 Weak Harnack inequality

A Weak Harnack inequality controls a positive integral mean of a nonnegative weak supersolution by its essential infimum. For example, let $n\ge 3$, let $A$ be a bounded measurable uniformly elliptic coefficient matrix, and let $u\in H_{\mathrm{loc}}^1$ be nonnegative with $\mathrm{div}(A\nabla u)\le 0$ in the weak sense. If $B_{4R}⋐\Omega$, then for $0<q<n/(n-2)$,The constant depends on dimension, ellipticity/coefficient bounds and the chosen exponent. The normalization makes the estimate invariant under spatial rescaling. Its infimum is essential because a function in a Sobolev space is initially an almost-everywhere equivalence class; changing a single representative value cannot invalidate the estimate.

This is a one-sided statement for supersolutions, distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions. Combining it with the subsolution bound from Moser iteration yields that latter estimate. If the essential infimum is zero, the left side forces vanishing almost everywhere on the larger ball, giving the mechanism of propagation of zeros by the weak Harnack inequality. Lower-order terms or forcing require the corresponding modified assumptions and error terms.