# Equivalent definition of subharmonic functions.

I know that a function $u \in C^2(\Omega)$ is subharmonic on $\Omega$ if $\Delta u(x) \ge 0$ for all $x \in \Omega$. But I have just acquainted with another definition of subharmonic functions which is given as follows $:$

"A function $u \in C^0 (\Omega)$ is subharmonic on $\Omega$ if for any $h \in C^2(\Omega) \cup C^0 (\bar \Omega)$ with $\Delta h = 0$ on $\Omega$ and for every ball $B \subset \subset \Omega$, $u \le h$ on $\partial B \implies u \le h$ on $B$."

To see that previous definition implies the later is immediate. For this we just need to apply Strong Maximum Principle on the function $w:=u-h$ by observing that $w \le 0$ on $\partial B$ and $h$ is superharmonic on $\Omega$ since it is harmonic on $\Omega$. But how can I do the converse part? Please give me some suggestions.

The "converse part", as you say, is false for regularity reasons. The function $u(x)=|x|$ is subharmonic on $\mathbb R$ in the sense of the second definition, but it is not in the sense of the first, because it is not differentiable.
• I think in order to hold the converse we need to restrict $u$ in $C^2 (\Omega)$ in the second definition. – Arnab Chattopadhyay. Apr 23 '18 at 8:48
• @GiuseppeNegro If $u$ is subharmonic then $u(x) \leq \fint u$. But the converse is obviously not true. – user586752 Aug 29 '18 at 9:45
• @OhDaeSu: Does \fint u mean $\frac{1}{|B|}\int_B u$? In that case, it is true that $u(x)\le \frac{1}{|B|}\int_B u$ for all balls centered at $x$ if and only if $u$ is subharmonic in some sense. I am not fresh in these things but they are in the linked book, I'm sure – Giuseppe Negro Aug 29 '18 at 9:57
• @OhDaeSu: Agreed, but the inequality must be satisfied for all points and for all balls centered at the point. In that example, this fails at $|x|\ge 5$. – Giuseppe Negro Aug 29 '18 at 10:18