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My guess is to transform into spherical coordinates, but I fail to prove that the mapping with it's inverse is uniformly bounded. Any suggestions? ----------updates------------ Sorry for the late update. I was off-internet in the last few days. The domain $C^k$ is a domain similar with Lipschitz domain, but the bijective function is $C^k$ continuous instead of Lipschitz continuous. |
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It's not entirely clear what you're asking, but $\mathbb{B} = \{ \|x\|^2 - 1 < 0 \}$, and the function $\rho(x) = \|x\|^2 - 1$ is $C^\infty$ (and its gradient vanishes only at the origin), so $\mathbb{B}$ has $C^\infty$ boundary. |
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