Expansion of Laplace operator

Question

Let $\nabla$ be the gradient operator and $\nabla ^2 = \Delta$ be the Laplace operator. Now I know that $$\| \nabla u \|^2 + \| \nabla v \|^2 = (u_x^2 + u_y^2 + v_x^2 + v_y^2).$$ My question is, what is $$\| \Delta u \|^2 + \| \Delta v \|^2?$$

Answer 1

Note that $\Delta u=u_{xx}+u_{yy}$, so $\Delta u$ is a real number, not a vector. Compare this to $\nabla u=(u_x,u_y)^T$. Hence, $||\Delta u||^2=|u_{xx}+u_{yy}|^2$ (just absolute values). Therefore, $||\Delta u||^2+||\Delta v||^2=|u_{xx}+u_{yy}|^2+|v_{xx}+v_{yy}|^2$.