Moser iteration

Moser iteration combines a Caccioppoli inequality with the Sobolev embedding theorem to raise an integrability exponent geometrically. On nested balls it turns an $L^p$ bound for a nonnegative subsolution of a uniformly elliptic divergence-form equation into a local supremum bound.

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.

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the Weak Harnack inequality implyfor a universal $0<\theta <1$. Iteration gives local Hölder continuity.

A nonnegative weak solution of a uniformly elliptic divergence-form equation satisfieswhenever a fixed larger concentric ball lies in the domain. Combine the subsolution bound from Moser iteration with the Weak Harnack inequality for supersolutions.