Three-sphere inequalities for second order singular partial differential equations

In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div(A ∇ u) – Vu = 0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation ∂tu – div(A ∇ u) + Vu = 0, where the singular potential V belonging to the Kato-Fefferman-Phong's class. Some applications are also discussed.