Timeline for Lebesgue differentiation theorem at boundary points for Sobolev traces
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 27, 2023 at 17:05 | comment | added | leo monsaingeon | Yes, I believe it should work out fine, but I'm waaaay too lazy for that. Anyway, thank you again. | |
| Jan 27, 2023 at 16:23 | comment | added | Willie Wong | But for your previous comment: yes, you'd need a little bit of gymnastics to account for the contributions from the two halves of the ball using extensions. For flat (or sufficiently smooth) boundaries basically the same reflection formula from Evans' PDE book should work out okay. | |
| Jan 27, 2023 at 16:16 | comment | added | Willie Wong | No prob; had I had my copy of Evans and Gariepy at hand, I would've posted the same answer as Nate. | |
| Jan 27, 2023 at 10:53 | comment | added | leo monsaingeon | Thank you Willie, I rather accepted Nate River's answer because the statement there is neat and exactly what I need. I still appreciate you taking the time. | |
| Jan 27, 2023 at 10:44 | comment | added | leo monsaingeon | Huum, I'm not too sure this is really what I need: how does the boundary trace come into play here? of course I may extend $u\in W^{1,p}(\Omega)$ to $\bar u\in W^{1,p}(R^n)$, but then it is absolutely not clear to me that $tr(u)=\bar u|_{\partial\Omega}$ (where the restriction is well-defined thanks to prop. 7.1 in question). What's more, I need to compare the average of $\bar u$ over balls with the average of $u$ over partial balls (intersected with $\Omega$). And again, I don't see how to do this, unless some particular trick (reflexion?) is implemented. Am I missing something? | |
| Jan 26, 2023 at 16:33 | comment | added | Willie Wong | I would also check Evans and Gariepy "Measure Theory and Fine Properties of Functions"; but my copy is not in my office right now (on my desk at home). This seems like something that may be treated in it. | |
| Jan 26, 2023 at 15:57 | comment | added | Willie Wong | The statement becomes then a $W^{1,p}$ function has a representative that satisfies the Lebsegue differentiation theorem condition outside of a set of $(n-1)$-dimensional Hausdorff measure 0. | |
| Jan 26, 2023 at 15:54 | comment | added | Willie Wong | Sorry, I mean proposition 7.1 in section 7.3. And yes, it is exactly what your question is asking. The notion of "strictly defined" is exactly "the formula in the Lebesgue differentiation theorem holds". $G_\alpha$ is the Bessel potensial, so $G_1*f$ given $f\in L^p$ is a $W^{1,p}$ function. In your case you should set $j = 0$ and $d = n-1$. | |
| Jan 26, 2023 at 15:52 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 15 characters in body
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| Jan 26, 2023 at 15:37 | comment | added | leo monsaingeon | Thank you Willie for your input. However, before spending some time trying to understand and navigating the heavy notations therein: I don't see a proposition 7.3 there, only proposition 7.1 (which doesn't see to be related to my question). Is this really the statement I should try to understand? | |
| Jan 26, 2023 at 15:23 | history | answered | Willie Wong | CC BY-SA 4.0 |