Timeline for Existence of unique critical points to second order elliptic PDEs
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2018 at 7:50 | comment | added | Mateusz Kwaśnicki | @VishnuM: I do not quite remember the problem, but I think the simplest way is to make the counter-example explicit: set $u(x,y) = (1-x^2-y^2)(1+2x^2)$ (it has three critical points in the unit disk), $f(x,y)=1$ and $Lu = (2+32x^2+4y^2)^{-1}(u_{xx}+2 u_{yy})$. However, this does not meet your condition about the coefficients having no humps. | |
| Oct 12, 2018 at 4:39 | comment | added | Andre of Astora | How could I make the counterexample you gave rigorous? | |
| Sep 9, 2018 at 15:16 | comment | added | Mateusz Kwaśnicki | @VishnuM: Thanks for the update. I do not think I can contribute more; I suppose one should have a look at the original dissertation to figure out what the author had on mind. | |
| Sep 8, 2018 at 13:01 | comment | added | Andre of Astora | I have added a condition in the question that hopefully solves this problem. | |
| Sep 8, 2018 at 10:02 | comment | added | Andre of Astora | I had a specific PDE in mind, but wanted to prove a more general result since my PDE was quite messy. However an additional condition on $L$ would be that the coefficient functions have nonzero gradient in the $\Omega$. | |
| Sep 8, 2018 at 9:04 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |