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.
