Intuition for the laplacian equation

Question

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.

Answer 1

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 $$ f(x,y)=xy $$

Answer 2

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.