I'm having a problem figuring out the answer to this question. Consider the problem $$ \begin{cases} -\Delta u = f & \mbox{in } \Omega = (0,1)^2, \newline \nabla u \cdot \mathbf{n} = g_N & \mbox{on } \Gamma_N = \{ (x,y) \in \mathbb{R}^2 | x = 0 \}, \newline u = 0 & \mbox{on } \Gamma_D = \partial \Omega \backslash \Gamma_N, \end{cases} $$ Suppose that the problem is solved with the same boundary conditions, but a different source term $\tilde{f}$. Call the corresponding solution $\tilde{u}$. Write the strong form of the problem that must be solved to determine $w := u - \tilde{u}$. Then using the Poincaré inequality, prove that there exists a constant $C > 0$ such that $$ \| w \|_{L^2(\Omega)} \leq C \| f - \tilde{f} \| _{L^2(\Omega)} $$
Personally I already figured out how to answer the first question related to the strong form to determine $w$ but I'm struggling a bit with the second part. I think that we should use the fact that $L^2$ norm is bounded by the $H^1$ norm but I really don't know how