Assume $u\in H^1(U)$ is a bounded weak solution of $$-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}=0 ~~~in~ U$$
Let $\phi:R\rightarrow R$ be convex and smooth,and set $w=\phi(u)$ Show $w$ is a weak subsolution; that is $$B[w,v]\leq 0$$
for all $v\in H^1_0(U),~v\geq0$ $$B=\int_U \sum_{i,j=1}^na^{ij}v_{x_i}w_{x_j}$$
I used integration by part and eliptic property, actually my problem is that I don't know when $\phi$ is convex $\phi'(u)$ is positive or not?or
$$\int_U \sum_{i,j=1}^na^{ij}u_{x_jx_j}v\phi'(u)dx$$ is positive?