Quadratic perturbation proof of the minimum principle ($\Delta (u-\epsilon |x{|}^2)<0$)

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