I have always visualized subharmonic functions as Ahlfors' Complex Analysis thaught me to do: in one dimension lines are harmonic functions and "convex" functions are subharmonic.
I actually just tried to generalized this to multidimensional domains but I find it a bit harder. My question is:
Can Harnack inequality help to see them better? Any other advice or analogies as Ahlfors' one?
Well, the broad intuition is still correct: convex functions are still subharmonic, concave functions are superharmonic, etc. It's just that you can now have functions which are subharmonic but not convex. Morally speaking, you can think thusly: the Hessian of a convex function is always positive definite, while for a subharmonic function it is only the trace that must be non-negative. For example $2x^2 - y^2$ is subharmonic but not convex - yet nonetheless, at each point the curve "convexwards" along the $x$ direction is stronger than the curve "concavewards" in the $y$ direction.
A harmonic function is one where the value at any interior point is comparable to any other interior point (this is what Harnack's inequality says) and has no interior minima or maxima (this is what the maximum principle says), so something "flat-ish" is still not a bad way of visualizing it for certain purposes, or the curvatures "concavewards" are balanced out by the curvatures "convexwards," morally speaking.
