Let $\space u \in C^{\infty}\left(\mathbb{R}^d \right)$ be a function such that all the 1st and 2nd order partial derivatives of $\space u \space$ (as well as the function $u$ itself) belong to $L^{2}\left(\mathbb{R}^d \right)$.
Under these assumptions I'm trying to prove that $$\lim_{r \rightarrow \infty} \int_{\partial B(r)}u \nabla u \space \mathrm{d} \textbf{S}=0$$ where $B(r)$ denotes a ball $\{x \in \mathbb{R}^d : ||x|| \leq r \}$.
I don't even know if it holds true, but I hope it does because I need it in order to prove certain inequality for Sobolev spaces.
I'll appreciate any help.