In Sobolev spaces $W^{1,p}(\Omega),\Omega\subseteq \mathbb{R}^n,p<n$, positive cone has empty interior. How does one prove that? Is it a consequence of some more general result like the embedding theorem?
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$\begingroup$ Take a test function $f(x)\ge0$, $0<s<-1+n/p$, and let $f_a(x)=a^{-s}f(x/a)$ for $a>0$. The Sobolev norm of $f_a$ is $\simeq a^{-s-1+n/p}\to0$ as $a\to0$, while the sup norm of $f_a$ tends to $\infty$. Thus we can modify locally the sign of a function by adding or subtracting a perturbation $f_a$ arbitrarily small in the Sobolev norm. $\endgroup$– Piero D'AnconaCommented Dec 16, 2015 at 15:14
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$\begingroup$ Sorry, now that I've written it down, I don't see how sobolev norm of $f_a$ is calculated? $\endgroup$– DeeCommented Dec 16, 2015 at 22:03
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$\begingroup$ Just by rescaling. The norm of $f_a$ is a power of $a$ times the norm of $f$. When $a\to0$ the behaviour is the one written above $\endgroup$– Piero D'AnconaCommented Dec 17, 2015 at 9:39
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