Discussion Board Lecture 11

Dear Christian,

`no' to both questions. The point is that for large $ k $ the support of $ \rho_k*\tilde u $ `contracts' to the support of $ u $, and therefore is a compact subset of $ \Omega $. (Well, in a sense you are right; our formulation is a short form of: $ \mathop{\rm spt}(\rho_k*\tilde u)\subseteq\Omega $ for large $ k $, and for these $ k $ one has $ (\rho_k*\tilde u)|_{\Omega}\in C_{\rm c}^\infty(\Omega) $.)

And for the second question: The point is that the approximating functions belong to $ C_{\rm c}^\infty(\Omega) $ and converge in $ H^1(\Omega) $ to $ u $. This shows that $ u $ belongs to the closure of $ C_{\rm c}^\infty(\Omega) $ in $ H^1(\Omega) $, i.e., to $ H^1_0(\Omega) $. (Then, naturally, the sequence converges to $ u $ also in $ H^1_0(\Omega) $.)

Best wishes, Jürgen

Dear Jürgen,

thank you very much for your comments.

Unfortunately, I miss-typed my second question, I meant to say: Shouldn't it read $ \rightarrow u $ instead of $ \rightarrow \tilde{u} $ in $ H^1(\Omega) $?

Best regards, Christian

Dear Christian,

there are so many possibilities to make mistakes! (I made this experience every week during the last time.) You are absolutely correct there; it should read $ \to u $. Thanks for pointing it out! (And I have to apologise for not reading your second comment more carefully.)

Best wishes, Jürgen

Dear Jürgen,

so at least I wasn't completely off in the end ;)

Best regards, Christian

Dear Christian,

actually, your first question was helpful, too! In the corrected version we will write that the restriction of $ \rho_k*\tilde u $ to $ \Omega $ belongs to $ C^\infty_c(\Omega) $. (Why not make it really correct, if it's possible?) Thanks!

Best wishes, Jürgen

Dear Anton,

thanks a lot for your comments.

In Lemma 11.1 there should be no $ v $, right!

In Section 11.2, do you mean almost every $ x\in\Omega $? Then I agree with you. If you do mean $ \xi $, then I don't understand that.

As for the proof of Prop. 11.9, I also don't see why we work with $ u_k^+ $. We want compact support, and for this we can (should!) just work with $ u_k $. Thanks for pointing that out!

Viele Grüße,  Hendrik

Dear Hendrik,

yes, I messed this up and meant almost every $ x\in\Omega $

Best wishes

Anton

Dear Clemens,

just note that $ \partial_j(u-1)^+ = \mathbf{1}_{[u\geq 1]} \partial_j(u-1)^+ $ and $ (u\wedge 1) \cdot \mathbf{1}_{[u\geq 1]} = \mathbf{1}_{[u\geq 1]} $. Does this help?

Best wishes, Christian

Dear Christian, I understand the proof now. Thank you for your reply. Best regards, Clemens.

Dear Jamil,

maybe you should have a look at Theorem 10.12. Thanks for the question.

Best wishes, and good success! Jürgen

Dear Jürgen,

Sure, Theorem 10.12(b) does the job. I overlooked it.

Thanks!