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I studied about the Laplacian at https://www.youtube.com/watch?v=EW08rD-GFh0 and I understand that it can be thought of as the second order derivative test where the value $ \triangle f(x,y) $ will be high at a local minima and low at a local maxima.

But what does it mean to solve for the laplacian i.e. $ \triangle f(x,y) = 0$ ? I have extremely little background in physics so it would be nice if someone, bearing that it mind, explain what it means to solve for the laplacian. I come from an image processing background and have only used the laplacian operator for edge detection so "solving" it sounds completely alien to me.

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  • $\begingroup$ When you say solve for the laplacian i.e., $\triangle f(x,y) = 0$, do you mean solve $\triangle f(x,y) = 0$ for $f$? When I read solve for I think find the unknown quantity. $\endgroup$
    – Matthew Leingang
    Feb 24, 2018 at 22:12
  • $\begingroup$ Suppose we video taped water flowing on the output graph in the video. Could we perhaps view the vector field as representing the direction and speed of water running uphill when the tape is played in reverse? This could explain why sinks appear at local maxima, and sources appear at local minima, instead of vice-versa. Could some intuition in this vein be constructed for the laplacian of every point on the output graph, perhaps with some special meaning when it equals 0? $\endgroup$
    – user10478
    Nov 11, 2018 at 20:17
  • $\begingroup$ Unfortunately I don't have enough reputation to post this as a comment, but Giacomo Cavallo's answer to this Quora question proved too good and I just have to share it. You will find there a high-school intuitive proof of the meaning of the Laplacian. Highly recommended. You don't even need to know what a derivative is and at the end you will know what the Laplacian is. $\endgroup$
    – cyau
    Dec 19, 2020 at 15:28

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I think a good fact to remember, and you can see it in that video, a function $f$ satisfying $$ \Delta f(x,y)=0 $$ has no local minima or maxima. One way to visualize this is that if you put a ball anywhere on the surface defined by $f$, it will roll off.

I encourage you to plot a few examples of such $f$, like $$ f(x,y)=x^2-y^2 $$ or $$ f(x,y)=xy $$

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  • $\begingroup$ Good answer, but I take issue with “boring.” Harmonic functions are far from boring! $\endgroup$
    – Matthew Leingang
    Feb 24, 2018 at 22:09
  • $\begingroup$ I guess this is silly, Riemannian geometers make their careers studying harmonic maps, i will remove my comment :) $\endgroup$
    – operatorerror
    Feb 24, 2018 at 22:14
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You can find several interpretations of the Laplacian in this related question. (One of the answers there relates it to the image-processing application with which you’re familiar.) With the interpretation of $\Delta$ as an averaging operator, the equation $\Delta u=0$ basically says that the value of $u$ at each point is equal to its average value over nearby points.

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