 Strong maximum principle for elliptic operators

For a locally uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with locally bounded coefficients and $c\le 0$, a $C^2$ function satisfying $Lu\ge 0$ on a connected domain cannot attain a nonnegative global maximum inside unless it is constant. If $c=0$, the maximum can have either sign. If the maximum set were proper, choose an interior ball in its complement tangent to that set. The Hopf boundary point lemma gives a nonzero normal derivative at the contact point, contradicting the vanishing gradient at an interior maximum.

The strong minimum principle for elliptic operators follows by applying this to $-u$. For nonnegative $u$ with $Lu\le 0$, replacing $c$ by $\min (c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.