Timeline for Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Current License: CC BY-SA 4.0
31 events
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| Jun 21 at 14:38 | comment | added | Nate River | @PietroMajer Oh sorry, I did not get notified for this reply somehow. I will email you soon then! There is still more work to be done to prove the corollary I think, so it would be great to have some help. 😅 | |
| Jun 19 at 14:16 | comment | added | Pietro Majer | BTW: I happen to discover that Prop 3 was proven by Leonida Tonelli, the father of modern Calculus of Variation! He even allows countably many non-differentiability points, always assuming v continuous and v' ≤ 0 a.e. | |
| Jun 19 at 14:08 | comment | added | Pietro Majer | Hi, I think an acknowledgment to MathOverflow would be nice (that's why it has been created). If you like the idea of a joint paper, why not, but perhaps I should give some more contribution. We may talk about this by mail. | |
| Jun 19 at 14:00 | comment | added | Nate River | @PietroMajer Hi, sorry to tag you on an old post. I believe I have proven the result linked here for all integer $0 \leq k < n$. I plan to write it up and submit it somewhere. As the proof rests heavily on the case $(k, n) = (0, 1)$ that you proved, would you like to be a coauthor on the paper with me? Or otherwise, how can I acknowledge your contribution? | |
| Jun 9 at 17:33 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Jun 4 at 18:21 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 31 at 8:06 | comment | added | Nate River | Glad you enjoyed the problem! By the way, the sharper form of the result you state in the Remark actually proves the result here, a result for which I was looking for an elementary proof for awhile. The linked paper is pretty heavy and I cannot gain much intuition from it. So thank you too! | |
| May 31 at 5:37 | comment | added | Pietro Majer | It is a very interesting question and it made me think about some old facts which are still a source of curiosity, so it's me who has to thank you :) | |
| May 31 at 5:32 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 30 at 8:33 | history | bounty ended | Nate River | ||
| May 30 at 8:31 | vote | accept | Nate River | ||
| May 30 at 8:31 | comment | added | Nate River | @PietroMajer No worries at all, I appreciate you spending so much effort on my problem. | |
| May 30 at 6:05 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 30 at 5:45 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 30 at 5:38 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 30 at 3:05 | comment | added | Pietro Majer | (my apologies for being late --I was taken by a sort of generalization frenzy and the short argument for the above Prop 3 has become a bit larger... I'll be back soon with the proof :) | |
| May 24 at 18:45 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 24 at 14:48 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 24 at 3:41 | comment | added | Nate River | @PietroMajer Take your time! | |
| May 24 at 3:20 | comment | added | Pietro Majer | As soon as the sun rises | |
| May 24 at 3:16 | comment | added | Pietro Majer | I have a short argument via Vitali covering to show the above MVT , I'll add it | |
| May 24 at 2:22 | comment | added | Nate River | @WillieWong Ahh, I see the reasoning now, thanks. | |
| May 24 at 1:57 | comment | added | Willie Wong | Incidentally, Pietro's comment was exactly my first thought when I saw this question last night, and I haven't been able to prove that claim. So I would be very interested to see a proof if that is true. | |
| May 24 at 1:51 | comment | added | Willie Wong | @NateRiver: Pietro's "claim" that the esssup |f'| = sup |f'|, if true, would imply that the scenario you envision is impossible, as it would imply that $f_n'$ is Cauchy under the uniform norm. Conversely, if you have a specific example of a sequence $f_n$ satisfying your hypotheses and a point $x$ at which $(f_n'(x))$ is not Cauchy, then it would disprove Pietro's conjecture. | |
| May 24 at 0:34 | comment | added | Nate River | @PietroMajer Hm that claim sounds plausible, but I don’t see how this implies $f’_n$ can converge to anything uniformly if there are some points $x$ at which $\{f’_n (x)\}$ is not Cauchy. | |
| May 23 at 22:49 | comment | added | Pietro Majer | I think (I’ll check) for an everywhere differentiable function f on [a,b] one has supess |f’| =sup |f’| (this is of course false if f is only continuous and differentiable a.e.). If so, up to modifying $g$ on a null set, condition (2) can be replaced by “ $f’_n\to g $ uniformly.” | |
| May 23 at 21:33 | comment | added | Nate River | @PietroMajer What if $f’_n$ is not Cauchy on the null set of non convergent points? | |
| May 23 at 21:15 | comment | added | Iosif Pinelis | In Theorem (8.6.3), $f_n'$ is required to converge locally uniformly, but that does not seem to easily follow from the conditions in the OP. In particular, the second condition in the OP only provides the uniform convergence of $f_n'$ on a set of full measure. | |
| May 23 at 17:13 | comment | added | Pietro Majer | It's a fantastic book from the roaring 60's | |
| May 23 at 17:01 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| May 23 at 16:45 | history | answered | Pietro Majer | CC BY-SA 4.0 |