Timeline for Solution to the differential equation $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle =-\frac{\partial g(X)}{\partial x_{1}} $?
Current License: CC BY-SA 4.0
6 events
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| Jun 28, 2022 at 0:42 | comment | added | Willie Wong | I don't know the most general form of the solution. But, observe that if $k\in \mathbb{C}^n$, then $$ g(X) = \sum_{j = 1}^n k_j x_j $$ solves the PDE iff (I almost got caught out by the double conjugate) $$ \sum_{j = 1}^n k_j^2 = - k_1 $$ This defines a variety in $\mathbb{C}^n$ and shows that there are infinitely solutions just under this ansatz. When $n \geq 3$ there are also multiple solutions with $k_1 = 0$. | |
| Jun 27, 2022 at 22:38 | comment | added | Schemer1 | Thank you @WillieWong. I corrected the statement of the problem to state that $ g(X): \mathbb{C}^{n} \to \mathbb{C} $ is a polynomial in $ n $-indeterminates $ x_{1},\dots,x_{n} $ and that I consider a solution $ g(X) $ to be equivalent to one of the form $ g(X)+c $ where $ c \in \mathbb{C} $. | |
| Jun 27, 2022 at 22:34 | history | edited | Schemer1 | CC BY-SA 4.0 |
The question about uniqueness was flawed. I corrected it. I added constraints on the domain of $ g(X) $
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| Jun 27, 2022 at 19:18 | comment | added | Willie Wong | But the solution shouldn't be unique, since if $g$ is a solution to the PDE, then so is $g + C$ for any constant. Are you prescribing some sort of boundary condition? | |
| Jun 27, 2022 at 19:17 | comment | added | Willie Wong | What are the range and domain of the function $g$? | |
| Jun 27, 2022 at 14:15 | history | asked | Schemer1 | CC BY-SA 4.0 |