Hölder continuity and Harnack's inequality for solutions of elliptic partial differential equation

In the first chapter we will exhibit a more generalized version of the De Giorgi method combined with the use of Harnack's inequality to show the iteration of Moser. Once proved the Hölder continuity we will conclude the solution to the Hilbert problem with the Calderon–Zygmund result. After exhibiting some applications of Harnack's inequality, in the last chapter we will shift our focus on proving a generalized version of Harnack's inequality for solutions of elliptic partial differential equation.