I have question on Theorem $1$ of Evan's book page $329$ or ($6.3$ Regularity).
At the part $6$, Evan wrote:
We finally combine $(12),(20)$ and $(22),$ to discover
$$\int_V |D^h_k Du|^2 dx \leq \int_U \zeta^2 |D^h_k Du|^2 dx \leq C \int_U f^2 +u^2 +|Du|^2 dx$$ for $k=1,...,n$ and all sufficiently small $|h| \not = 0.$
In view of Theorem $3(ii)$ in $\S 5. 8.2$, we deduce $Du \in H^1_{loc} (U; \mathbb{R}^n),$ and thus $u \in H^2_{loc}(U),$ with the estimate
$$\|u\|_{H^2(V)} \leq C (\|f\|_{L^2(U)}+\|u\|_{H^1(U)}).$$
I know why $Du \in H^1_{loc} (U; \mathbb{R}^n).$ But I do not understand why we will have the estimate $\|u\|_{H^2(V)} \leq C (\|f\|_{L^2(U)}+\|u\|_{H^1(U)}).$ Can someone explain to me?