De Giorgi's theorem on the isoperimetric property of the hypersphere

The aim of the thesis is to prove the isoperimetric property of the hypersphere as formulated by Ennio De Giorgi in his 1958 article. We will introduce the necessary elements of measure and integration  theory as well as some basic properties of convolution and the general version of the  Gauss-Green theorem. Subsequently, we will define functions of bounded variation in several variables and sets with finite perimeter. Finally, we will discuss some important properties of sets of finite perimeter such as approximation by polygonal domains and the monotonicity of perimeter under Steiner symmetrization. With these tools, we will show the isoperimetric property of the  hypersphere following  De Giorgi's original proof.