$Pointwise estimate for solutions of the laplace equation on bounded domains$

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$Pointwise estimate for solutions of the laplace equation on bounded domains$

Let $\Omega$ be a bounded open subset of $\mathbb{R}^ n$ . Prove that there exists a constant $C$, depending only on $\Omega$, such that
$$\max |u| \leq C (\max_{\partial \Omega }|g|+\max_{\overline{\Omega} }|f|), $$
where $u$ is a solutions of the PDE
$$ -\Delta u=f $$ with $u=g$ on $\partial \Omega$.

Partial Differential Equations Elliptic PDEs Laplace Equation

Alejandraramirez

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Let $v=u+\frac{|x|^2}{2n}\lambda$, where $\lambda=\max_{x\in \overline{\Omega}}$. Then
$$-\Delta v=-\Delta u-\lambda=f-\max_{x\in \overline{\Omega}} |f| \leq 0.$$
Hence $v$ is submharmonic, and by the weak maximum principle for subharmonic functions we have $$\max_{x\in \overline{\Omega}} u \leq \max_{x\in \overline{\Omega}} v =\max_{x\in \partial \Omega} v \leq \max_{x\in \partial \Omega } (u+ \frac{|x|^2}{2n}\lambda) \leq \max_{x \in \partial \Omega}|g|+C_1 \lambda.$$
Hence $$\max_{x\in \overline{\Omega}} u \leq C_2( \max_{x \in \partial \Omega}|g|+ \lambda)=C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|),$$ for some constant $C_1, C_2$ which only depend on $\Omega$. By replacing $u$ with $-u$, we get
$$\max_{x\in \overline{\Omega}} (-u) \leq C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|).$$
Combining the above two inequalities we get $$\max_{x\in \overline{\Omega}} |u| \leq C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|).$$

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$Pointwise estimate for solutions of the laplace equation on bounded domains$

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