Is $w =\text{ max}(v_1, v_2)$ subarmonic if $v_1$ and $v_2$ are?

Question

I am studying Perron method to prove the existence of solution to \begin{equation} \Delta u = 0 \quad \text{in } \Omega \\ \ u = g \quad \text{in } \partial \Omega \end{equation} In the proof they define for $v_1$ and $v_2$ two subarmonic functions in $\Omega$ the function $w(x) = \text{max}(v_1(x), v_2(x))$. and use that $w$ is subarmonic ($\Delta w \geq 0$). Is this true?

Answer 1

Yes, that is correct. The easiest way to see this is using the sub-mean value characterisation of subharmonic functions. It's clear that the maximum of two upper semicontinuous (usc) functions is usc. Furthermore, since

$$v_1(x) \le \int_{\partial B} v_1\,d\sigma \le \int_{\partial B} \max\{ v_1, v_2 \}\,d\sigma$$ and similarly for $v_2$, it follows that $$ \max\{v_1(x), v_2(x)\} \le \int_{\partial B} \max\{ v_1, v_2 \}\,d\sigma $$ for every ball $B$ centered at $x$ (where $\sigma$ is normalized surface measure) which implies that $\max \{ v_1, v_2 \}$ is subharmonic.