Comparison results for PDEs with a singular potential

Abstract

Several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential. Solutions to these equations are singular in many cases, and our results roughly say that they all have comparable singularities, provided that they belong to an appropriate space. We formulate the hypothesis on the potential in terms of an inequality, which in the case of the well-known inverse-square potential, is a consequence of an improvement of Hardy's inequality due to Vázquez and Zuazua.