Weak formulation of a differential equation

For $-\Delta u=f$ on an open set, the distributional formulation for locally integrable $u,f$ is $\int u(-\Delta\phi)=\int f\phi$ for every compactly supported smooth $\phi$. A variational formulation may instead require a Sobolev space and use $\int\nabla u\cdot\nabla\phi=\int f\phi$. Initial and boundary conditions need their own trace or test conventions. Nonlinear products must exist in the stated function spaces; an integral identity alone does not supply their integrability.