Let $U \subset \mathbb{R}^N$ be a bounded open set, and let $h \in C^\infty(U)$.
Since $h \in L^p(U)$ for every $1 \leq p \leq \infty$, there exists a sequence $(h_j) \subset C_c^\infty(U)$ such that $$h_j \rightarrow h \quad \text{strongly in} \quad L^p(U)$$ provided $p < \infty$.
Does $\nabla h_j \rightarrow \nabla h$ in $L^p(U)$ ?
I know that if $(f_n) \subset L^1_{\mathrm{loc}}(U)$ is such that in $L^1_{\mathrm{loc}}(U)$, $$f_n \rightarrow f, \quad \partial_j f_n \rightarrow g_j \quad \forall j \in [1,N],$$ then $G = \nabla f$, where $G$ has components $g_j$, $1 \leq j \leq N$.
Does there exists $H$ such that $\nabla h_j \rightarrow H$ in $L^p(U;\mathbb{R}^N)$ ?