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### Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$

In $d \geq 3$ the answer is no from scaling argument. WLOG we can assume $0\in \Omega$ (by translation) and that $B(0,r_0)\subset\Omega$. Take $\phi\in C^\infty_0(B(0,r_0))\subset C^\infty_0(\Omega)$. ...
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