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.
On
Laplace started with Newton's law of gravitation, which says that the force with witch a unit mass body is attracted to another body of mass $M$ is proportional to $M$ and inversely proportional to the square of the distance between them. Instead of the force one can consider the following force potential: $$ u=\gamma\,\frac{M}{r}\,, $$ where $$ r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}. $$ The components of the force field are given by $\nabla u=(u_x,u_y,y_z)$.
If there are several bodies with masses $M_i$ then $u=\sum_i u_i$.
The key idea of Laplace was to consider not the potential $u$ itself but the differential equation it satisfies. I will leave it as an exercise in taking derivatives to find out that for $u_i$ one finds $$ \frac{\partial^2 u_i}{\partial x^2}+\frac{\partial^2 u_i}{\partial y^2}+\frac{\partial^2 u_i}{\partial z^2}=0. $$ Since $u=\sum_i u_i$ one ends up with $$ u_{xx}+u_{yy}+u_{zz}=\Delta u=0, $$ which is the Laplace equation.
Remark. It is interesting to note that in most modern textbooks the Laplace equation is motivated by either introducing first Maxwell equation, or diffusion equation, and then consider the stationary distributions whereas historically I can say that it was Maxwell who was motivated by Laplace's approach.
Ref. Godunov, Equations of mathematical physics, 1979 (I am not sure whether an English translation exists)
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.