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Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).

Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,\cdot>$ be inner product between two vectors.

Consider the PDE $<{g},{\nabla^2(g)}> = <{g},{g}>$.

$(A)$ What is such a class of equation formally called in the literature (it seems to be inner product of a field with its vector Laplacian)?

$(B)$ What are the solutions to the above pde?

$(C)$ What are the solutions of $g$ if $g_{i}(x_{1},\cdots,x_{n}) \in [0,1]$ $\forall i$?

$(D)$ What are the solutions for the special case $g_{i}(x_{1},\cdots,x_{n}) = g_{i}(x_{i})$?

$(E)$ What happens if I replace $\mathbb{C}^{n}$ by:

$(1)$ a torus $\mathbb{C}^{n}/L$ where $L$ is a lattice

$(2)$ a sphere centered at $(\frac{1}{2}, \frac{1}{2}, \cdots,\frac{1}{2})$ and radius $\frac{\sqrt{n}}{2}$.

$(3)$ a cube given by the $0-1$ combinations of the standard bases $e_{i}$ (or its closest smooth approximation) enclosing the above sphere.

$(F)$ Does anything interesting happen as limit $n\rightarrow\infty$.

I feel this is a standard pde. However, since I am not in the math field, I do not know the keywords or whether there are standard solutions? Where should I look for them?

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math.stackexchange.com may be a better venue for this question. Also, if this question 'sticks' here, please specify what sort of functions f and g are (smooth? holomorphic? do you allow poles? –  David Roberts Jun 30 '11 at 6:57
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I thought because of its restriction to different cases overflow might be better. I have modified the question. It also came from a certain research problem. –  user16007 Jun 30 '11 at 7:05
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On a compact manifold, you can integrate your equation by parts and get that $\int (\nabla g, \nabla g) dvol = 0$, which, by the positive-definiteness of the inner product, implies that $g$ is a parallel vector field. Depending on manifold, such a field may not exist. The same is true if you restrict to finite energy solutions on $\mathbb{C}^n$. –  Willie Wong Jun 30 '11 at 9:06
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You can also re-write the equation as $\triangle |g|^2 = |\nabla g|^2$. Then by strong maximum principle, $|g|^2$ cannot attain any local maximum. Why are you interested in this PDE, btw? –  Willie Wong Jun 30 '11 at 9:23
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Note that there does not exist a 'closest' smooth approximation, as a cube can be arbitrarily approximated by a smoothly embedded sphere. –  David Roberts Jun 30 '11 at 9:36
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