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.
Best Answer
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 $$\displaystyle \int_{U} D v_k \cdot D w_k \vert D w_k \vert ^ {p-2} \to \int_{U} \vert D u \vert ^p.$$