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?
Best Answer
Im gonna use here the Einstein summation convention and $\frac{\partial v}{\partial x_i}=v_i$. Also, Im gonna assume ellipticity on $(a_{ij})$ and that $w\in H^1$, because that only with your hypothesis this is not always true. Note that $(v\in C_0^1(U),\ v\geq 0)$\begin{eqnarray} B[w,v] &=& \int_U a_{ij}v_iw_j \nonumber \\ &=& \int_U a_{ij}\phi'(u)u_iv_j \nonumber \\ &=& \int_U a_{ij}u_i(\phi'(u)v)_j-\int_U(a_{ij}u_iu_j)v\phi''(u) \end{eqnarray}
To conclude, you have to show that $\phi'(u)v\in H_0^1$ and $-(a_{ij}u_iu_j)v\phi''(u)\leq 0$, then you use the fact that $C_0^1$ is dense in $H_0^1$. Can you do this?