Suppose that $u\in H^{1}(U)$ is the unique weak solution of the boundary value problem
$-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}+2u=f$ in $U$
$u=g$ on $\partial U$, where $g$ be the trace on $\partial U$ of an $H^1(U)$ function $w$.
Prove that i) $\|u\|_{H^1{(U)}}\leq C(\|f\|_{L^2{(U)}}+\|w\|_{H^1{(U)}}$.
ii)$\|u\|_{H^2{(U)}}\leq C(\|f\|_{L^2{(U)}}+\|w\|_{H^2{(U)}}$, where $w,u \in {H^2{(U)}}$.
Can anyone suggest to me some hints for this question?
I am going to make some small additional assumptions. We assume that the problem is elliptic (this requires assumptions on $a^{ij}$). We fruther assume that $f\in L^2(U)$.
Then we can use the following arguments. Let $L$ denote the differential operator on the left-hand side, so that we currently have $Lu=f$ on $U$, $u=g$ on $\partial U$. Using $w$, this is equivalent to $$ L(u-w) = f- Lw\text{ in }U,\quad u-w=0\text{ on }\partial U. $$ This is an elliptic PDE with zero boundary conditions, and with the right-hand side $f-Lw\in H^{-1}(U)$.
Then one can apply standard estimates for elliptic PDEs, which leads to $$ \|{u-w}\|_{H^1(U)} \leq C\|{f-Lw}\|_{H^{-1}(U)}. $$ Here, we can apply the triangle inequalities, norm estimates for $L$, and embedding inequalities to obtain $$ \|{u}\|\leq \|{w}\|+\|{u-w}\| \leq \|{w}\|+C\|{f-Lw}\|_{H^{-1}(U)} \leq (C\|{L}\|+1)\|{w}\|+C\|{f}\|_{H^{-1}(U)} \leq (C\|{L}\|+1)\|{w}\|_{H^1(U)}+C^2\|{f}\|_{L^2(U)}, $$ where $C>0$ is a suitable constant.
Similar, one can obtain estimates for $H^2(U)$. This time, we have $f-Lw\in L^2(U)$. Using the standard estimates yields $$ \|{u-w}\|_{H^2(U)} \leq C\|{f-Lw}\|_{L^2(U)}. $$ This implies $$ \|{u}\|_{H^2(U)}\leq \|{w}\|_{H^2(U)}+\|{u-w}\|_{H^2(U)} \leq \|{w}\|_{H^2(U)}+C\|{f-Lw}\|_{L^2(U)} \leq (C\|{L}\|+1)\|{w}\|_{H^2(U)}+C\|{f}\|_{L^2(U)}. $$ Overall, I have used other constants than you.