Maximum and minimum principles for plane harmonic functions

Suppose $u$ has an interior local maximum at $a\in \mathrm{\Omega }$, with value $M=u(a)$. By [L1], some disc $D\subseteq \mathrm{\Omega }$ and some holomorphic $F$ on $D$ satisfy $\mathrm{Re}F=u$ on $D$; the real part of $F$ has a local maximum at $a$, so [L2] makes $F$ constant on $D$, and therefore $u\equiv M$ on $D$.

Let $S:=\{z\in \mathrm{\Omega }:u\equiv M\text{ on some neighbourhood of }z\}$. Step 1.1 gives $a\in S$, and $S$ is open by definition. If $b\in \bar{S}$, choose a disc $D_b\subseteq \mathrm{\Omega }$ and a holomorphic $G$ on $D_b$ with $\mathrm{Re}G=u$ there by [L1]; since $S\cap D_b$ contains a nonempty open set on which $\mathrm{Re}G=M$, both $\mathrm{Re}(G-M)$ and $\mathrm{Re}(M-G)$ have local maxima there, so [L2] makes $G$ constant on $D_b$, hence $u\equiv M$ on $D_b$ and $b\in S$. Thus $S$ is closed in $\mathrm{\Omega }$.

Because $\mathrm{\Omega }$ is connected by [L3], the nonempty set $S$ that is open and closed in $\mathrm{\Omega }$ must equal $\mathrm{\Omega }$. Thus a local interior maximum forces $u$ to be constant on $\mathrm{\Omega }$. Applying the same argument to $-u$, which is harmonic because $(-u{)}_{xx}+(-u{)}_{yy}=-(u_{xx}+u_{yy})=0$, gives the local minimum statement as well.

Now assume $\mathrm{\Omega }$ is bounded and $u$ is continuous on $\bar{\mathrm{\Omega }}$. The closure $\bar{\mathrm{\Omega }}$ is nonempty, closed and bounded in ${\mathbb{R}}^2$, hence compact by [L4], so [L5] says that the continuous extension of $u$ attains a maximum and a minimum there. If either extremum were attained at an interior point and $u$ were nonconstant, step 3.1 would force $u$ to be constant. Therefore both extremal values are realized on $\mathrm{\partial }\mathrm{\Omega }$, and the displayed equalities follow.