Continuous dependence on forcing

Solutions depend continuously on their forcing when convergence of the forcing in a specified norm implies convergence of solutions in a specified solution norm, with boundary conditions fixed. A uniform estimate $\Vert u-v{\Vert }_X\le C\Vert f-g{\Vert }_Y$ is a stronger quantitative form, namely Lipschitz continuity of the solution map. The monotonicity energy estimate for a semilinear Dirichlet problem is an example.