 Weak Harnack inequality

The weak estimate for supersolutions is distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions.

For a nonnegative weak supersolution of a uniformly elliptic divergence-form equation, a positive $L^q$ mean on a ball is bounded by a constant times the infimum on a smaller concentric ball. The admissible exponent and constant depend only on dimension and the ellipticity bounds.