Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$: \begin{equation} L\rho = -\nabla \cdot \left(f\nabla \rho\right) \end{equation} where $f$ is some positive function. Under what circumstances can $f$ and the principal eigenvector $\rho^+$ of $L$ have the same level-surfaces? i.e. \begin{equation} \nabla \rho^+ = g \nabla f \end{equation} for some function $g$.

Can this only occur whenever $U$ is a ball centered on some point $\mathbf{x}\in \mathbb{R}^n$ while $f$ is invariant under rotations around this point?

This question is related to https://math.stackexchange.com/questions/1790088/static-self-gravitating-gas-spherically-symmetric


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