Physical intuition behind no extremum of a function During many of the courses (my background is fluid dynamics), I have seen that if a function $\phi(x,y)$ is smooth and continuous and satisfies a diffusion/Laplace equation of the form:
$$\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$$
over a closed region, $R$, bounded by a curve, $P$, with the boundary value held constant at $\phi_P$ (again, smooth and continuous). How can I physically argue that the function $\phi$ would not have a local maxima or minima in the interior of $R$?
I am able to reason it through numerical methods e.g. finite differences. But what would be the physical explanation behind this?
 A: The Laplace equation $\Delta u=0$ describes equilibrium solutions (or time-independent, steady-state, or whatever terminology you prefer) to the heat equation $u_t = \Delta u$. And you wouldn't expect an equilibrium solution to have a “hotspot” (i.e., a local maximum) somewhere in the middle, would you? That situation would not last, since heat wants to flow from hot to cool regions, to even out the temperature as much as possible.
A: 
What is the physical meaning of the Laplace operator?
A not too complicated road towards insight is to consider the well known Finite Difference
stencil at a uniform rectangular grid with spacing $h$.
$$
\frac{\partial^2 \phi}{\partial x^2} = \frac{(\phi_{i+1,j}-\phi_{i,j})/h-(\phi_{i,j}-\phi_{i-1,j})/h}{h} \\
\frac{\partial^2 \phi}{\partial y^2} = \frac{(\phi_{i,j+1}-\phi_{i,j})/h-(\phi_{i,j}-\phi_{i,j-1})/h}{h} \\
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \quad \Longrightarrow \\
\left(\phi_{i-1,j}-2\phi_{i,j}+\phi_{i+1,j}\right)+\left(\phi_{i,j-1}-2\phi_{i,j}+\phi_{i,j+1}\right) = 0
\\ \Longrightarrow \quad \phi_{i,j} = \frac{1}{4}\left(\phi_{i-1,j}+\phi_{i+1,j}+\phi_{i,j-1}+\phi_{i,j+1}\right)
$$
It is observed that any value of $\phi$ in the Laplace domain is the mean of its surrounding values.
Quite in general: the Laplace operator $\nabla^2$ is a mean value generator. This becomes even more obvious
if the function $\phi(x,y)$ is identified with a temperature distribution in a heat conducting medium, as
exemplified in the answer by Hans Lundmark.
