= Quadratic perturbation proof of the minimum principle
{title2=$\Delta(u-\varepsilon|x|^2)<0$}
For $u\in C^2(D)\cap C(\overline D)$ on a bounded domain in $\mathbb R^n$ with $\Delta u\leq0$, the minimum lies on the boundary. Define $u_\varepsilon=u-\varepsilon|x|^2$, so $\Delta u_\varepsilon\leq-2n\varepsilon<0$. An interior minimum would have positive semidefinite <Hessian matrix> and nonnegative <trace>, a contradiction. Thus $u_\varepsilon$ attains its minimum on the boundary. Boundedness controls $|u-u_\varepsilon|$ uniformly by a constant times $\varepsilon$; letting $\varepsilon\downarrow0$ proves the weak minimum principle for the original <superharmonic function>. The perturbation handles the non-strict inequality, including constant <functions>.
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