I am on a course about viscosity theory and we had this as an exercise:
Assume $u: \Omega \to R$ is bounded and upper semicontinuous. Fix $x_0$. Prove that for every $r>0$ there is $\phi\in C^2(B(x_0,r))$ such that for some $y\in B(x_0,r)$ $\phi(y)=u(y)$ and $\phi(x)\geq u(x)$ for every $x\in B(x_0,r)-\{y\}$.
The hint was to choose $\phi(x)=a+b|x-x_0|^2$ and I know that I need to fix $a$ to be such that $\phi(y)=u(y)$ and b such that $\phi(x)\geq u(x)$, but I don't know how.