This principle is from the harmonic function theory. Let $\Omega \subset \mathbb{R}^d$ be a bounded, open subset of $\mathbb{R}^d$ with compact closure $\overline{\Omega}$ for some $d \in \mathbb{N}$ and $u \in C^2(\Omega) \cap C(\overline{\Omega})$ (function of class $C^2$ and $C$ on $\Omega$ and $\overline{\Omega}$, respectively). Given $c \in C(\Omega)$ a nonpositive function satisfying $\Delta u + cu \geq 0$. Prove that if $u_{\mid \partial \Omega} \leq 0$ then $u_{\mid \Omega} \leq 0$.

It is closely related to the weak maximum principle (WXP) ($c=0$ we get $u$ is a subharmonic function) but I do not know how to mimic the proof of the WXP.


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