Timeline for $W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2022 at 22:44 | vote | accept | Tian LAN | ||
| Dec 6, 2022 at 21:26 | answer | added | Hannes | timeline score: 4 | |
| Dec 6, 2022 at 13:32 | comment | added | Tian LAN | @RomainGicquaud I don't think there exists such a strong elliptic regularity. Remember we only assume $a_{ij}$ are uniformly elliptic without smoothness or even continuity assumption | |
| Dec 6, 2022 at 13:18 | comment | added | Romain Gicquaud | If you are in a situation where you can apply standard elliptic regularity, yes. You will have $v \in W^{2, q}$ for any $q = \frac{1}{p} - \frac{1}{n}$. So, by the Sobolev embedding theorem, you have $Dv \in L^r$ with $\frac{1}{r} = \frac{1}{p} - \frac{2}{n}$. If $p$ is close enough to $2$, you have $r > 2$ which is exactly what you want. | |
| Dec 6, 2022 at 13:03 | comment | added | Tian LAN | @RomainGicquaud To make $v$ a test function, we need $\int |Du|\,|Dv| $ to be well-defined. Is it possible to show $v\in W^{1,p}$ for some $p>2$? | |
| Dec 6, 2022 at 10:20 | comment | added | Romain Gicquaud | If you are capable of solving the problem $\partial_i (a_{ij} \partial_j v) = f$ with $v$ sufficiently regular, you can solve the equation $\partial_i (a_{ij} \partial_j v) = u$ and use this $v$ as a test function. This gives you $\int u^2 = 0$ so $u \equiv 0$. See arxiv.org/pdf/2210.09823.pdf proposition 2.12 | |
| Dec 6, 2022 at 1:02 | history | edited | LSpice | CC BY-SA 4.0 |
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| Dec 5, 2022 at 23:50 | history | edited | YCor |
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| Dec 5, 2022 at 23:45 | history | edited | Tian LAN | CC BY-SA 4.0 |
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| Dec 5, 2022 at 23:34 | comment | added | Tian LAN | @GiorgioMetafune because the Laplacian is so special that if $\Delta u=0$ in the distributional sense, then it must be a harmonic function. So for a fixed $x\in B$, we can apply the Poisson integral formula in $B_r$ for $r<1$, and then let $r\rightarrow 1$. Use the trace of $u$ on $\partial B_r$ converges to the trace on $\partial B$ in $L^1$. | |
| Dec 5, 2022 at 21:27 | comment | added | Giorgio Metafune | How you prove it for the Laplacian? Usually Poisson formula requires classical solutions. | |
| Dec 5, 2022 at 18:13 | comment | added | Tian LAN | @DanielShapero Of course we can, but how to use that to prove $u=0$? | |
| Dec 5, 2022 at 17:06 | comment | added | Daniel Shapero | Can you assume not and then approximate $u$ by functions in $W_0^{1, 2}$? | |
| Dec 5, 2022 at 14:28 | history | edited | Tian LAN | CC BY-SA 4.0 |
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| Dec 5, 2022 at 12:56 | comment | added | CommunityBot | Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. | |
| Dec 5, 2022 at 5:52 | review | Close votes | |||
| Dec 10, 2022 at 3:03 | |||||
| Dec 4, 2022 at 23:43 | history | edited | Tian LAN | CC BY-SA 4.0 |
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| S Dec 4, 2022 at 23:41 | review | First questions | |||
| Dec 5, 2022 at 12:56 | |||||
| S Dec 4, 2022 at 23:41 | history | asked | Tian LAN | CC BY-SA 4.0 |