Harmonic function is a function which its Laplacian is equal zero:
$$ {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}} =0$$ Harmonic functions have the mean value property which states that the average value of the function over a ball or sphere is equal to its value at the center.
So my questions are:
1) Could harmonic function be bounded or has extrema ?
2) If harmonic functions couldn't be bounded or have extrema, then could it represent some real "physical" system ?
Thanks
I tried to apply this thought on single variable function as following: $$f''(x)=0$$ $$f'(x)=c_1$$ $$f(x)=c_1x+c_2$$
which is unbounded function, is there more single variable harmonic functions ?
Quoting from https://en.wikipedia.org/wiki/Harmonic_function#Liouville.27s_theorem:
"If $f$ is a harmonic function defined on all of ${\bf R}^n$ which is bounded above or bounded below, then $f$ is constant (compare Liouville's theorem for functions of a complex variable).
"Edward Nelson gave a particularly short proof [2] of this theorem, using the mean value property mentioned above:
"Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume. Since f is bounded, the averages of it over the two balls are arbitrarily close, and so f assumes the same value at any two points."
The reference is to Edward Nelson, A proof of Liouville's theorem, Proceedings of the AMS, 1961.