in this case $\|f\|_p \le C\|f\|_q$ by Holder's inequality. Any reasonable book on PDEs/Sobolev Spaces should have a proof of Friedrich's inequality, although its not too hard to cook one up yourself. Hint: integration by parts.

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    I think the OP's inequality is supposed to hold with $C$ _not depending_ on $u$. If you allow for dependency on the size and shape of the support, then you could also say "for each $u$ the inequality holds for some $C$", which would be trivial. (Of course the version you propose is not trivial.)2012-05-30
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    @HendrikVogt Yes, the problem is that the OP only specified $u \in C_c(\mathbb{R}^n)$,rather than say $u \in C_c(\Omega)$ for some fixed bounded $Omega$ in which case $C$ could be chosen independently of $u$.2012-05-30
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    Yep, for bounded $\Omega$ that's Friedrich's inequality. I'm pretty sure, however, that the OP wants the Gagliardo–Nirenberg–Sobolev inequality after all.2012-05-30
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    @HendrikVogt Hm I just checked Wikipedia, and yes that looks more like what the OP wants. I never realized one applied to more cases than the other...2012-05-30