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I was wondering if any of you had ever encountered operators on $L^2(\mathbb{R}^d)$ of the form $$ - \nabla \cdot A(x)\nabla $$ where $A(x)$ is some matrix field (viewed as $L^2(\mathbb{R}^{d^2}$)), and if so where can I find some literature about it ?

More precisely, I'm trying to get a lower bound, $i.e$ hopefully some function $h(x)$ such that $- \nabla \cdot A(x)\nabla \geq h(x)$ (in the sense of quadratic forms), when $A(x) = (2|x| - |x|^{-1})xx^T$.

Thanks a lot!

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Indeed, such operators are often encountered in differential geometry, where a Laplace operator on a Riemannian manifold with metric $g$ is given by $$ \Delta f = \frac{1}{\sqrt{\det g}} \nabla\cdot(g^{-1}\sqrt{\det g}\nabla f)$$ If matrix $g$ isn't positive definite, but it's non-degenerate, it's a slightly more complicated case of pseudo-Riemannian manifolds.

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