I saw this statements in a proof, but I would like to see the details.
Let $U \subset \mathbb{R}^{n}$ and $u$ sperharmonic function in the sense that \begin{equation} \int_{D} \langle \nabla u(x) ,\nabla \varphi(x) \rangle dx \quad \mbox{for any nonnegative} \quad \varphi \in C^{\infty}_{0}(U). \end{equation}
Is $u$ lower semicontinuous?
if $u$ is lower semicontinuous and $\varphi \in C^{\infty}_{0}(U)$. Is $\{u > \varphi\}$ an open set?
Thank you.