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Math Help - Inclusion in Sobolev spaces

  1. #1
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    Inclusion in Sobolev spaces

    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).
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    Re: Inclusion in Sobolev spaces

    Quote Originally Posted by kierkegaard View Post
    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.

    -----

    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).)
    Last edited by johnsomeone; September 13th 2012 at 03:45 PM.
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