TITLE: About the smoothness of the boundary of support set
QUESTION [1 upvotes]: If $\varphi \in C_0^2 (\Omega)$, where $\Omega \in \mathbf{R}^n$. Can we infer the boundary of supp $\varphi$ is $C^2$? How to prove?

REPLY [1 votes]: No. Let $\psi \in C^k(\mathbb R)$ be a bump function with support $[0,1]$ and define $\varphi \in C^k (\mathbb R^2)$ by $\varphi(x,y) = \psi(x)\psi(y)$. Then $\text{supp }\varphi = [0,1]^2$ is not even $C^1$ - it has sharp corners.
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