Timeline for Regularity of solutions to $-\Delta u = \operatorname{div} F$, $F\in L^1$
Current License: CC BY-SA 3.0
16 events
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| Feb 13 at 16:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 1, 2018 at 17:43 | history | edited | Martin Sleziak |
added top-level tag; https://meta.mathoverflow.net/questions/1457/why-are-mo-tags-formatted-as-they-are
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| May 1, 2018 at 17:42 | comment | added | Michael Hardy | When you use \operatorname{div} then the spacing to the right and left depends on the context, so that, for example, in $a\operatorname{div} b$ you see more space to the right of div than in $a\operatorname{div}(b),$ and you don't need to add space manually. I edited this question accordingly. | |
| May 1, 2018 at 17:41 | history | edited | Michael Hardy | CC BY-SA 3.0 |
added 13 characters in body; edited title
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| Apr 30, 2018 at 17:54 | answer | added | Bazin | timeline score: -1 | |
| Apr 30, 2018 at 16:56 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
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| Apr 30, 2018 at 11:52 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
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| Apr 30, 2018 at 10:33 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
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| Apr 30, 2018 at 8:55 | comment | added | Hannes | There is a correspondence between functionals acting on $W^{1,p'}$ and $\mathop{div} F$ for $L^p$ vector fields for $1 \leq p < \infty$, so you are essentially looking at right hand sides from $W^{-1,p}$ there (Maz'ya: Sobolev Spaces, Ch.1.1.14). For these, maximal elliptic regularity would give solutions in $W^{1,p}$, at least for $p>1$. (I think the $\mathop{div} F$ representation is classically used in the Ladyzhenskaja book(s).) I'd also be interested about $p=1$, though.. | |
| Apr 30, 2018 at 3:37 | comment | added | Piotr Hajlasz | @DeaneYang $p=2$ is straightforward and you get a solution in $W_0^{1,2}$. I am not sure what are the optimal results for other values of $p$. It is probably well known and I am just an ignorant. | |
| Apr 30, 2018 at 3:27 | comment | added | Deane Yang | Oops. I assumed that the $p >1$ case was more straightforward using, say, elliptic regularity estimates for Sobolev spaces. Surely that’s true at least for $p=2$? | |
| Apr 30, 2018 at 3:03 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
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| Apr 30, 2018 at 3:01 | comment | added | Piotr Hajlasz | @DeaneYang Good point. I will modify my question. | |
| Apr 30, 2018 at 2:50 | comment | added | Deane Yang | And presumably you also know the answer when $F\in L^p$, $p>1$? | |
| Apr 30, 2018 at 2:43 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
edited title
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| Apr 30, 2018 at 2:36 | history | asked | Piotr Hajlasz | CC BY-SA 3.0 |