Timeline for How far can a continuous, almost everywhere differentiable function be from being a Sobolev function?
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| Aug 30, 2021 at 13:12 | comment | added | Nate River | @Alexandre Eremenko Yes agreed there! Very nice. | |
| Aug 30, 2021 at 13:07 | comment | added | Alexandre Eremenko | When $n=1$ the answer is certainly 0. Indeed in this case $\nabla f=f'$ a. e., and by your condition $|f'|\leq 1$ a. e., so $f'\in L^1$, but $C_0^\infty$ is dense in $L^1$. | |
| Aug 30, 2021 at 12:39 | comment | added | mlk | My first idea would be using the devil's staircase. If you would have required $f$ and $g$ to have the same boundary values, then you could actually have an arbitrarily large value by using multiples of that. But the way you phrase it, e.g. in 2d you would have to alternate between staircases going up and ramps going down along a circle to achieve something that cannot be approximated by gradients of smooth functions. So while the answer is certainly not 0, I am not sure how close to 1 it can be easily pushed. | |
| Aug 30, 2021 at 9:17 | history | edited | Nate River | CC BY-SA 4.0 |
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| Aug 30, 2021 at 8:54 | history | edited | Nate River | CC BY-SA 4.0 |
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| Aug 30, 2021 at 8:36 | history | asked | Nate River | CC BY-SA 4.0 |