$H^{1}_0(\Omega)$ norm is invariant under orthogonal maps?

Question

How to prove that $H^{1}_0(\Omega)$ norm is invariant under orthogonal maps? That IS, given $u \in H^{1}_0(\Omega)$ and $T \in O(N)$, How to obtain $$ ||u\circ T||_{H^{1}_0(\Omega)} = ||u||_{H^{1}_0(\Omega)} ? $$

Answer 1

You can proceed in two steps:

1)If $u$ is weakly differentiable, then the weak gradient of $u \circ T$ is given by $T^T \nabla u \circ T$.

2)Now use the fact that $T^TT=I$ and use a change of variable. Note that $|\det(T)|=1$.

Answer 2

Use a change of variables and the fact that $\vert \text{det}\, T \vert = 1$.