Timeline for Integral identity for critical points of the Ginzburg-Landau functional
Current License: CC BY-SA 4.0
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| Feb 13, 2023 at 22:50 | comment | added | Igor Khavkine | @LeoMoos About your updated question. Using the divergence theorem, you can shrink the integration boundary $\partial B$ as long as you don't pass through any of the "bad" points. For instance, $\partial B$ could become say the level set $\rho = \delta$ around one of the $x_j$. Then the factor $\rho^2 = \delta^2$ comes out of the integral. If you make some regularity assumptions about $v$ near its zero set, scaling $\delta \to 0$ might be enough to show that the boundary integral vanishes. Again, this argument depends on what regularity assumptions you are willing to make. | |
| Feb 13, 2023 at 10:52 | comment | added | Igor Khavkine | @LeoMoos You're right, in that case integration by parts is illegal and the identity doesn't hold. Perhaps I misread your question, but I didn't see these irregular cases as your focus. I didn't look up the [CM96] reference to see in which context they were trying to use the integral identity. | |
| Feb 13, 2023 at 8:55 | comment | added | Leo Moos | Right, but the issue is that $\varphi$ is not single-valued, which makes the integration by parts illegal. (Take for example the harmonic function $\log r$, which has $\int_{\partial B_R} \frac{\partial}{\partial \nu} \ln r = 2 \pi \neq \int_{B_R} \mathrm{div} \nabla \ln r$.) | |
| Feb 12, 2023 at 18:49 | history | answered | Igor Khavkine | CC BY-SA 4.0 |