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Let $\Omega \subset \mathbb{R}^{d}$ be a smooth domain. Let $\mu_{i}: \Omega \rightarrow \mathbb{R}$ , $i=1,2$, $\mu_{i} \in C^{2}(\Omega) \cap C(\bar{\Omega}),f_{i} \in C(\bar{\Omega}) $ and $$ \begin{cases} u_i - \Delta u_i=f_i, \: in \: \Omega\\ u_i=0 \: in \: \partial \: \Omega \end{cases}$$

I want to show that $||u_1-u_2||_{L^2} \leq ||f_1-f_2||_{L^2}$. In my computation I need to prove that $||\Delta u_1- \Delta u_2||_{L^2}=0$ and I did not know how to show this. Any suggestion?

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Let $v=u_1-u_2$. $v$ is a solution of $v-\Delta v=f_1-f_2$ with null boundary conditions. Multiply the equation by $v$ and integrate: $$ \int_\Omega|v|^2+\int_\Omega|\nabla v|^2=\int_\Omega v(f_1-f_2)\le\|v\|_2\,\|f_1-f_2\|_2. $$

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