I am learning about weak derivatives and sobolev space. In particular I need help to learn the proving strategy/technique.

I have trouble proving on how to show a solution belongs to some sobolev space. A particular problem I came across is to prove $$ u\in W^{k,p}(\mathbb{R}^d)$$ if and only if $\exists$ a sequence of functions $\{u_m\}\subset C^{\infty}(\mathbb{R}^d)\cap L^p (\mathbb{R}^d)$ such that

  1. $$ \|u_m-u\|_{L^p(\mathbb{R}^d)}\to 0\quad\text{as}\quad m\to\infty$$ and
  2. $$\|D^{\alpha}u_m-D^{\alpha}u_n\|_{L^p(\mathbb{R}^d)}\to 0\quad\text{as}\quad m,n\to\infty\quad\text{for each}\quad |\alpha|\le k $$

Questions Could anyone sketch a proof of above? What books contain proofs of this kind? So that I can pick up the proving technique quickly...

  • 1
    $\begingroup$ I think this is not about a criterion for testing whether something is in a given Sobolev space, but, rather, the assertion (for you to prove) that smooth functions (in fact, test functions) are dense in every Sobolev space. $\endgroup$ – paul garrett Dec 8 '15 at 0:33
  • $\begingroup$ @paulgarrett: On"smooth functions are dense in every sobolev space". How do I show this? Or in what book can I look this up? $\endgroup$ – math101 Dec 8 '15 at 4:59

Almost all graduate-level PDE texts will have this classical result, my personal favorite is Evans'. If you just google "global approximation of smooth functions", any number of proofs will pop up, and here was a previous post in which a user had questions about Evans' proof.

EDIT: Broad strokes of iff proof: Forward direction: If $u\in W^{k,p}$, then we have a sequence of smooth functions satisfying 1.,2. by the global approximation theorem. For the reverse direction, if we have a sequence $u_m$ for which $D^\alpha u_m\rightarrow D^\alpha u$ for each $\alpha$, then by completeness of $W^{k,p}$ it must be that $u\in W^{k,p}$.

  • $\begingroup$ I came across this proof in Evan's book. However I am looking for help in writing if and only if type of proof. I often get lost when people ask me to prove necessary and sufficient conditions. Would be helpful if I can find more examples of this kind. $\endgroup$ – math101 Dec 8 '15 at 1:40
  • $\begingroup$ Only one direction of the iff is nontrivial: $W^{k,p}$ is complete (a separate, but nontrivial important result) so any sequence in $W^{k,p}$ that converges will be in $W^{k,p}$... Thus the other direction is where the only work happens. $\endgroup$ – charlestoncrabb Dec 8 '15 at 1:46
  • $\begingroup$ I cannot understand what you mean? 1) which direction ? 2) How does this "iff" type of proof relates to the the sentence you wrote $W^{k,p}$ is complete? Do you mean...the proof of $u\in W^{k,p}$ is done if we can show the limit of $u_m$ and the limit of $D^{\alpha} u_m$ can be found in $L^p?$ $\endgroup$ – math101 Dec 8 '15 at 3:53
  • $\begingroup$ Looking at your edited answer, "The forward direction" can be regarded as the proof of necessity, which is much of 5.3.3.Evans' proof about? And the reverse implication is the proof of sufficiency? $\endgroup$ – math101 Dec 8 '15 at 5:30
  • $\begingroup$ Also, on "completeness of $W^{k,p}$, do you mean completeness of $L^p$? $\endgroup$ – math101 Dec 8 '15 at 6:14

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