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PDES - from Vector fields whose inner product with their vector Laplacian vanishesequalsnormofthevectorfield

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

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Asimplesecondorder PDE givenbyinnerproductbetweenavectorfield and its restrictiontoatorus,sphereora(closestapproximationtoa)cubevectorlaplacian

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