"Schwarz symmetrization" on annulus

Question

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we find a $u^* \in W_0^{1,2}(\Omega)$ such that 1) $u^*$ is radial, 2) $u$ and $u^*$ have the same distribution function, and 3) $\int_\Omega|\nabla u|^2dx \geq \int_\Omega|\nabla u^*|^2dx $?

Answer 1

Such a construction is not possible.

One way to see this is to note that if such a construction was possible, then you would have, by the embeddings for Sobolev spaces of radial functions $$ \Vert u^* \Vert_{L^\infty} \le C \Vert \nabla u^* \Vert_{L^2}. $$ By the assumptions that you have on your rearrangement, this would imply in turn that $$ \Vert u \Vert_{L^\infty} \le C \Vert \nabla u \Vert_{L^2}, $$ which is known to fail.