PDE and hypersurfaces with prescribed mean curvature

These are lecture notes for the mini-course \textit{PDE and hypersurfaces with prescribed mean curvature} held in Federal University of São Carlos at the Workshop on Submanifold Theory and Geometric Analysis, August 05 -- 09, 2019. The aim of these notes is to introduce to the geometers useful tools from the \textit{Theory of partial differential equations} which are used in order to obtain hypersurfaces with prescribed mean curvature. For instance, graphs with prescribed mean curvature can be obtained by solving the Dirichlet problem for a particular quasilinear elliptic partial differential equation of second order.