Timeline for $f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2023 at 15:19 | vote | accept | No-one | ||
| Nov 13, 2023 at 15:01 | history | edited | No-one | CC BY-SA 4.0 |
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| Nov 13, 2023 at 14:35 | history | edited | No-one | CC BY-SA 4.0 |
added 269 characters in body
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| Nov 11, 2023 at 11:17 | answer | added | Nate River | timeline score: 5 | |
| Nov 10, 2023 at 22:46 | comment | added | Giorgio Metafune | In 1d let $A=\{f\neq 0\} =\cup (a_n,b_n)$ and for a test function $\phi$, $\int u\phi'=\int_A u\phi'=\sum_n \int_{a_n}^{b_n} u\phi'=-\sum_n \int_{a_n}^{b_n} u'\phi=-\int_A u' \phi$, since $u(a_n)=u(b_n)=0$. In 2d it should follow from this and a sectional argument...and so on (I admit that I did not check all details). | |
| Nov 10, 2023 at 22:04 | comment | added | No-one | @PietroMajer Can you please elaborate? They look the same to me. | |
| Nov 10, 2023 at 21:44 | comment | added | Pietro Majer | The question in the title looks slightly different from the question in the text | |
| Nov 10, 2023 at 20:59 | history | edited | No-one | CC BY-SA 4.0 |
edited title
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| Nov 10, 2023 at 20:23 | comment | added | No-one | @IosifPinelis It's the unit ball, I have edited to make it clear. | |
| Nov 10, 2023 at 20:23 | comment | added | No-one | @leomonsaingeon You are right the question didn't make sense as it was written, I was missing two crucial hypothesis (continuity and that I already know that the weak derivative exists outside $\{f=0\})$! I have now edited, thanks for pointing it out. | |
| Nov 10, 2023 at 20:19 | history | edited | No-one | CC BY-SA 4.0 |
added 121 characters in body; edited title
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| Nov 10, 2023 at 20:14 | comment | added | leo monsaingeon | A perhaps naive question: what do you mean by $Df\mathbb 1_{f\neq 0}$? Unless I am mistaken $Df$ is the weak derivative, which is a distribution. What is the meaning of multiplying the distribution $Df\in\mathcal D'(B_1)$ by the (posisbly) nonsmooth function $\mathbb 1_{f\neq 0}(x)$? The answer is: nothing, this is not well-defined. Unless of course you already know that $Df$ is actually a function, but then this is actually your claim somehow (if you knew already that $Df\in L^p(B_1)$ for some $p$ you would be done by standard properties of Sobolev functions) | |
| Nov 10, 2023 at 20:13 | comment | added | Iosif Pinelis | How is $B_1$ defined? | |
| Nov 10, 2023 at 20:04 | history | asked | No-one | CC BY-SA 4.0 |