Theory of Elliptic PDE

Abstract

In this dissertation we present a brief introduction to theory of elliptic partial differential equations (PDE). First we review theory of Sobolev spaces. After that we discuss existence, regularity and other qualitative properties of weak solutions to the second order linear elliptic PDE. Afterwards, we discuss various standard variational and non-variational techniques to study nonlinear elliptic pde, mainly existence/nonexistence and various qualitative properties. Finally, in the last two chapters we mention various regularity results for weak solutions to elliptic equations in divergence form, in particular well-known theory of De Giorgi-Nash-Moser.