Generalized Laplace--Beltrami operators Given a smooth surface $M$, we have the well-known Laplace--Beltrami operator which in coordinate-free form is given by $\mathrm{div}_M\,(\mathrm{grad}_M)$. However, we can also consider the more generalized form $\mathrm{div}_M\,(D\,\mathrm{grad}_M)$, where $D$ is a symmetric positive-definite matrix acting on tangent vectors. In fact, this operator appears in many diffusion processes (see e.g., here). Unfortunately, while this form is discussed in the PDE literature, I couldn't find any mathematical treatment of this operator in the differential (Riemannian) geometry community. Does someone know of a textbook about this particular object? 
As a remark, I am aware of the $p$-Laplacian. However, I am interested in the case when $D$ is a general SPD matrix.
 A: To some extent you can think of $D$ as changing the metric with which you are calculating the Laplace-Beltrami operator, which is intuitive if you think of the effect of $D$ on diffusion or heat flow. 
In fact you can always pick a new metric $\tilde g$ on the surface so that your operator is of the form
$$L_D = \nabla_{\tilde g} \cdot e^{s(q)} \nabla_{\tilde g}$$
for a scalar function $s: M\to \mathbb{R}$, with $s$ and $\tilde g$ unique up to shifting $s$ by a constant. I wrote down a calculation for this a bit ago at
https://www.dropbox.com/s/o2b01gm5pjhsnhh/smoothL.pdf?dl=0.
As an operator $L_D$ has more or less all of the properties of the ordinary Laplacian: it is elliptic, self-adjoint, obeys the maximum principle, vanishes on constants, has a discrete negative spectrum, etc. It does differ from the usual $L$ in that applying it to the embedding function no longer gives the mean curvature normal (i.e. for $M$ flat, $L_Df$ no longer vanishes when $f$ is linear).
If you do find references that discuss $L_D$ I'd also be interested in reading about them.
EDIT: The comments correctly point out that my calculation in the PDF is specific to two dimensions. But the above decomposition still holds in arbitrary dimension $n$, with
\begin{align*}
\tilde{g}(v,w) &= (\det D)^{1/n} g(v, D^{-1}w)\\
s &= \frac{1}{n} \log \det D.
\end{align*}
