Timeline for Weak solution of elliptic differential equation of divergence type
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 12, 2023 at 15:00 | answer | added | Giuseppe Di Fazio | timeline score: 1 | |
| Jan 11, 2023 at 8:27 | comment | added | Dejv | Giorgio Metafune. It seems you are right. Thank you very much. | |
| Jan 10, 2023 at 23:22 | comment | added | Giorgio Metafune | Solve in $W^{2,p}$ and you get a solution in $H^1$ which, by uniquesess in $H^1$, coincides with your original solution. This is what I have in mind. | |
| Jan 10, 2023 at 21:42 | comment | added | Dejv | @Giorgio Metafune, how do you know $\|u\|_{2,p}\le C\|f\|_p$? Maybe u is not in W^{2,p}?? | |
| Jan 10, 2023 at 21:17 | comment | added | Giorgio Metafune | If $p^* \leq 2$ the inequality is obvious (omitting $\|f\|_p$ on the RHS. In the other case, since the coefficients are smooth, you can use the estimate $\|u\|_{2,p} \leq C\|f\|_p$ and then Sobolev embedding $\|Du\|_{p^*} \leq C\|u\|_{2,p}$ to obatain the inequality without $Du$ on the right hand side (when $p^* >2$ the $W^{2,p}$ solution is in $H^1$). | |
| Jan 10, 2023 at 18:07 | history | asked | Dejv | CC BY-SA 4.0 |