A modern approach to De Giorgi's Elliptic Regularity theory and its applications to Navier-Stokes system and Stochastic PDE's

We show how De Giorgi's Elliptic Regularity theory can be extended to different problems and it represents the strongest regularity result for PDEs with just measurable coefficients. In particular, we analyse the Caffarelli-Kohn-Nirenberg theorem for Navier Stokes equations and we prove the Holderianity of the solution to the Stochastic heat equation.