I've just begun studying PDE.
In order to prove that $w \in C^1(D)$ is a weak solution of \begin{equation} \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; \partial D\end{cases} \end{equation} should I prove that \begin{equation*} \int_{D}{\bigtriangledown w\cdot\bigtriangledown \phi\: dx}=\lambda\:\int_{D}{w\:\phi \:dx} \quad{\rm for\; all\;\phi\in{C}_0^\infty }\quad? \end{equation*} Is this the definition? Thanks!