Timeline for Is $(e^{u}+1)\Delta u+u=0$ the Euler-Lagrange equation of a functional energy?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 25, 2019 at 20:28 | comment | added | Michael Engelhardt | $Li_2 $ is the dilogarithm function. It's still not clear what you're asking. You don't seem to be interested in the Euler-Lagrange equation per se, but somehow a functional which yields specifically $e^u \Delta u $ when you take a functional derivative. It would help if you would be clear about what it is you actually want. | |
| Nov 25, 2019 at 18:53 | comment | added | liding | @MichaelEngelhardt, What is the mean of $Li_{2}(-e^{u})$? | |
| Nov 25, 2019 at 18:52 | comment | added | liding | @MichaelEngelhardt , I met a more complex equation than this. This is a simplified version. The essential difficulty is the energy functional of $e^{u}\Delta u$, without changing the equation. | |
| Nov 25, 2019 at 17:18 | comment | added | AlexArvanitakis | @MichaelEngelhardt agreed; I was just puzzled over the change | |
| Nov 25, 2019 at 14:53 | comment | added | Michael Engelhardt | @AlexArvanitakis Well, if you admit sufficiently fancy functions in your potential ... $V(u)=u^2 /2 - u\ln (1+e^u ) - Li_2 (-e^u ) $ ... | |
| Nov 25, 2019 at 14:06 | comment | added | AlexArvanitakis | @liding you seem to have changed the question so the manipulation I suggested doesn't immediately work | |
| Nov 25, 2019 at 4:48 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
edited title
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| Nov 25, 2019 at 3:25 | comment | added | Michael Engelhardt | If you don't allow for a rewriting of the equation, then you haven't formulated your question accurately. Then it sounds like you're asking specific terms in the Euler-Lagrange equation to each correspond to specific functional derivatives of a functional - as though you were really positing two separate equations rather than one. | |
| Nov 25, 2019 at 1:12 | comment | added | liding | Thank you! If we do not rewritten the equation, is there energy function? | |
| Nov 25, 2019 at 1:09 | history | edited | liding | CC BY-SA 4.0 |
edited title
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| Nov 25, 2019 at 0:31 | comment | added | AlexArvanitakis | Try writing this as $\Delta u+ ue^{-u}=0$ | |
| Nov 25, 2019 at 0:00 | history | asked | liding | CC BY-SA 4.0 |