Mathematics

Generally speaking a Liouville-type theorem says that bounded global solutions of some PDE are constant. A standard method to derive this kind of result is to obtain suitable a-priori decay estimates on balls and then 'sending the radius to infinity'. In the Euclidean space it is  well understood that also a sort of converse of this holds, meaning that one can obtain local a-priori estimates from Liouville theorems. In this talk we will  investigate this strategy in Riemannian manifolds and use it to prove local estimates for some class of nonlinear elliptic equations. We will also see that the analysis of non-smooth metric spaces is naturally required and plays a key role in the argument.