Rearrangement of Laplacian function

Question

It is known that if $\nabla u$, in the sense of distributions, is a function that satisfies $\|\nabla u\|_{L^2(\mathbb{R}^n)}<\infty$, then its Symmetric decreasing rearrangement $\nabla u^\ast$ satisfies $$ \|\nabla u^\ast\|_{L^2(\mathbb{R}^n)}\leq\|\nabla u\|_{L^2(\mathbb{R}^n)}. $$ Is there some inequality for the higher-order derivative of $u$, for example is there any relation between $\|\Delta u^\ast\|_{L^2(\mathbb{R}^n)}$ and $\|\Delta u\|_{L^2(\mathbb{R}^n)}$?

Answer 1

Hint see that $$\Delta u=div(\nabla u)$$ Apply the given inequality first for each partial derivative then on the gradient.