Timeline for Interior elliptic regularity in W^{k,1} spaces
Current License: CC BY-SA 3.0
25 events
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| Jan 24, 2018 at 8:27 | 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. | |
| Dec 25, 2017 at 7:47 | 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. | |
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| Aug 27, 2017 at 4:20 | 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. | |
| Jul 28, 2017 at 7:59 | comment | added | Hannes | Concerning your last paragraph, maybe this paper, Section 2 and the references, is helpful (although it originally considers nonlinear elliptic equations). | |
| Jul 28, 2017 at 6:19 | comment | added | Fan Zheng | @MarcNardmann If $\Delta$ is surjective, then there is a uniform estimate. This follows from applying the open mapping theorem (en.wikipedia.org/wiki/…) to $\Delta$. | |
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| Feb 28, 2017 at 2:09 | 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. | |
| Jan 29, 2017 at 1:21 | 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. | |
| Dec 31, 2016 at 3:09 | comment | added | Marc Nardmann | As expected, elliptic regularity (not just the estimate) fails for $p=1$; see Rother 1987. Next week I'll write a detailed answer to my own question(s). | |
| Dec 30, 2016 at 0:39 | history | edited | Marc Nardmann | CC BY-SA 3.0 |
more detailed questions, added remarks
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| Dec 29, 2016 at 2:07 | comment | added | Marc Nardmann | The reference for the failure of the $p=1$ elliptic estimate is D. Ornstein: A non-inequality for differential operators in the $L_1$ norm, doi:10.1007/BF00253928, link. The analogue for $p=\infty$ is K. de Leeuw / H. Mirkil: A priori estimates for differential operators in $L_\infty$ norm, link. | |
| Dec 28, 2016 at 1:05 | comment | added | Marc Nardmann | @M. Renardy: That the elliptic estimates do not hold for p=1 is good to know; a reference would be very helpful. But this information alone does not even answer my first question. It is conceivable (but admittedly seems unlikely) that every solution u lies in $W^{1,1}$ even when no uniform elliptic estimate forces it to do so. It all depends on why/how the estimates fail. | |
| Dec 27, 2016 at 16:01 | answer | added | Bazin | timeline score: 1 | |
| Dec 27, 2016 at 5:55 | comment | added | Deane Yang | Or provide a reference. | |
| Dec 27, 2016 at 5:04 | comment | added | Yemon Choi | @MichaelRenardy Why not write an answer giving or sketching a counterexample, or addressing the follow-up questions? | |
| Dec 27, 2016 at 2:31 | review | Close votes | |||
| Dec 27, 2016 at 9:42 | |||||
| Dec 27, 2016 at 2:18 | comment | added | Michael Renardy | It is well known that elliptic regularity estimates do not work for p=1 and $p=\infty$. | |
| Dec 27, 2016 at 1:33 | history | asked | Marc Nardmann | CC BY-SA 3.0 |