Integral of Laplacian of Compactly Supported Function

Question

Let's say I'm interested in integrating an expression of the form: $$ \int_{\mathbb{R}^d} \Delta u \mathrm{d}x $$ Where $u$ is some compactly supported function with well-defined laplacian. Is it valid to say that: $$ \int_{\mathbb{R}^d}\Delta u \mathrm{d}x = \lim_{r \to \infty}\int_{B_r}\Delta u \mathrm{d}x = \lim_{r \to \infty}\int_{\partial B_r}\nabla u \cdot n \mathrm{d}x $$ (Applying the divergence theorem to obtain the last equality) And because $\nabla u \rightarrow 0 $as $r \rightarrow \infty$ due to the fact that $u$ has compact support, it follows that: $$ \int_{\mathbb{R}^d}\Delta u \mathrm{d}x = \lim_{r \to \infty}\int_{ \partial B_r}\nabla u \cdot n \mathrm{d}x = 0 $$ Is this valid reasoning?

Answer 1

It seems this line of reasoning is correct. Thanks everyone!