The problem stated in the Evan's book is
Integrate by parts to prove:$$\int_U|Du|^p\;dx\le C \left(\int_U|u|^p\;dx\right)^\frac12\left(\int_U|D^2u|^p\;dx\right)^\frac12$$ for $2\le p< \infty$ and all $u\in W^{2,p}(U)\cap W^{1,p}_0(U)$.
I have managed to get the result for all $v\in C^\infty_c(U)$ and would like to generalise the results to $u\in W^{2,p}(U)\cap W^{1,p}_0(U)$. Let $\{v_k\}$ be a sequence in $C^\infty_c$ converging to $u$ in the $W^{1,p}$ sense, and $\{w_k\}$ be a sequence in $C^\infty$ converging to $u$ in the $W^{2,p}$ sense. My attempt is to make the following estimate: $$ \begin{align*} \int_UD{v_k}\cdot D{w_k}|Dw_k|^{p-2}-Du\cdot Du|Du|^{p-2}&\le \int_U|Dv_k||Dw_k-Du||Dw_k|^{p-2}\;dx\\ &\phantom{\le\ }+\int_U|Du||Dv_k-Du||Du|^{p-2}\;dx\\ &\phantom{\le\ }+\int_U|Dv_k||Du|\left||Dw_k|^{p-2}-|Du|^{p-2}\right|\;dx. \end{align*} $$
If I can bound $\left||Dw_k|^{p-2}-|Du|^{p-2}\right|$ in some form of $\left|Dw_k-Du\right|^{p-2}$, then by holder's inequality and the fact that $|v_k-u|$ and $|w_k-u|$ are uniformly bounded, I can get the desired result. Yet, I have been stuck on bounding the last part very long. May I know some ideas on how to bound it?
Update on Mar 2, 2024: I am aware of the use of generalised Holder's inequality, which is extremely useful in controlling the first two summands of the RHS. However, I still fail to obtain a desired bound for the last summand, i.e., $\int_U|Dv_k||Du|\left||Dw_k|^{p-2}-|Du|^{p-2}\right|\;dx$. It seems that using generalised Holder's inequality to bound the last term directly is too rough.
