Physical interpretation of Laplace equation

Can someone please help me, how from a physical point of view the Laplace equation $$ \Delta u=0 $$ can be constructed. I want to know some physical applications from where it can be constructed mathematically.

Some explanation or some suitable reference would be fine. Thanks in advance.

We are dealing here with harmonic functions:

$u$ is called a harmonic function if $\Delta u(x,y) = 0$ for all $(x,y) \in D$ where $D$ is an open domain of $\mathbb{R}^2$.

I am not certain that what I am going to present is answering exactly your question about these functions ; I would like to give you an example issued not from physics but from image processing. This example has helped me to understand the fundamental "mean value property" of harmonic functions which is :

Whatever $P_0(x_0,y_0) \in D$, the value $u(x_0,y_0)$ is the mean of the values taken by $u$ on an infinitesimal vicinity of $P_0$, more precisely on an infinitesimal circle centered in $P_0$. Otherwise said, thinking to the surface with equation $z=u(x,y)$, for a given point $(x,y,z)$ on this kind of circle, having $z \ge u(x_0,y_0)$ is as likely as having $z \le u(x_0,y_0)$.

This property can be given a very natural image by the following discrete analogy. Consider $u$ as a function defined on a square grid with values $u(r,c)$ ($r$ for row, $c$ for column). As the laplacian operator is defined by the sum of the second partial derivatives of $u$ with respect to horizontal and vertical directions:

$$\Delta u:=\partial^2 u/\partial x^2+\partial^2 u/\partial y^2$$

Its classical (see for example here) approximation is, up to a certain non essential factor here) :

$$\Delta u:=[u(r,c-1)-2 u(r,c)+u(r,c+1)]+[u(r-1,c)-2 u(r,c)+u(r+1,c)]\tag{1}$$

Let us gather differently the terms of (1):

$$\Delta u:=u(r,c-1)+u(r,c+1)+u(r-1,c)+u(r+1,c)-4 u(r,c)$$

Therefore, constraining $\Delta u=0$ is equivalent to this property:

$$u(r,c)=\dfrac14(u(r,c-1)+u(r,c+1)+u(r-1,c)+u(r+1,c))$$

which means (as said above) that the center value of the cross is the mean of the values taken in its 4 neighbor sites.