PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

2011-06-30T06:45:52Z

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?

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field - MathOverflow

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field