Let $f \in C^0 (\Omega), g \in C^0 (\partial \Omega) $ and let $u \in C^2 (\overline{\Omega}) $ solve the equation $$ \int_{\Omega} \langle \nabla u, \nabla w \rangle = \int_{\Omega} fw + \int_{\partial \Omega} gw \\ \forall w \in C^{\infty}_c (\Omega) $$
What could be a possible differential equation and its boundary conditions?
An example could be the following Poisson equation with inhomogeneous Neumann boundary condition: \begin{aligned} -\Delta u &= f \, ,\\ (\nabla u\cdot n)|_{\partial\Omega} &= g \, . \end{aligned} Indeed, using the relation $w \Delta u = \nabla\cdot(w \nabla u) - \nabla w\cdot \nabla u$ and the divergence theorem, one obtains the weak formulation \begin{aligned} -\int_\Omega \Delta u\, w &= \int_\Omega \nabla w\cdot \nabla u - \int_\Omega \nabla\cdot(w \nabla u) \\ &= \int_\Omega \nabla w\cdot \nabla u - \int_{\partial\Omega} w \nabla u\cdot n \\ &= \int_\Omega \nabla w\cdot \nabla u - \int_{\partial\Omega} wg \\ &= \int_\Omega fw\, . \end{aligned}