# Could a non-constant harmonic function be bounded or has extrema ? Could it exist in the physical world?

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 ?

"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).
• Mathematics provides models for physics. An unbounded harmonic function might be a useful model for a bounded physical system. Also, note the requirement in the theorem about the function being harmonic on all of ${\bf R}^n$. Is ${\bf R}^n$ a physical object? a perfect model for the Universe? – Gerry Myerson Jul 17 '16 at 10:37