Timeline for Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2021 at 18:48 | answer | added | Willie Wong | timeline score: 1 | |
| Jan 5, 2021 at 18:09 | vote | accept | BigbearZzz | ||
| Jan 5, 2021 at 17:03 | answer | added | mlk | timeline score: 4 | |
| Jan 5, 2021 at 17:03 | comment | added | mlk | @WillieWong Well, since you asked nicely, though it isn't the precise statement. | |
| Jan 5, 2021 at 15:00 | comment | added | BigbearZzz | I agree. If he posts an answer based on this comment I'll accept it. | |
| Jan 5, 2021 at 14:48 | comment | added | Willie Wong | @mlk since the OP tagged reference request, your comment is probably good as an answer. | |
| Jan 5, 2021 at 13:54 | comment | added | BigbearZzz | @mlk Thank you, I'll have a look at that and perhaps just prove it myself. The formula in my question just seem so natural hence it surprised me a bit to learn that it's not already well-documented in the literatures. | |
| Jan 5, 2021 at 13:21 | comment | added | mlk | If you look in Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., you see that they define the trace on BV (which includes $W^{1,1}$) for Lipschitz domains in a similar fashion and prove a similar convergence. You should be able to adapt this proof. | |
| Jan 5, 2021 at 13:20 | comment | added | Giorgio Metafune | This is true with some regularity on $\partial \Omega$ and follows for $u \in W^{1,1}$ from the half-space case. If $Q$ is a cube in the $x$ variable, then $\int_Q |u(x,y_2)-u(x,y_1)| dx \le \int_{Q\times (y_1,y_2)}|u_y(x,y)|dxdy$ and the function $y \to \int_Q u(x,y)dx$ is continuous. | |
| Jan 5, 2021 at 10:37 | history | edited | BigbearZzz | CC BY-SA 4.0 |
added 140 characters in body
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| Jan 5, 2021 at 2:29 | history | edited | BigbearZzz |
edited tags
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| Jan 4, 2021 at 19:22 | history | asked | BigbearZzz | CC BY-SA 4.0 |