Let $B\Subset\Omega$ and let $h$ be the harmonic function whose boundary values are $\max(u_1,u_2)$. The harmonic extensions $(u_i)_B$ satisfy $(u_i)_B\leq h$ by the <maximum principle for harmonic functions>, while subharmonicity gives $u_i\leq(u_i)_B$ in $B$. Hence $\max(u_1,u_2)\leq h$, proving that the maximum of two <subharmonic function>[subharmonic functions] is subharmonic.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 4 / b / i / Solution

 Solution (source code)