Timeline for Estimates for an elliptic PDE
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2020 at 9:44 | comment | added | Jakob Möller | @GiorgioMetafune Thanks! | |
| Nov 30, 2020 at 22:48 | comment | added | Giorgio Metafune | Well, the apriori bound must be turned into an existence proof. This should work (details are omitted): approximate $u$ by $u_\epsilon$, smooth and with compact support in the $H^1$ norm and solve $\Delta A_\epsilon-A_\epsilon (u^2_\epsilon+\epsilon)=u_\epsilon\nabla u_\epsilon$ using Lax-Milgram to get existence in $H^1$. Then the above computations give the boundedness of $\|\nabla A_\epsilon\|_2 $ and $\|A_\epsilon\|_6$. Lettting $\epsilon \to 0$ you get existence in $L^6$ (with gradient in $L^2$) | |
| Nov 30, 2020 at 19:43 | comment | added | Jakob Möller | A little question (maybe trivial): Why can we perform integration by parts? | |
| Nov 30, 2020 at 19:35 | vote | accept | Jakob Möller | ||
| Nov 30, 2020 at 16:36 | comment | added | Leo Moos | Oops, thank you Giorgio! | |
| Nov 30, 2020 at 15:46 | comment | added | Giorgio Metafune | First estimate $\|Au\|_2$ neglecting $\nabla A$ which has the same sign. Then insert this estimate in the RHS. | |
| Nov 30, 2020 at 15:40 | comment | added | Leo Moos | I'm being a bit obtuse, but how does the integral give a bound $\lVert \nabla A \rVert_2$? The same trick as for $\lVert Au \rVert_2$ (dividing both sides) isn't available. | |
| Nov 30, 2020 at 15:37 | history | edited | Giorgio Metafune | CC BY-SA 4.0 |
added 1 character in body
|
| Nov 30, 2020 at 15:26 | history | answered | Giorgio Metafune | CC BY-SA 4.0 |