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?


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.
