Let $D$ be an open, bounded, connected region in $\Bbb R^2$, and suppose $u$ is a continuous real-valued function on $\bar{D}$ that is $C^2$ in $D$. Also suppose that $\Delta u>0$ in $D$. I have heard that the maximum principle holds in this case, i.e., $u$ attains its maximum on the boundary of $D$, but how can I show this? For harmonic functions, we may use the mean value property, but I cannot apply the same thing in this case, so I got stuck.
P.S. I don't know so much about harmonic function theory(only a little bit), but I know quite of complex function theory, so it may be good to use complex function theory.
Best Answer
A subharmonic function satisfies an averaging inequality: If $x\in D$ and if $B\subset D$ is a ball centered at $x$, then $|B|^{-1}\int_B u\,dx \ge u(x)$. (Here $|B|$ denotes the area of $B$.) [Given your interest in complex function theory, Ransford's book Potential Theory in the Complex Plane will be of interest.)