= Strong maximum principle for elliptic operators
= Strong maximum principle
{synonym}
For a locally <uniformly elliptic operator> $L=a^{ij}D_{ij}+b^iD_i+c$ with locally bounded coefficients and $c\leq0$, a $C^2$ function satisfying $Lu\geq0$ 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 $Lu\leq0$, replacing $c$ by $\min(c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.
Back to article page
