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Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$?

I know when $\Omega$ is bounded, $\|\nabla u \|_2$ defines a equivalent norm on $W_0^{1,2}(\Omega)$ with $\|u\|_{1,2}$.

And to me it seems that $\|\nabla u\|_2$ does define a norm on $W^{1,2}(\mathbb{R}^N)$, but since Poincare inequality fails on $\mathbb{R}^N$ so this norm is weaker than the $\|u\|_{1,2}$ norm on $W^{1,2}(\mathbb{R}^N)$.

So what are the properties of this norm? Is $W^{1,2}(\mathbb{R}^N)$ complete w.r.t. this norm? What about uniformly convexity? Why do we never use this norm?

Thank you very much!

Xiao

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It's clearly a norm: If $\|\nabla u\|_{L^2}=0$ then $u$ is constant, and the only constant in $L^2$ is $0$. On the other hand, if $H^1=W^{1,2}$ were complete under this norm we would have, by the bounded inverse theorem, that both norms are equivalent, which we know to be false (the scaling below should give you an idea of how to build explicit counterexamples), in fact $\| \nabla \cdot \|$ is the norm of a Hilbert space, which is very important in applications.

The completion of $C_c^\infty$ with respect to this norm I've seen called $\mathcal{D}^{1,2}=\dot{H}^1=L^{1,2}$. In any case, call it $X$, it is the space $$ X=\{ u\in L^{2^*}: \nabla u\in L^2\}, \qquad 2^*=\frac{2n}{n-2}. $$ This is a consequence of the Gagliardo-Nirenberg-Sobolev inequality that says,
$$ \| u\|_{L^{2^*}} \leq C \| \nabla u\|_{L^2}, \qquad u\in X. $$ Although usually this inequality is stated for functions in $H^1$, it's true in the more general $X$, and it turns out that this latter space is the appropriate one if we are looking for minimizers, i.e. functions $u$ that give an equality in the preceding (this functions statisfy $u\in X\setminus H^1$). The existence of such functions has very important implications, see for example this work of Kenig and Merle.

The fact that this inequality is invariant under the scaling $u\mapsto u_\lambda=\lambda^{(n-2)/2}u(\lambda \cdot)$, i.e. both norms are the same for $u$ and $u_\lambda$ provides a criterion for 'energy criticality' in dispersive PDE. In this setting a google search for "Energy critical Schrödinger" should give you plenty of examples where $X$ is used.

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  • $\begingroup$ Thank you very much! It is very helpful. $\endgroup$ – Xiao Feb 13 '15 at 6:20
  • $\begingroup$ @Jose27: Is it that $\dot{H}^1 = \{u \in L^{2*}: \nabla u \in L^2\}$? Thanks.. $\endgroup$ – abcd Mar 18 '17 at 11:01
  • $\begingroup$ @abcd: I've seen $\dot{H}^1$ to denote that space, but there may be other definitions depending on who's writing. $\endgroup$ – Jose27 Mar 18 '17 at 15:02
  • $\begingroup$ @Jose27: Thanks. But if we define $A$ as the completion of $c_c^{\infty}$ with respect to $\|\nabla u\|_{L^2}$ norm then $A=X$ (where $X$ is as you have defined above)? $\endgroup$ – abcd Mar 18 '17 at 15:09
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    $\begingroup$ @abcd: Yes, that is correct. $\endgroup$ – Jose27 Mar 18 '17 at 15:22

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