Evan's PDE question Chapter 5 problem 9

Question

I've been driven crazy by this problem.

Question $5.9$ - Evans PDE $2$nd edition

(Thanks and yes, I have read this answer, but my question is actually how should I proceed next)

Question:

Integrate by parts to prove: $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ and all $u \in W^{2,p}(U) \cap W^{1,p}_{0}(U)$.

So far, I have proven the result assuming $u\in C_c^{\infty}(U)$, and then had trouble to generalize it for $u \in W^{2,\ p}(U)∩W_0^{1,\ p}(U)$.

In the link above, someone said "one can conclude the theorem by density". What is the meaning of density here? I'm sorry but I really couldn't understand this. Hopefully somebody can help.

As in the answer below, I'm also trying to prove something like: $$\int_U Dv_k\cdot Du|Dw_k|^{p−2}dx=\int_U |Du|^p dx$$ but shamefully just don't have much idea.

Answer 1

Ideed, because $$ v_k\to u \;\; \mbox{in}\;\;W^{1,p}(U),$$ $$ w_k \to u \;\; \mbox{in} \;\; W^{2,p}(U) $$ we can suppose that $$D v_k\to D u, $$ $$ D w_k \to D u$$ pointwise. Then, pointwise we have $$D v_k \cdot D w_k \vert D w_k \vert ^ {p-2} \to \vert Du \vert ^p\in L^1. $$ By Young’s inequality associated to Swhartz’s Inequality $$\vert D v_k \cdot D w_k \vert D w_k \vert ^ {p-2}\vert \le C (\vert Dw_k \vert^p+\vert Dv_k \vert^p)$$ and as $$\int_U (\vert Dw_k \vert^p+\vert Dv_k \vert^p) \to 2\int_U \vert Du\vert^p,$$ the General Lebesgue Dominated Theorem implies that $$\displaystyle \int_{U} D v_k \cdot D w_k \vert D w_k \vert ^ {p-2} \to \int_{U} \vert D u \vert ^p.$$

Answer 2

Let $u \in W^{2,p}(U) \cap W_0^{1,p}(U)$.

Suppose that

1. $\{v_k\} \subset C_c^\infty (U)$ satisfies $\|v_k - u\|_{W^{1,p}(U)} \to 0$, and
2. $\{w_k\} \subset C^\infty (U)\cap W^{2,p}(U)$ satisfies $\|w_k - u\|_{W^{2,p}(U)} \to 0$.

Try to prove that $$\int_U Dv_k \cdot Dw_k |Dw_k|^{p-2} \, dx \to \int_U Du \cdot Du |Du|^{p-2} \, dx = \int_U |Du|^p \, dx.$$

If you can do that, it is just a matter of following the steps in the linked answer using $v$ and $w$. . For instance, \begin{align*} \int_U Dv_k \cdot Dw_k |Dw_k|^{p-2} \, dx &= \sum_{j=1}^n \int_U (v_k)_{x_j} (w_k)_{x_j} |Dw_k|^{p-2} \, dx \\ &= - \sum_{k=1}^n \int_U v_k \left[ (w_k)_{x_j} |Dw_k|^{p-2} \right]_{x_j} \, dx \end{align*} from which you can make the appropriate estimates and take limits.