Suppose I have a general elliptic operator $$Lu = \nabla \cdot M\nabla u$$ on $\mathbb{R}^2$, where $M(x,y)$ is a symmetric positive-definite $2\times 2$ matrix at every point $(x,y)$ and varies sufficiently smoothly over the plane.
Does there always exist a metric $g$ on the plane such that $Lu = \Delta_g u$, the Laplace-Beltrami operator with respect to the metric $g$?
Even for the case where $M = f(x,y)I$ is everywhere a multiple of the identity matrix, it's not obvious to me. Willie Wong, in his comment here, suggests looking at $g$ conformal to the Euclidean metric, but as far as I can tell this doesn't actually help: all terms depending on the conformal factor inside the divergence cancel.
