The problem is related to the proof of the interior $H^2$ regularity theorem from Evans's book Partial Differential Equations (sec 6.3).
In the proof, I am having difficulty in understanding how the following inequality was deduced,
$$|A_2|\leq C \int_U \zeta |D_k^hDu||D_k^hu| + \zeta|D_k^hDu||Du| + \zeta|D_k^hu||Du|dx,$$ where $$ A_2 = \sum_{i,j=1}^n \int_U [a^{ij,h}D_k^hDuD_k^hu2\zeta D\zeta + (D_k^ha^{ij}) D_k^hDu\zeta^2 + (D_k^ha^{ij}) D_k^hu2\zeta D\zeta ]dx $$ and $D_k^h$ is the difference quotient.
I thought triangular inequality for integrals was used here, but I do not understand how terms like $\zeta^2$ disappears.
I am also having difficulty in understanding on how after application of Cauchy's inequality, the Integral space changes from $U$ to $W$?
Here are snapshots to the Theorem and Problem.
Thanks in advance!
I hope the following clarifications help: you want to bound $$ a^{ij,h}D_k^hu_{x_i}D_k^hu2\zeta\zeta_{x_j}+\left(D_k^ha^{ij}\right)u_{x_i}D_k^hu_{x_j}\zeta^2+\left(D_k^ha^{ij}\right)u_{x_i}D_h^ku2\zeta\zeta_{x_j}=I_1+I_2+I_3. $$ I dropped the summation signs and the integrals since they are not playing any role right now. Also, careful with how you wrote $A_2$, since you don't have full gradient but rather partial derivatives on $u$ and $\zeta$ and more importantly you are missing a $u_{x_i}$ in the last two terms.
Now, we now that $a^{ij}\in C^1(U)$ with implies uniform bounds on $a^{ij,h}$ and $D_k^ha^{ij}$ (by Section 5.8.2 in Evans' book). Also, $0\leq\zeta\leq 1$ (which implies $\zeta^2\leq\zeta$) and $|\nabla \zeta|$ is of the order of $\frac{1}{\text{dist}(V,\mathbb{R}^n\setminus W)}$. And of course, $|D_k^hu_{x_i}|\leq|D_k^hDu|$. This readily yields $$ |I_1|\leq C_1\zeta|D_k^hDu||D_k^hu|, \quad |I_2|\leq C_2\zeta|D_k^hDu||Du|,\quad |I_3|\leq C_3\zeta|D_k^hDu||Du| $$ where $C_k,k=1,2,3$ are universal constants. Then you just take $C=\max_{k=1,2,3}\{C_k\}$ sum over $i,j$ and integrate to get the desired bound.
As for the integral space, it's not written in the clearest of ways, but the idea is that you have, say $\int_{U}\zeta|D_k^hDu||D_k^hu|\:dx$. Since $\zeta$ vanishes outside $W$ you can write $$ \int_{U}\zeta|D_k^hDu||D_k^hu|\:dx=\int_{W}\zeta|D_k^hDu||D_k^hu|\:dx\leq \epsilon\int_{W}\zeta^2|D_k^hDu|^2\:dx+\frac{1}{4\epsilon}\int_W|D_k^hu|^2\:dx $$ but $\zeta^2|D_k^hDu|^2\geq 0$ so I can integrate in the larger domain $U$, therefore $$ \int_{U}\zeta|D_k^hDu||D_k^hu|\:dx\leq \epsilon\int_{U}\zeta^2|D_k^hDu|^2\:dx+\frac{1}{4\epsilon}\int_W|D_k^hu|^2\:dx $$ the point being that you want the first integral in the whole domain to absorb it with the left hand side and the second integral in $W$ so you can apply the bound on the difference quotients. Applying this idea you get the desired bounds.
Does all of this make sense?