Inclusion in Sobolev spaces

**kierkegaard** · September 9th 2012, 06:55 AM

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Prove the inclusion $W^{1,n}(\Omega)\subset L^{p}(\Omega)$ for any $p\in[1,\infty)$ .

**johnsomeone** · September 13th 2012, 03:29 AM

Originally Posted by kierkegaard

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Prove the inclusion $W^{1,n}(\Omega)\subset L^{p}(\Omega)$ for any $p\in[1,\infty)$ .

It's been a while, so this is just to see if I've unpacked the problem correctly.

$u \in W^{1,n}(\Omega) \Rightarrow u \in L^{n}(\Omega)$ and $D_iu \in L^{n}(\Omega)$ for $i \in \{1, 2, ... N \}$ , where $D_i$ is the weak partial derivative in the $x_i$ coordinate.

Want to show that if for some $u$ that's true for some $n$ (I assume $n \ge 1$ ??), then $u \in L^{p}(\Omega) \ \forall p \in [1,\infty)$ .

Since $|\Omega| < \infty$ , it follows that $||u||_p \rightarrow ||u||_{\infty}$ .

It would suffice to prove the inclusion if could show $||u||_{\infty} < \infty$ (although that might not be the case - it's just an initial thought as to how to begin.)

What $D_iu$ exists means is that there are functions $D_iu \ni \int_{\Omega} u\frac{\partial \phi}{\partial x_i} = - \int_{\Omega} \phi D_iu \ \ \ \forall \phi \in C_{c}^{\infty}(\Omega)$ .

Anyway, I just wanted to see if I have the setup correct. I don't know the common techniques for Sobolev spaces, but what I'll be thinking about is: Can show $u$ is bounded a.e.? What about showing just $||u||_1 < \infty$ ? What if $u \in C^1(\Omega)$ , since it's "close" to that? What about approximating $u$ by smooth functions? Can show $||u||_p < \infty$ on compact subsets that expand to $\Omega$ in a controllable way so that that the limit is also finite? Surely $\Omega$ boounded is important - how should that be exploited? I dunno - it's just initial random thoughts.

I'd be interested to see how it's done, since I imagine it's done using common techniques for this topic. If I get anywhere, I'll post it.

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One more observation. The condition that $D_iu \in L^{n}(\Omega)$ for $i \in \{1, 2, ... , N \}$ , the Sobolev-ness, is vital, since without it, the claim fails.

Let $\Omega = ( 0, 1 ), u(x) = \frac{1}{\sqrt{x}}$ . Then $u \in C^{\infty}(\Omega) \cap L^1(\Omega)$ , but $u \notin L^2(\Omega)$ .

$u$ failed to be in $W^{1,1}(\Omega)$ because $u'(x) = \frac{-1}{2\sqrt{x^3}} \notin L^1(\Omega)$ .

In the same vein, suppose $f \in W^{1,n}(0,1)$ is of the form $f(x) = x^{\alpha}$ . Then:

$f'(x) = \alpha x^{\alpha - 1} \in L^n(0,1) \Rightarrow n(\alpha - 1)>-1 \Rightarrow \alpha >1 - 1/n \geq 0$

$\Rightarrow p\alpha \geq 0 \ \forall \ p \geq 1 \Rightarrow f(x)^p = x^{p\alpha} \in L^1(0,1) \ \forall \ p \geq 1$ .

Thus $f(x) = x^{\alpha} \in W^{1,n}(0,1) \Rightarrow f \in L^p(0,1) \ \forall \ p \geq 1$ . It also implies that $||f||_{\infty} < \infty$ .

Perhaps an the idea for the proof is found by: $D_1u \in L^n$ somehow gets you, using $u \in L^n$ , that $u \in L^m$ for some m > n. And then repeat that reasoning with this bigger m, "staircasing" your way up to higher and higher values so that can eventually conclude $u \in L^p \ \forall p$ ?
(If this approach even worked, I wouldn't expect $D_1u$ to "staircase" up - it would remain stuck in $L^n$ - since otherwise the whole of $W^{1,n}$ would staircase up to be a subset of $W^{1,p} \ \forall p$ , unless of course that's actually true, or true for bounded domains (I simply don't know -but I suspect it's not so).)

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