I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this
Assume $u\in H^1(U)$ is a bounded weak solution of
\begin{equation}-\sum_{i,j=1}^n (a^{ij}u_{x_i})_{x_j}=0\end{equation}
Furthermore let $\phi :\mathbb{R} \rightarrow \mathbb{R}$ be a convex and smooth function, and define $w=\phi (u)$. Now the problem is to show that $B[w,v]\leq 0$ for all $v\in H_0^1(U), v\geq 0$.
My idea so far goes like this:
\begin{align} B[w,v]&=\int_U \sum_{i,j=1}^n a^{ij}w_{x_i}v_{x_j}\\ &= \sum_{i,j=1}^n \int_U a^{ij}\phi'(u)u_{x_i}v_{x_j} \\ &=\sum_{i,j=1}^n \left[a^{ij}\phi'(u)u_{x_i}v \right] -\int_U a^{ij}\frac{\partial}{\partial x_j}(\phi'(u)u_{x_i})v \\ &=-\sum_{i,j=1}^n\int_U a^{ij}(\phi''(u)u_{x_i}u_{x_j} + \phi'(u)u_{x_ix_j})v \end{align}
Now I want to say that $\phi''(u)u_{x_i}u_{x_j}\geq0$ because of ellipticity and that $a^{ij}\phi'(u)u_{x_ix_j}=0$ because $u$ solves the pde. Are these arguments correct? And I think the extra term from the integration by parts should be 0, but im not sure why to be honest. Can anyone help?