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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.

nearly_phd
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  • It is not clear what that $D$ is. I suppose that your surface has to be in $\mathbb R^3$ in order that $D$ is defined (treat of $grad f$ as an vector in $\mathbb R^3$. But then $div_M$ is not so well defined. –  May 21 '15 at 04:57
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    Thank you. I have edited my question so that all of the operators are intrinsic. Does it make sense now? – nearly_phd May 21 '15 at 06:08
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    http://www.math.cornell.edu/~lsc/uniformly-jdg.pdf (Is that what you are looking for?) –  May 21 '15 at 06:23
  • Thank you @John. Indeed, it seems relevant and for sure will serve (at least) as a starting point. – nearly_phd May 21 '15 at 08:31
  • If the answer below is correct, which means that all operator of the form you wrote all given by $L_D$, and is called a drift Laplacian. Quite a lot of studies can been done on this operator (quite recently) –  May 21 '15 at 11:12

1 Answers1

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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*}

user7530
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  • Um.... quite interesting. From the proof it seems that it works only for dimension $=2$. Did you try to generalize that? Also, that $L_D$ is in general called a drift Laplacian, in case anyone are interested. –  May 21 '15 at 07:29
  • @John Yes I think it work in any dimension (I was interested in this for a project involving elastic membranes so was focused mostly on the $n=2$ case). – user7530 May 21 '15 at 07:38
  • One more question: In your note, is $g_1(X, Y) = g(M^{-1}X, M^{-1} Y)$? (Up to a positive constant)? –  May 21 '15 at 10:48
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    No it would be $\sqrt{\det M} g(X, M^{-1}Y)$. – user7530 May 21 '15 at 15:13
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    @John I've worked out the correct formulas for arbitrary dimension and edited them into my post. – user7530 May 21 '15 at 21:11
  • @user7530 if it is still relevant, I find these notes to be extremely relevant and helpful. – nearly_phd May 31 '15 at 07:32