We say $u\in\operatorname{W}^{1,2}(\Omega)$ is the weak subsolution to the elliptic PDE \begin{equation}\label{1} -D_i(a_{ij}D_j u)=0\quad \text{in}\,\, \Omega\,, \end{equation} if there holds $$\int_\Omega a_{ij}D_iuD_j\varphi\,{\operatorname d}x\leq0\,,\quad\forall\,\varphi\in \operatorname{W}_0^{1,2}(\Omega)\,,\varphi\geq0.$$ Let $u_1,u_2$ be two weak subsolutions, set $v:=\max\{u_1,u_2\}.$ The question is $v$ still a weak subsolution?
My attempt
For any $\varphi\in {C}_0^{\infty}(\Omega),\varphi\geq0,$ there is $$ \begin{aligned}\int_\Omega a_{ij}D_i v D_j\varphi\, {\operatorname d} x&=\int_{\{u_1<u_2\}} a_{ij}D_iu_2 D_j\varphi \,{\operatorname d} x+\int_{\{u_1\geq u_2\}} a_{ij}D_i u_1 D_j \varphi \,{\operatorname d} x\\&=\int_{\Omega} a_{ij}D_i u_2D_j\varphi\cdot 1_{\{u_1<u_2\}}\, {\operatorname d}x+ \int_{\Omega} a_{ij}D_i u_1D_j\varphi\cdot 1_{\{u_1\geq u_2\}}\, {\operatorname d}x\,. \end{aligned}$$ But I don’t know if $\varphi1_{\{u_1<u_2\}}\,,\varphi1_{\{u_1\geq u_2\}}\in \operatorname{W}_0^{1,2}(\Omega),$ since the character function may be so wired that the weak derivative does not exist.
I want to know is this proposition right? If right, how to prove it? Looking forward to your answer, thanks in advance.
$1_{u1<u2}$ is at least a measurable function. We can use continuous function to approach it, particularly, polynomial function. The Dominated convergence theorem thus guarantees the reasonableness .