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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.

Write $w=u-v$ and $r=f-g$. Subtraction of the two boundary value problems gives $-w^{\prime \prime }+w+u^3-v^3=r$, with $w(0)=w(1)=0$. Multiply by $w$ and apply integration by parts:Consequently the required energy identity isThe nonlinear term is nonnegative pointwise, sinceThis is the monotonicity energy estimate for a semilinear Dirichlet problem for the monotone function $s\mapsto s^3$. Apply Young's inequality for products, $rw\le (r^2+w^2)/2$, inside the last integral and discard the nonnegative nonlinear term. Moving $\frac{1}{2}{\int }_0^1{w}^{2}dx$ to the left givesIn particular, equal forcing gives $w=0$, proving uniqueness and a quantitative continuous dependence on forcing estimate for this boundary value problem.