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I am trying to prove the following statement:

If $v_k\in C_c^{\infty}(U)$ converges to $u$ in $W^{1,\ p}(U)$, and $w_k\in C^{\infty}(U)$ converges to $u$ in $W^{2,\ p}(U)$, where $2\le p < \infty$, then $$\int_U Dv_k\cdot Dw_k |Dw_k|^{p-2}dx\rightarrow \int_U |Du|^{p}dx$$ as $k\rightarrow \infty$.

And since this is only part of the problem that I am solving, I am also wondering that if we can weaken the constraints to $v_k, w_k\in C^{\infty}(U)\rightarrow u$ in $W^{1,\ p}(U)$ as $k\rightarrow \infty$ for the above statement to be true.

And the original problem if you are interested is here.

Qiuyi Li
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1 Answers1

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Note that the map $\mathbb R^n \ni W \mapsto |W|^{p-2}W \in \mathbb R^n$ has all derivatives bounded by $C |W|^{p-2}$, so by mean value theorem $$ \left| |W_1|^{p-2}V - |W_2|^{p-2} W \right| \le C(|W_1|^{p-2}+|W_2|^{p-2}) \cdot |W_1-W_2|. $$ If one assumes $W_k \to w$ in $L^p$, then by the above and Holder's inequality $$ \left\| |W_k|^{p-2}W_k - |W|^{p-2} W \right\|_{\frac{p}{p-1}} \le C \left\| |W_k|^{p-2}+|W|^{p-2} \right\|_{\frac{p}{p-2}} \left\| W_k-W \right\|_{p} \to 0, $$ hence $|W_k|^{p-2}W_k \to |W|^{p-2}W$ in $L^{\frac{p}{p-1}}$. If moreover $V_k \to V$ in $L^p$, then the product is convergent in $L^1$, in particular $$ \int V_k \cdot |W_k|^{p-2}W_k \to \int V \cdot |W|^{p-2}W. $$

This answers your question, as $V_k = \nabla v_k$, $W_k = \nabla w_k$ both converge to $\nabla u$ in $L^p$ and $\nabla u \cdot |\nabla u|^{p-2} \nabla u = |\nabla u|^p$.

Michał Miśkiewicz
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